From 6a1c2d56891e4d2d45a228e1e615ab7b5eacb50c Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Sat, 28 Jun 2025 20:00:51 -0500 Subject: [PATCH 01/14] import stereo from from-wordpress --- posts/stereo/1/index.qmd | 405 ++++++++++++++++++++++++++++++++++++++ posts/stereo/2/index.qmd | 411 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 816 insertions(+) create mode 100644 posts/stereo/1/index.qmd create mode 100644 posts/stereo/2/index.qmd diff --git a/posts/stereo/1/index.qmd b/posts/stereo/1/index.qmd new file mode 100644 index 0000000..d23e2f0 --- /dev/null +++ b/posts/stereo/1/index.qmd @@ -0,0 +1,405 @@ +--- +format: + html: + html-math-method: katex +--- + + +Algebraic Rotations and Stereography +==================================== + +Expressing rotation mathematically can be rather challenging, even in two dimensional space. Typically the subject is approached from complicated-looking rotation matrices, with the occasional accompanying comment about complex numbers. The challenge only increases when examining three dimensional space, where everyone has different ideas about the best implementations. In what follows, I attempt to provide a derivation of 2D and 3D rotations based in intuition and without the use of trigonometry. + + +Complex Numbers: 2D Rotations +----------------------------- + +Multiplication between complex numbers is frequently analyzed as a combination of scaling (a ratio of two scales) and rotation (a point on the complex unit circle). However, the machinery used in these analyses usually expresses complex numbers in a polar form involving the complex exponential, concepts which leverage knowledge of trigonometry and some calculus. In fact, they obfuscate the algebraic properties of complex numbers when neither concept is truly necessary to understand points on the complex unit circle, or complex number multiplication in general. + + +### A review of complex multiplication + +A complex number $z = a + bi$ multiplies with another complex number $w = c + di$ in the obvious way: by distributing over addition. + +$$ +zw = (a+bi)(c+di) = ac + bci + adi + bdi^2 = (ac -\ bd) + i(ad + bc) +$$ + +*z* also has associated to it a conjugate $z^* = a -\ bi$. The product $zz^*$ is the *norm* $a^2 + b^2$, a positive real quantity whose square root is typically called the *magnitude* of the complex number. Taking the square root is frequently unnecessary, as many of the properties of this quantity do not depend on this normalization. One such property is that for two complex numbers *z* and *w*, the norm of the product is the product of the norm. + +$$ +\begin{align*} +(zw)(zw)^* &= (ac -\ bd)^2 + (ad + bc)^2 \\ +&= a^2c^2 -\ 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2 \\ +&= a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 = a^2(c^2 + d^2) + b^2(c^2 + d^2) \\ +&= (a^2 + b^2)(c^2 + d^2) +\end{align*} +$$ + +Conjugation also distributes over multiplication, and complex numbers possess multiplicative inverses. Also, the norm of the multiplicative inverse is the inverse of the norm. + +$$ +\begin{gather*} +z^*w^* = (a -\ bi)(c -\ di) = ac -\ bci -\ adi + bdi^2 \\ += (ac -\ bd) -\ i(ad + bc) = (zw)^* \\ +{1 \over z} = {z^* \over zz^*} = {a -\ bi \over a^2 + b^2} \\ +\left({1 \over z}\right) +\left({1 \over z}\right)^* += {1 \over zz^*} = {1 \over a^2 + b^2} \\ +\end{gather*} +$$ + +A final interesting note about complex multiplication is the product $z^*w$ + +$$ +z^* w = (a -\ bi)(c + di) = ac -\ bci + adi -\ bdi^2 = (ac + bd) + i(ad -\ bc) +$$ + +The real part is the same as the dot product $(a, c) \cdot (b, d)$ and the imaginary part looks like the determinant of $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$. This shows the inherent value of complex arithmetic, as both of these quantities have important interpretations in vector algebra. + + +### Onto the Circle + +If $zz^* = a^2 + b^2 = 1$, then *z* lies on the complex unit circle. Since the norm is 1, a general complex number *w* has the same norm as the product *zw*. Thus, multiplication by *z* results in the rotation that maps 1 to *z*. But how do you get such a point on the unit circle? + +Simple. There are four complex numbers directly related to any complex number *w* that are guaranteed to share its norm: $w, -w, w^*$, and $-w^*$. Therefore, dividing any of these numbers by any other of these numbers results in a complex number on the unit circle. Division by -*w* is boring and just results in -1. However, division by $w^*$ turns out to be important: + +$$ +\begin{align*} +z &= {w \over w^*} = +\left( {a + bi \over a -\ bi} \right) \left( {a + bi \over a + bi} \right) = 1 \\[8pt] +&= {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} +\end{align*} +$$ + +Dividing the numerator and denominator of this expression through by $a^2$ yields an expression in $t = b/a$, where t can be interpreted as the slope of the line joining 0 and *w*. + +$$ +\begin{align*} +{(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} +&= {1/a^2 \over 1/a^2} \cdot {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} \\ +&= {(1 -\ b^2/a^2) + (2b/a)i \over 1 + b^2/a^2} += {(1 -\ t^2) + (2t)i \over 1 + t^2} \\ +&= {1 -\ t^2 \over 1 + t^2} + i{2t \over 1 + t^2} = z(t) +\end{align*} +$$ + +Dividing the numerator and denominator of this expression through by $a^2$ yields an expression in $t = b/a$, where *t* can be interpreted as the slope of the line joining 0 and *w*. + +$$ +\begin{align*} +{(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} +&= {1/a^2 \over 1/a^2} \cdot {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} \\ +&= {(1 -\ b^2/a^2) + (2b/a)i \over 1 + b^2/a^2} += {(1 -\ t^2) + (2t)i \over 1 + t^2} \\ +&= {1 -\ t^2 \over 1 + t^2} + i{2t \over 1 + t^2} = z(t) +\end{align*} +$$ + +Slopes range from $-\infty$ to $\infty$, which means that the variable *t* does as well. Moreover, if *z* is on the unit circle, then its reciprocal is: + +$$ +{1 \over z} = {z^* \over zz^*} = z^* +$$ + +In other words, for points already on the unit circle, $z / z^* = z^2$ describes the rotation from 1 to $z^2$. This demonstrates the decomposition of the rotation into two parts. For a general complex number *w* which may not lie on the unit circle, multiplication by this number dilates the complex plane as well rotating 1 toward the direction of *w*. Then, division by $w^*$ undoes the dilation and performs the rotation again. The proper description of the rotation described by $w/w^*$ is the combination of these half steps. + + +### Two Halves, Twice Over + +Plugging in some values for t reveals something else: $z(0) = 1 + 0i$, $z(1) = 0 + 1i$, and $z(-1) = 0 -\ 1i$. This shows that the semicircle in the right half-plane exists where $t \in (-1, 1)$. The other half of the circle comes from *t* with magnitude greater than 1. The point where $z(t) = -1$ does not properly exist unless it is allowed that $t = \pm \infty$. + +Is it possible to get the other half into the nicer interval? No problem, just double the rotation. + +$$ +\begin{align*} +z(t)^2 &= (\Re[z]^2 -\ \Im[z]^2) + i(2\Re[z]\Im[z]) \\[8pt] +&= {(1 -\ t^2)^2 -\ (2t)^2 \over (1 + t^2)^2} + i{2(1 -\ t^2)(2t) \over (1 + t^2)^2} \\[8pt] +&= {1 -\ 6t^2 + t^4 \over 1 + 2t^2 + t^4} ++i {4t -\ 4t^3 \over 1 + 2t^2 + t^4} \\ +\end{align*} +$$ + +Now the entire circle fits in the interval $t \in (-1, 1)$, and $z(-1)^2 = z(1)^2 = -1$. This action of squaring *z* means that as *t* ranges from $-\infty$ to $\infty$, the point $z^2$ loops around the circle twice, since the rotation *z* is applied twice. + +I'd now like to go on a brief tangent and compare these expressions to their transcendental trigonometric counterparts. Graphing the real and imaginary parts separately shows how much they resemble $\cos(\pi t)$ and $\sin(\pi t)$ around 0. + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +![]() +::: +:::: + +Incidentally, an approximation of a function as the ratio of two polynomials is called a [Padé approximant](https://en.wikipedia.org/wiki/Pad%C3%A9_approximant). Specifically, the real part (which approximates cosine) has order \[4/4\] and the imaginary part (which approximates sine) has order \[3/4\]. Ideally, the zeroes of the real part should occur at $t = \pm 1/2$, since $\cos(\pm \pi/2) = 0$. Instead, they occur at + +$$ +0 = 1 -\ 6t^2 + t^4 = (t^2 -\ 2t -\ 1)(t^2 + 2t -\ 1) +\implies t = \pm {1 \over \delta_s} = \pm(\sqrt 2 -\ 1) \approx \pm0.414 +$$ + +With a bit of polynomial interpolation, this can be rectified. A cubic interpolation is given by: + +$$ +\begin{gather*} +P(x) = ax^3 + bx^2 + cx + d \\ +\begin{pmatrix} +1 & 0 & 0 & 0 \\ +1 & 1/2 & 1/4 & 1/8 \\ +1 & -1/2 & 1/4 & -1/8 \\ +1 & 1 & 1 & 1 \\ +\end{pmatrix} \begin{pmatrix} +d \\ c \\ b \\ a +\end{pmatrix} = \begin{pmatrix} +0 \\ \sqrt 2 -\ 1 \\ -\sqrt 2 + 1 \\ 1 +\end{pmatrix} \\ +\implies \begin{pmatrix} +d \\ c \\ b \\ a + \end{pmatrix} = \begin{pmatrix} +0 \\ -3 + 8\sqrt 2/3 \\ 0 \\ 4 -\ 8\sqrt 2/3 \end{pmatrix} +\implies P(x) \approx 0.229 x^3 + 0.771x +\end{gather*} +$$ + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +![]() +::: +The error over the interval [-1, 1] in these approximations is around 1% (RMS). +:::: + +Notably, this interpolating polynomial is entirely odd, which gives it some symmetry about 0. Along with being a very good approximation, it has the feature that a rotation by an *n*^th^ of a turn is about $z(P(1/n))^2$. The approximation be improved further by taking *z* to higher powers and deriving another interpolating polynomial. + +It is impossible to shrink this error to 0 because the derivative of the imaginary part at $t = 1$ would be $\pi$ with no error. But the imaginary part is a rational expression (technically with the interpolating polynomial, a ratio of two irrational numbers), so its derivative is also a rational expression. Since $\pi$ is transcendental, there must always be some error. + +Even though it is arguably easier to interpret, there probably isn't a definite benefit to this approximation. For example, + +- Padé approximants can be calculated directly from the power series for sine and cosine. +- There are no inverse functions like `acos` or `asin` to complement these approximations. + - This isn't that bad, since I have refrained from describing rotations with an angle. Even so, the inverse functions have their use cases. +- Trigonometric functions can be hardware-accelerated (at least in some FPUs). +- If no such hardware exists, approximations of `sin` and `cos` can be calculated by software beforehand and stored in a lookup table (which is also probably involved at some stage of the FPU). + +Elaborating on the latter two points, using a lookup table means evaluation happens in roughly constant time. A best-case analysis of the above approximation, given some value *t* is + +$$ +\begin{matrix} +\text{Expression} & \text{Additions} & \text{Multiplications} & \text{Divisions} \\ \hline +p = t \cdot (0.228763834 \cdot t \cdot t + 0.771236166) & +1 & 3 \\[4pt] +q = p \cdot p & +& 1 \\[4pt] +r = 1 + q & +1 \\[4pt] +c = {1 -\ q \over r} & +1 & & 1 \\[4pt] +s = {2 \cdot p \over r} & +2 & 1 & 1 \\[4pt] +real = c \cdot c -\ s \cdot s & +1 & 2 \\[4pt] +imag = 2 \cdot c \cdot s & +& 2 \\ +\hline +\Sigma & +6 & 9 & 2 \\ +\end{matrix} +$$ + +Or 17 FLOPs. On a more optimistic note, a Monte Carlo test of the above approximation on my computer yields promising results when compared with the GCC `libmath` implementations of sin and cos (also tested with `cexp`). + +```{} +Timing for 100000 math.h sin and cos: 5409619ns +Timing for 100000 approximations: 2408645ns +Approximation faster, speedup: 3000974ns (2.25x) +Squared error in cosines: + Average: 0.000051 (0.713743% error) + Largest: 0.000174 (1.320551% error) + Input: 0.729202 + Value: -0.659428 + Approximation: -0.672634 +Squared error in sines: + Average: 0.000070 (0.835334% error) + Largest: 0.000288 (1.698413% error) + Input: 0.729202 + Value: 0.475669 + Approximation: 0.458685 +``` + +For the source which I used to generate this output, see the repository linked below. + +Even though results can be inaccurate, this exercise in the algebraic manipulation of complex numbers is fairly interesting since it requires no calculus to define, unlike sine, cosine, and their Padé approximants (and to a degree, $\pi$). + + +From Circles to Spheres +----------------------- + +The expression for complex points on the unit circle coincides with the stereographic projection of a circle. This method is achieved by selecting a point on the circle, fixing *t* along a line (such as the *y*-axis), and letting *t* range over all possible values. + +![]() + +The equations of the line and circle appear in the diagram above. Through much algebra, expressions for *x* and *y* in *t* can be formed. + +$$ +\begin{align*} +y &= t(x + 1) \\ +1 &= x^2 + y^2 = x^2 + (tx + t)^2 \\ +&= x^2 + t^2x^2 + 2t^2 x + t^2 \\ +{1 \over t^2 + 1} -\ {t^2 \over t^2 + 1} &= +x^2 + {t^2 \over t^2 + 1} 2x \\ +{1 -\ t^2 \over t^2 + 1} + \left( {t^2 \over t^2 + 1} \right)^2 &= +\left(x + {t^2 \over t^2 + 1} \right)^2 + {t^2 \over t^2 + 1} \\ +{1 -\ t^4 \over (t^2 + 1)^2} + {t^4 \over (t^2 + 1)^2} &= +\left(x + {t^2 \over t^2 + 1} \right)^2 \\ +{1 \over (t^2 + 1)^2} &= +\left(x + {t^2 \over t^2 + 1} \right)^2 \\ +{1 \over t^2 + 1} &= +x + {t^2 \over t^2 + 1} \\ +x &= {1 -\ t^2 \over 1 + t^2} \\ +y &= t(x + 1) = +t\left( {1 -\ t^2 \over 1 + t^2} + {1 + t^2 \over 1 + t^2} \right) \\ +&= {2t \over 1 + t^2} +\end{align*} +$$ + +These are exactly the same expressions which appear in the real and imaginary components of the ratio between $1 + ti$ and its conjugate. Compared to the algebra using complex numbers, this is quite a bit more work. Therefore, one might also ask whether, given a proper arithmetic setting, the projection of a sphere might be obtained in the same manner as the circle with respect to complex numbers. + +::: {} +![]() +What we are not doing: intersecting the sphere with a line through a pole and a point in the xy plane +::: + + +### Algebraic Projection + +The simplest thing to do is try it out. Let's say an extended number $h = 1 + si + tj$ has a conjugate of $h^* = 1 -\ si -\ tj$. It's a bit of an assumption to be able to take the reciprocal of the latter (we don't know whether *i* and *j* create a field or division ring), but following our noses: + +$$ +\begin{gather*} +{h \over h^*} = {1 + si + tj \over 1 -\ si -\ tj} = +\left({1 + si + tj \over 1 -\ si -\ tj}\right) +\left({1 + si + tj \over 1 + si + tj}\right) = \\[8pt] { +1 + si + tj + si + s^2i^2 + stij + tj + stji + t^2j^2 +\over +1 -\ si -\ tj + si -\ s^2 i^2 -\ stij + tj -\ stji -\ t^2j^2 +} +\end{gather*} +$$ + +While there is some cancellation in the denominator, both products are rather messy. To get more cancellation, we apply some nice algebraic properties between *i* and *j*. For the $stij$ terms to cancel, we let i and j be *anticommutative*, meaning that $ij = -ji$. So that the denominator is totally real (and therefore can be guaranteed to divide), we also apply the constraint that $i^2$ and $j^2$ are both real. Then the expression becomes + +$$ +{1 + si + tj \over 1 -\ si -\ tj} = +{1 + s^2i^2 + t^2j^2 \over 1 -\ s^2 i^2 -\ t^2j^2} + +i{2s \over 1 -\ s^2 i^2 -\ t^2j^2} + +j{2t \over 1 -\ s^2 i^2 -\ t^2j^2} +$$ + +If it is chosen that $i^2 = j^2 = -1$, this produces the correct equations ([Wikipedia](https://en.wikipedia.org/wiki/Stereographic_projection#First_formulation)) parametrizing the unit sphere. One can also check that the squares of the real, *i*, and *j* components sum to 1. + +If both *i* and *j* anticommute, then their product also anticommutes with both *i* and *j*. + +$$ +\textcolor{red}{ij}j = -j\textcolor{red}{ij} ~,~ +i\textcolor{red}{ij} = -\textcolor{red}{ij}i +$$ + +Calling this product *k* and noticing that $k^2 = (ij)(ij) = -ijji = i^2 = -1$, this completely characterizes the [quaternions](https://en.wikipedia.org/wiki/Quaternion). + +Choosing different values for $i^2$ and $j^2$ yield different shapes than a sphere. If you know a little group theory, you might know there are only two nonabelian (noncommutative) groups of order 8: the [quaternion group](https://en.wikipedia.org/wiki/Quaternion_group) and the [dihedral group of degree 4](https://en.wikipedia.org/wiki/Examples_of_groups#dihedral_group_of_order_8). In the latter group, *j* and *k* are both imaginary, but square to 1 (*i* still squares to -1). I'm being a bit careless with the meanings of "1" and "-1" here, but these objects come from the additive structure of the algebra, rather than the multiplicative. + +Changing the sign of one (or both) of the imaginary squares in the expression $h/h^*$ above switches the multiplicative structure from quaternions to the dihedral group. In this group, picking $i^2 = -j^2 = -1$ parametrizes a hyperboloid of one sheet, and picking $i^2 = j^2 = 1$ parametrizes a hyperboloid of two sheets. + + +Quaternions and Rotation +------------------------ + +Complex numbers are useful for describing 2D rotations and elegantly describe the stereographic projection of a circle. Consequently, since quaternions elegantly describe the stereographic projection of a sphere, they are useful for 3D rotations. + +Since the imaginary units *i*, *j*, and *k* are anticommutative, a general quaternion does not commute with other quaternions like complex numbers do. This embodies a difficulty with 3D rotations in general: unlike 2D rotations, they do not commute. + +Only three of the four components in a quaternion are necessary to describe a point in 3D space. The three imaginary axes are symmetric in some way (i.e., they all square to -1), so we use the *ijk* subspace. Quaternions in this subspace are called *vectors*. Another reason for using the *ijk* subspace comes from considering a point $u = ai + bj + ck$ on the unit sphere ($a^2 + b^2 + c^2 = 1$). Then the square of this point is: + +$$ +\begin{align*} +u^2 &= (ai + bj + ck)(ai + bj + ck) \\ +&= a^2i^2 + abij + acik + abji + b^2j^2 + bcjk + acki + bckj + c^2k^2 \\ +&= (a^2 + b^2 + c^2)(-1) + ab(ij + ji) + ac(ik + ki) + bc(jk + kj) \\ +&= -1 + 0 + 0 + 0 +\end{align*} +$$ + +This property means that *u* behaves similarly to a typical imaginary unit. If we form pseudo-complex numbers of the form $a + b u$, then ${1 + tu \over 1 -\ tu} = \alpha + \beta u$ specifies *some kind of rotation* in terms of the parameter t. Identically as with complex numbers, the inverse rotation is the conjugate $1 -\ tu \over 1 + tu$. + +Another useful feature of quaternion algebra deals with symmetry transformations of the imaginary units. If an imaginary unit (one of *i*, *j*, *k*) is left-multiplied by a quaternion $q$ and right-multiplied by its conjugate $q^*$, then the result is still imaginary. In other words, if p is a vector, then $qpq^*$ is also a vector since *i*, *j*, and *k* form a basis and multiplication distributes over addition. I will not demonstrate this fact, as it requires a large amount of algebra. + +These two features combine to characterize a transformation for a vector quaternion. For a vector *p*, if $q_u(t)$ is a "rotation" as above for a point *u* on the unit sphere, then another vector (the image of *p*) can be described by + +$$ +\begin{gather*} +qpq^* = qpq^{-1} += \left({1 + tu \over 1 -\ tu}\right) p \left({1 -\ tu \over 1 + tu}\right) += (\alpha + \beta u)p(\alpha -\ \beta u) \\[4pt] += (\alpha + \beta u)(\alpha p -\ \beta pu) += \alpha^2 p -\ \alpha \beta pu + \alpha \beta up -\ \beta^2 upu +\end{gather*} +$$ + + +Testing a Transformation +------------------------ + +If this transformation is a rotation, then it should recreate the 2D case which can be described more simply by complex numbers. For example, if $u = k$ and $p = xi + yj$, then this can be expanded as: + +$$ +\begin{align*} +qpq^{-1} &= (\alpha + \beta k)(xi + yj)(\alpha -\ \beta k) \vphantom{\over} \\ +&= \alpha^2(xi + yj) -\ \alpha \beta (xi + yj)k + \alpha \beta k(xi + yj) -\ \beta^2 k(xi + yj)k +\vphantom{\over} \\ +&= \alpha^2(xi + yj) -\ \alpha \beta(-xj + yi) + \alpha \beta(xj -\ yi) -\ \beta^2(xi + yj) +\vphantom{\over} \\ +&= (\alpha^2 -\ \beta^2)(xi + yj) + 2\alpha \beta(xj - yi) +\vphantom{\over} \\ +&= [(\alpha^2 -\ \beta^2)x -\ 2\alpha \beta y]i + [(\alpha^2 -\ \beta^2)y + 2\alpha \beta x]j +\end{align*} +$$ + +If *p* is rewritten as a vector (in the linear algebra sense), then this final expression can be rewritten as a linear transformation of *p*: + +$$ +\begin{pmatrix} +(\alpha^2 -\ \beta^2)x -\ 2 \alpha \beta y \\ +(\alpha^2 -\ \beta^2)y + 2 \alpha \beta x +\end{pmatrix} = +\begin{pmatrix} +\alpha^2 -\ \beta^2 & -2\alpha \beta \\ +2\alpha \beta & \alpha^2 -\ \beta^2 +\end{pmatrix} +\begin{pmatrix} x \\ y \end{pmatrix} +$$ + +The matrix which appears above is the square of the matrix $\begin{pmatrix}\alpha & -\beta \\ \beta & \alpha \end{pmatrix}$. This is isomorphic to the complex number $\alpha + \beta i$. Since $\alpha$ and $\beta$ were chosen so to lie on the (pseudo-) complex unit circle, this means that the vector (*x*, *y*) is rotated by $\alpha + \beta i$ twice. This also demonstrates that u specifies the axis of rotation, or equivalently, the plane of a great circle. This means that the "some kind of rotation" specified by $q = {1 + tu \over 1 -\ tu}$ is a rotation around the great circle normal to *u*. + +The double rotation resembles the half-steps of rotation in the complex numbers, where $1 + ti$ performs a dilation, ${1 \over 1 -\ ti}$ reverses it, and together they describe a rotation. Since quaternions do not commute, sandwiching *p* between *q* and its conjugate makes the algebra simpler. The product of a vector with itself should be totally real, as shown above (and equal to the negative of the norm of *p*, the sum of the squares of its components). The rotated *p* shares the same norm as *p*, so it should equal $p^2$. Indeed, this is the case, as + +$$ +(qpq^{-1})^2 = (qpq^{-1})(qpq^{-1}) = q p p q^{-1} = p^2 q q^{-1} = p^2 +$$ + +As a final remark, due to rotation through quaternions being doubled, the interpolating polynomial used above to smooth out double rotations can also be used here. That is, $q_u(P(t))pq_u^{-1}(P(t))$ rotates *p* by (approximately) the fraction of a turn specified by t through the great circle which intersects the plane normal to *u*. + + +Closing +------- + +It is difficult to find a treatment of rotations which does not hesitate to use the complex exponential $e^{ix} = \cos(x) + i \sin(x)$. The true criterion for rotations is simply that a point lies on the unit circle. Perhaps this contributes to why understanding the 3D situation with quaternions is challenging for many. Past the barrier to entry, I believe them to be rather intuitive. I have outlined some futher benefits of this approach in this post. + +As a disclaimer, this post was inspired by [this video](https://www.youtube.com/watch?v=d4EgbgTm0Bg) by 3blue1brown on YouTube (at least subconsciously; I had not seen the video in years before checking that I wasn't just plagiarizing it). It *also* uses stereographic projections of the circle and sphere to describe rotations in the complex plane and quaternion vector space. However, I feel like it fails to provide an algebraic motivation for quaternions, or even stereography in the first place. Hopefully, my remarks on the algebraic approach can be used to augment the information in the video. + +Diagrams created with GeoGebra. Repository with approximations (as well as GeoGebra files) available [here](https://github.com/queue-miscreant/approx-trig). diff --git a/posts/stereo/2/index.qmd b/posts/stereo/2/index.qmd new file mode 100644 index 0000000..56f2f13 --- /dev/null +++ b/posts/stereo/2/index.qmd @@ -0,0 +1,411 @@ +--- +format: + html: + html-math-method: katex +--- + + +Further Notes on Algebraic Stereography +======================================= + +In my previous post, I discussed the stereographic projection of a circle as it pertains to complex numbers, as well as its applications in 2D and 3D rotation. In an effort to document more interesting facts about this mathematical object (of which scarce information is immediately available online), I will now elaborate on more of its properties. + + +Chebyshev Polynomials +--------------------- + +[Previously](), I derived the [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials) with the archetypal complex exponential. These polynomials express the sines and cosines of a multiple of an angle from the sine and cosine of the base angle. + +$$ +\begin{gather*} +\cos(n\theta) = T_n(\cos(\theta)) \\ +\sin(n\theta) = U_{n-1}(\cos(\theta)) \sin(\theta) +\end{gather*} +$$ + +Where $T_n(t)$ are Chebyshev polynomials of the first kind and $U_n(t)$ are those of the second kind. The complex exponential derivation begins by squaring and developing a second-order recurrence. + +$$ +\begin{align*} +(e^{i\theta})^2 &= (\cos + i\sin)^2 \\ +&= \cos^2 + 2i\cos \cdot \sin -\ \sin^2 + (0 = \cos^2 + \sin^2 -\ 1) \\ +&= 2\cos^2 + 2i\cos \cdot \sin -\ 1 \\ +&= 2\cos \cdot (\cos + i\sin) -\ 1 \\ +&= 2\cos(\theta)e^{i\theta} -\ 1 \\ +(e^{i\theta})^{n+2} &= 2\cos(\theta)(e^{i\theta})^{n+1} -\ (e^{i\theta})^n +\end{align*} +$$ + +This recurrence relation can then be used to obtain the Chebyshev polynomials, and hence, the expressions using sine and cosine above. Presented this way with such a simple derivation, it appears as though these relationships are inherently trigonometric. However, these polynomials actually have *nothing* to do with sine and cosine. For one, they appear in graph theory, and for two, the importance of the complex exponential is overstated. + +$e^{i\theta}$ really just specifies a point on the complex unit circle. This property is used on the second line to coax the equation into a quadratic in $e^{i\theta}$. This is also the *only* property upon which the recurrence depends; all else is algebraic manipulation. + + +### Back to the Stereograph + +Knowing this, let's start over with the stereographic projection of the circle: + +$$ +o_1(t) = {1 + it \over 1 -\ it} += {1 -\ t^2 \over 1 + t^2} + i {2t \over 1 + t^2} += \text{c}_1 + i\text{s}_1 +$$ + +The subscript "1" is because as *t* ranges over $(-\infty, \infty)$, a single loop around the circle is made. Thanks to the properties of complex numbers, it is impossible to grow the norm by exponentiating, and therefore impossible for points to stray from the unit circle. Thus, more loops can be made around the circle by raising this expression to higher powers: + +$$ +\begin{align*} +o_n &= (o_1)^n += \left( {1 + it \over 1 -\ it} \right)^n \\ +\text{c}_n + i\text{s}_n +&= (\text{c}_1 + i\text{s}_1)^n +\end{align*} +$$ + +This mirrors raising the complex exponential to a power, i.e., increasing the number of loops *it* makes in the interval $(-\pi, \pi)$. The final line is analogous to de Moivre's formula, but in a form where everything is a ratio of polynomials in *t*. This also means that the Chebyshev polynomials can be obtained directly from these rational expressions: + +$$ +\begin{align*} +o_2 = (o_1)^2 &= (\text{c}_1 + i\text{s}_1)^2 \\ +&= \text{c}_1^2 + 2i\text{c}_1\text{s}_1 -\ \text{s}_1^2 + +(0 = \text{c}_1^2 + \text{s}_1^2 -\ 1) \\ +&= 2\text{c}_1^2 + 2i\text{c}_1\text{s}_1 -\ 1 \\ +&= 2\text{c}_1(\text{c}_1 + i\text{s}_1) -\ 1 \\ +&= 2\text{c}_1 o_1 -\ 1 \\ +o_2 \cdot (o_1)^n &= 2\text{c}_1 o_1 \cdot (o_1)^n -\ (o_1)^n \\ +o_{n+2} &= 2\text{c}_1 o_{n+1} -\ o_n \\ +\end{align*} +$$ + +This matches the earlier recurrence relation with the complex exponential and therefore the recurrence relation of the Chebyshev polynomials. This also means that the the rational functions obey the same relationship as sine and cosine + +$$ +\begin{matrix} +\begin{gather} +\text{c}_n = T_n(\text{c}_1) \\ +\text{s}_n = U_{n-1}(\text{c}_1) \text{s}_1 +\end{gather} & +\text{where } +\text{c}_1 = {1 -\ t^2 \over 1 + t^2}, & +\text{s}_1 = {2t \over 1 + t^2} +\end{matrix} +$$ + +Thus it is shown that the Chebyshev polynomials are tied to (coordinates on) circles, rather than explicitly to trigonometry. It is a bit strange that these polynomials are in terms of rational functions, but no stranger than them being in terms of *ir* rational functions like sine and cosine. + + +Calculus +-------- + +Since these functions behave similarly to sine and cosine, one might wonder about the nature of these expressions in the context of calculus. For comparison, the complex exponential (as it is a parallel construction) has a simple derivative: + +$$ +\begin{align*} +e^{it} &= \cos(t) + i\sin(t) \\ +{d \over dt} e^{it} &= {d \over dt} \cos(t) + {d \over dt} i\sin(t) \\ +i e^{it} &= -\sin(t) + i\cos(t) \\ +i[\cos(t) + i\sin(t)] &\stackrel{\checkmark}{=} -\sin(t) + i\cos(t) +\end{align*} +$$ + +By applying the chain rule, one can easily differentiate the complex exponential. Since the exponential function is its own derivative, the expression acquires an imaginary coefficient through the chain rule. Meanwhile, the complex stereograph has derivative + +$$ +\begin{align*} +{d \over dt} o_1(t) &= {d \over dt} {1 + it \over 1 -\ it} += {i(1 -\ it) + i(1 + it) \over (1 -\ it)^2} \\ +&= {2i \over (1 -\ it)^2} += {2i(1 + it)^2 \over (1 + t^2)^2} += {2i(1 -\ t^2 + 2it) \over (1 + t^2)^2} \\ +&= {-4t \over (1 + t^2)^2} + i {2(1 -\ t^2) \over (1 + t^2)^2} \\ +&= {-2 \over 1 + t^2}s_1 + i {2 \over 1 + t^2}c_1 \\ +&= -(1 + c_1)s_1 + i(1 + c_1)c_1 \\ +&= i(1 + c_1)o_1 +\end{align*} +$$ + +Just like the complex exponential, an imaginary coefficient falls out. However, the expression also accrues a $1 + c_1$ term, almost like an adjustment factor for its failure to be the complex exponential. Sine and cosine obey a simpler relationship with respect to the derivative, and thus need no adjustment. + + +Complex Analysis +---------------- + +Since $o_n$ is a curve which loops around the unit circle *n* times, that possibly suits it to showing a simple result from complex analysis. Integrating along a contour which wraps around a sufficiently nice function's pole (i.e., where its magnitude grows without bound) yields a familiar value. This is easiest to see with $f(z) = 1 / z$: + +$$ +\oint_\Gamma {1 \over z} dz = +\int_a^b {\gamma'(t) \over \gamma(t)} dt += 2\pi i +$$ + +In this example, $\Gamma$ loops once counterclockwise around the pole at *z* = 0; more loops will scale this by a factor according to the number of loops. Normally this is demonstrated with the complex exponential, but will $o_1$ work just as well? If $\Gamma$ is the unit circle, the integral is: + +$$ +\oint_\Gamma {1 \over z} dz = +\int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt += \int_{-\infty}^\infty i(1 + c_1(t)) dt += 2i\int_{-\infty}^\infty {1 \over 1 + t^2} dt +$$ + +If one has studied their integral identities, the indefinite version of the final integral will be obvious as $\arctan(t)$, which has horizontal asymptotes of $\pi / 2$ and $-\pi / 2$. Therefore, the value of the integral is indeed $2\pi i$. + +If there are *n* loops, then naturally there are *n* of these $2\pi i$s. Fortunately, powers of *o* are more loops around the circle, and the chain and power rules can be used to show: + +$$ +\begin{gather*} +{d \over dt} (o_1)^n = n(o_1)^{n-1} {d \over dt} o_1 \\ \\ +\oint_\Gamma {1 \over z} dz = +\int_{-\infty}^\infty {n o_1(t)^{n-1} o_1'(t) \over o_1(t)^n} dt += n \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt += 2 \pi i n +\end{gather*} +$$ + +It is certainly possible to perform these contour integrals along straight lines; in fact, integrating along lines from 1 to *i* to -1 to -*i* similarly deals with an indefinite integral involving arctangent. However, the best one can do to construct more loops with lines is to count each line multiple times, which isn't extraordinarily convincing. + +Perhaps the use of $\infty$ in the integral bounds is also unconvincing, but the integral can be shifted back into the realm of plausibility by considering simpler bounds on $o_2$: + +$$ +\begin{align*} +\oint_\Gamma {1 \over z} dz &= +\int_{-1}^1 {2 o_1(t) o_1'(t) \over o_1(t)^2} dt \\ +&= 2 \int_{-1}^1 {o_1'(t) \over o_1(t)} dt \\ +&= 2(2i\arctan(1) -\ 2i\arctan(-1)) \\ +&= 2\pi i +\end{align*} +$$ + +This has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating, one obtains the series form of the arctangent. This series converges for $-1 \le t \le 1$, which happens to match the bounds of integration. The convergence of this series is fairly important, since it is tied to formulas for $\pi$, in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80). + +Were one to integrate with the complex exponential, the bounds $(0, 2\pi)$ would be invoked, since at this point a full loop has been made. But think to yourself -- how do you know that $2\pi$ radians is equivalent to 0 radians? How do you know the period of the complex exponential? The result using stereography relies on neither of these prior results and is directly pinned to a formula for $\pi$ instead an apparent detour through the number e. + + +Polar Curves +------------ + +Polar coordinates are useful for expressing for which the distance from the origin is a function of the angle with respect to the positive *x*-axis. They can also be readily converted to parametric forms: + +$$ +\begin{gather*} +r(\theta) &\Longleftrightarrow& +\begin{matrix} +x(\theta) = r \cos(\theta) \\ +y(\theta) = r \sin(\theta) +\end{matrix} +\end{gather*} +$$ + +Polar curves frequently loop in on themselves, and so choosing appropriate bounds for $\theta$ (usually as multiples of $\pi$) is necessary for plotting. Evidently, this is due to the use of sine and cosine in the above parametrization. Fortunately, $s_n$ and $c_n$ (as shown by the calculus above) potentially have much simpler bounds. So what happens when one substitutes the rational functions in place of the trig ones? + + +### Polar Roses + +[Polar roses](https://en.wikipedia.org/wiki/Rose_(mathematics)) are beautiful shapes which have a simple form when expressed in polar coordinates. + +$$ +r(\theta) = \cos \left( {p \over q} \cdot \theta \right) +$$ + +The ratio $p/q$ in least terms uniquely determines the shape of the curve. + +If this weren't this post, you might assume this curve is transcendental since it uses cosine, but you probably know better at this point. The Chebyshev examples above demonstrate the resemblance between $c_n$ and $\cos(n\theta)$. The subscript of $c$ is easiest to work with as an integer, so let $q = 1$. + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +$$ +x(t) = c_p(t) c_1(t) \qquad y(t) = c_p(t) s_1(t) +$$ + +will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$. Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0). +::: +:::: + +$q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder whether the other polar curves can be obtained by allowing it to vary as well. And you'd be right. + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +$$ +x(t) = c_p(t) c_q(t) \qquad y(t) = c_p(t) s_q(t) +$$ + +will plot a $p/q$ polar rose as t ranges over $(-\infty, \infty)$. +::: +:::: + +Just as with the prior calculus examples, doubling all subscripts of *c* and s will only require t to range over $(-1, 1)$, which removes the ugly bite mark. Perhaps it is also slightly less satisfying, since the fraction $p/q$ directly appears in the typical polar incarnation with cosine. On the other hand, it exposes an important property of these curves: they are all rational. + +This approach lends additional precision to a prospective pseudo-polar coordinate system. In the next examples, I will be using the following notation for compactness: + +$$ +\begin{gather*} +R_n(t) = f(t) &\Longleftrightarrow& +\begin{matrix} +x(t) = f(t) c_n(t) \\ +y(t) = f(t) s_n(t) +\end{matrix} +\end{gather*} +$$ + + +### Conic Sections + +The polar equation for a conic section (with a particular unit length occurring somewhere) in terms of its eccentricity $\varepsilon$ is: + +$$ +r(\theta) = {1 \over 1 -\ \varepsilon \cos(\theta)} +$$ + +Correspondingly, the rational polar form can be expressed as + +$$ +R_1(t) = {1 \over 1 -\ \varepsilon c_1} +$$ + +Since polynomial arithmetic is (slightly) easier to work with than trigonometric identities, it is a matter of pencil-and-paper algebra to recover the implicit form from a parametric one. + + +#### Parabola ($|\varepsilon| = 1$) + +The conic section with the simplest implicit equation is the parabola. Since $c_n$ is a simple ratio of polynomials in *t*, it is much simpler to recover the implicit equation. For $\varepsilon = 1$, + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +$$ +\begin{align*} +1 -\ c_1 &= 1 -\ {1 -\ t^2 \over 1 + t^2} = {2 t^2 \over 1 + t_1} \\ +y &= {s_1 \over 1 -\ c_1} = {2t \over 1 + t^2} {1 + t^2 \over 2 t_1} = {1 \over t} \\ +x &= {c_1 \over 1 -\ c_1} = {1 -\ t^2 \over 1 + t^2} \cdot {1 + t^2 \over 2 t^2} = {1 -\ t^2 \over 2t^2} \\ +&= {1 \over 2t^2} -\ {1 \over 2} = {y^2 \over 2} -\ {1 \over 2} +\end{align*} +$$ +::: +:::: + +*x* is a quadratic polynomial in *y*, so trivially the figure formed is a parabola. Technically it is missing the point where $y = 0 (t = \infty)$, and this is not a circumstance where using a higher $c_n$ would help. It is however, similar to the situation where we allow $o_1(\infty) = -1$, and an argument can be made to waive away any concerns one might have. + + +#### Ellipse ($|\varepsilon| < 1$) + +Ellipses are next. The simplest fraction between zero and one is 1/2, so for $\varepsilon = 1/2$, + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +$$ +\begin{align*} +1 -\ {1 \over 2}c_1 &= 1 -\ {1 \over 2} \cdot {1 -\ t^2 \over 1 + t^2} = {3 t^2 + 1 \over 2 + 2t^2} \\ +y &= {s_1 \over 1 -\ {1 \over 2}c_1} = {4t \over 3t^2 + 1} \\ +x &= {c_1 \over 1 -\ {1 \over 2}c_1} = {2 -\ 2t^2 \over 3t^2 + 1} \\ \\ +\end{align*} +$$ +::: +:::: + +There isn't an obvious way to combine products of *x* and *y* into a single equation. However, we do know that such an equation must have total degree 2. Any monomial of degree 2 must has a squared denominator, but must be able to add with the lower-degree terms. Therefore, the numerator of the sum of these degree 2 terms has the (degree 1) denominator as a factor. + +Since the coefficient of $t^4$ in $x^2$ is 4, it must be multiplied by a multiple of 3 (the coefficient of $t^2$ in the denominator) for any factorization to occur. + +$$ +\begin{align*} +x^2 &= {4 -\ 8t^2 + 4t^4 \over (3t^2 + 1)^2} \qquad +y^2 = {16t^2 \over (3t^2 + 1)^2} \\ \\ +3x^2 + 4y^2 &= +{12 -\ 24t^2 + 12t^4 + 64t^2 \over (3t^2 + 1)^2} \\ +&= {12 -\ 40t^2 + 12t^4 \over (3t^2 + 1)^2} \\ +&= {(4t^2 + 12) (3t^2 + 1) \over (3t^2 + 1)^2} \\ +&= {4t^2 + 12 \over 3t^2 + 1} +\end{align*} +$$ + +The linear term *t* occurs nowhere in the result, so this is clearly some combination of *x* and a constant. By the previous line of thought, the constant term must be a multiple of 4, and picking the smallest option finally results in the implicit form: + +$$ +\begin{align*} +4 &= {4(3t^2 + 1) \over 3t^2 + 1} = {12t^2 + 4) \over 3t^2 + 1} \\ +{4t^2 + 12 \over 3t^2 + 1} -\ 4 &= {8 -\ 8t^2 \over 3t^2 + 1} = 4x \\ \\ +3x^2 + 4y^2 &= 4x + 4 \\ +3x^2 + 4y^2 -\ 4x -\ 4 &= 0 +\end{align*} +$$ + +Notably, the coefficients of x and y are 3 and 4; simultaneously, $o_1(\varepsilon) = o_1(1/2) = {3 \over 5} + i{4 \over 5}$. This binds together three concepts: the simplest case of the Pythagorean theorem, the 3-4-5 right triangle; the coefficients of the implicit form; and the role of eccentricity with respect to stereography. + + +#### Hyperbola ($|\varepsilon| > 1$) + +As evidenced by the bound on the eccentricity above, hyperbolae are in some way the inverses of ellipses. Since $o_1(2)$ is a reflection of $o_1(1/2)$, you might think the implicit equation for $\varepsilon = 2$ to be the same, but with a flipped sign or two. Unfortunately, you'd be wrong. + +:::: {layout-ncol = "2"} +::: {} +![]() +::: + +::: {} +$$ +\begin{align*} +x &= {c_1 \over 1 -\ 2c_1} = {1 -\ t^2 \over 3t^2 -\ 1} \\ +y &= {s_1 \over 1 -\ 2c_1} = {2t \over 3t^2 -\ 1} \\ \\ +3x^2 -\ y^2 &= {3 -\ 6t^2 + 3t^4 -\ 4t^2 \over 3t^2 -\ 1} \\ +&= {3 -\ 10t^2 + 3t^4 \over 3t^2 -\ 1} \\ +&= {(t^2 -\ 3)(3t^2 -\ 1) \over (3t^2 -\ 1)^2 } \\ +&= {t^2 -\ 3 \over 3t^2 -\ 1 } \\ +&= {3t^2 -\ 1 + t^2 -\ 3 \over 3t^2 -\ 1 } -\ 1 \\ +&= {4t^2 -\ 4 \over 3t^2 -\ 1 } -\ 1 += -4x -\ 1 +\end{align*} +$$ + +$$ +3x^2 -\ y^2 + 4x + 1 = 0 +$$ +::: +:::: + +At the very least, the occurrences of 1 in the place of 4 have a simple explanation: 1 = 4 - 3. This reinforces the previous example's link to the Pythagorean triple 3-4-5. + + +### Archimedean Spiral + +Arguably the simplest (non-circular) polar curve is $r(\theta) = \theta$. This is also the unit [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral). Since the curve is defined by a constant turning, this is a natural application of the properties of sine and cosine. The closest equivalent in rational polar coordinates is $R_1(t) = t$. But this can be converted to an implicit form: + +$$ +\begin{gather*} +x = tc_1 \qquad y = ts_1 \\ \\ +x^2 + y^2 = t^2(c_1^2 + s_1^2) = t^2\\ +y = {2t^2 \over 1 + t^2} = {2(x^2 + y^2) \over 1 + (x^2 + y^2)} \\ \\ +(1 + x^2 + y^2)y = 2(x^2 + y^2) +\end{gather*} +$$ + +The curve produced by this equation is a [right strophoid](https://mathworld.wolfram.com/RightStrophoid.html) with a node at (0, 1) and asymptote $y = 2$. This form suggests something interesting about this curve: it approximates the Archimedean spiral (specifically the one with polar equation $r(\theta) = \theta/2$). Indeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate the (unit) spiral for larger *n*, as can be seen in the following video. + +![]() + +Since R necessarily defines a rational curve, the curves will never be equal, just as any stretching of $c_n$ will never exactly become cosine. + + +Closing +------- + +There is nothing wrong with using sine, cosine, the exponential function, or any of the entailing math, especially in a calculus setting. The benefit to using these functions is their constant "velocity" around the circle as their parameter varies continuously. Indeed, velocity, change, and continuity are some of the concepts which calculus deals with in the first place. Further, nearly every modern scientific calculator in the world features buttons for trigonometric functions. + +We can however be misled by their apparent omnipresence. Stereographic projection has been around for *millennia*; just because Newton invented calculus 3.5 centuries ago does not mean every formula needs to be rewritten in its language. For example (and as previously mentioned), the Chebyshev polynomials are tied to the multiplication of two complex numbers whose norm cannot grow. This explanation requires only an understanding of complex number arithmetic, and not of trigonometry and dividing angles. Many other instances of sine and cosine merely rely on a number (or ratio) of loops around a circle, rather than their properties in calculus, and for these instances it will obviously do to "stay rational". + +One of my favorite things to plot as a kid were polar roses, so it was somewhat of a shock to see that they are, in fact, rational curves. On the other hand, their rationality follows immediately from the rationality of the circle (which itself follows from the existence of Pythagorean triples). If I were more experienced with manipulating Chebyshev polynomials or willing to set up a linear system in (way too) many terms, I might have considered attempting to find an implicit form for them as well. However, only the latter approach would involve polynomial arithmetic and therefore be relevant to the contents of this post. + +Diagrams created with Sympy and Matplotlib. From 7dd5f5a44efce374fdd5d4b9269fd612dde9530f Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Sun, 29 Jun 2025 23:53:02 -0500 Subject: [PATCH 02/14] revisions and images to stereo.1 --- posts/stereo/1/index.qmd | 716 ++++++++++++++++++++++--------- posts/stereo/1/stereo_circle.png | Bin 0 -> 196771 bytes posts/stereo/1/stereo_sphere.png | Bin 0 -> 120577 bytes 3 files changed, 509 insertions(+), 207 deletions(-) create mode 100644 posts/stereo/1/stereo_circle.png create mode 100644 posts/stereo/1/stereo_sphere.png diff --git a/posts/stereo/1/index.qmd b/posts/stereo/1/index.qmd index d23e2f0..de9a4ea 100644 --- a/posts/stereo/1/index.qmd +++ b/posts/stereo/1/index.qmd @@ -1,218 +1,379 @@ --- +title: "Algebraic Rotations and Stereography" +description: | + How do you rotate in 2D and 3D without standard trigonometry? format: html: html-math-method: katex +jupyter: python3 +date: "2021-09-26" +date-modified: "2025-06-28" +categories: + - geometry + - symmetry --- + -Algebraic Rotations and Stereography -==================================== - -Expressing rotation mathematically can be rather challenging, even in two dimensional space. Typically the subject is approached from complicated-looking rotation matrices, with the occasional accompanying comment about complex numbers. The challenge only increases when examining three dimensional space, where everyone has different ideas about the best implementations. In what follows, I attempt to provide a derivation of 2D and 3D rotations based in intuition and without the use of trigonometry. +Expressing rotation mathematically can be rather challenging, even in two dimensional space. +Typically, the subject is approached from complicated-looking rotation matrices, + with the occasional accompanying comment about complex numbers. +The challenge only increases when examining three dimensional space, + where everyone has different ideas about the best implementations. +In what follows, I attempt to provide a derivation of 2D and 3D rotations based in intuition + and without the use of trigonometry. Complex Numbers: 2D Rotations ----------------------------- -Multiplication between complex numbers is frequently analyzed as a combination of scaling (a ratio of two scales) and rotation (a point on the complex unit circle). However, the machinery used in these analyses usually expresses complex numbers in a polar form involving the complex exponential, concepts which leverage knowledge of trigonometry and some calculus. In fact, they obfuscate the algebraic properties of complex numbers when neither concept is truly necessary to understand points on the complex unit circle, or complex number multiplication in general. +As points in the plane, multiplication between complex numbers is frequently analyzed + as a combination of scaling (a ratio of two scales) and rotation (a point on the complex unit circle). +However, the machinery used in these analyses usually expresses complex numbers in a polar form + involving the complex exponential. +This leverages prior knowledge of trigonometry and some calculus, and in fact obfuscates + the algebraic properties of complex numbers. +Neither concept is truly necessary to understand points on the complex unit circle, + or complex number multiplication in general. ### A review of complex multiplication -A complex number $z = a + bi$ multiplies with another complex number $w = c + di$ in the obvious way: by distributing over addition. +A complex number $z = a + bi$ multiplies with another complex number + $w = c + di$ in the obvious way: by distributing over addition. $$ -zw = (a+bi)(c+di) = ac + bci + adi + bdi^2 = (ac -\ bd) + i(ad + bc) +zw = (a + bi)(c + di) = ac + bci + adi + bdi^2 = (ac - bd) + i(ad + bc) $$ -*z* also has associated to it a conjugate $z^* = a -\ bi$. The product $zz^*$ is the *norm* $a^2 + b^2$, a positive real quantity whose square root is typically called the *magnitude* of the complex number. Taking the square root is frequently unnecessary, as many of the properties of this quantity do not depend on this normalization. One such property is that for two complex numbers *z* and *w*, the norm of the product is the product of the norm. +*z* also has associated to it a conjugate $z^* = a - bi$. +The product $zz^*$ is the *norm* $a^2 + b^2$, a positive real quantity whose square root + is typically called the *magnitude* of the complex number. +Taking the square root is frequently unnecessary, as many of the properties of + this quantity do not depend on this normalization. + +One such property is that for two complex numbers *z* and *w*, + the norm of the product is the product of the norm. $$ \begin{align*} -(zw)(zw)^* &= (ac -\ bd)^2 + (ad + bc)^2 \\ -&= a^2c^2 -\ 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2 \\ -&= a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 = a^2(c^2 + d^2) + b^2(c^2 + d^2) \\ -&= (a^2 + b^2)(c^2 + d^2) + (zw)(zw)^* &= (ac - bd)^2 + (ad + bc)^2 + \\ + &= a^2c^2 - 2abcd + b^2d^2 + a^2d^2 + 2abcd + b^2c^2 + \\ + &= a^2c^2 + b^2d^2 + a^2d^2 + b^2c^2 + = a^2(c^2 + d^2) + b^2(c^2 + d^2) + \\ + &= (a^2 + b^2)(c^2 + d^2) \end{align*} $$ -Conjugation also distributes over multiplication, and complex numbers possess multiplicative inverses. Also, the norm of the multiplicative inverse is the inverse of the norm. +Conjugation also distributes over multiplication, and complex numbers possess multiplicative inverses. +Also, the norm of the multiplicative inverse is the inverse of the norm. $$ \begin{gather*} -z^*w^* = (a -\ bi)(c -\ di) = ac -\ bci -\ adi + bdi^2 \\ -= (ac -\ bd) -\ i(ad + bc) = (zw)^* \\ -{1 \over z} = {z^* \over zz^*} = {a -\ bi \over a^2 + b^2} \\ -\left({1 \over z}\right) -\left({1 \over z}\right)^* -= {1 \over zz^*} = {1 \over a^2 + b^2} \\ + z^*w^* = (a - bi)(c - di) = ac - bci - adi + bdi^2 + \\ + = (ac - bd) - i(ad + bc) = (zw)^{*} + \\ + {1 \over z} = {z^{*} \over z z^{*}} = {a - bi \over a^2 + b^2} + \\[14pt] + \left({1 \over z}\right) + \left({1 \over z}\right)^{*} + = {1 \over zz^{*}} = {1 \over a^2 + b^2} \end{gather*} $$ -A final interesting note about complex multiplication is the product $z^*w$ +A final interesting note about complex multiplication is the product $z^{*} w$ $$ -z^* w = (a -\ bi)(c + di) = ac -\ bci + adi -\ bdi^2 = (ac + bd) + i(ad -\ bc) +z^{*} w = (a - bi)(c + di) = ac - bci + adi - bdi^2 = (ac + bd) + i(ad - bc) $$ -The real part is the same as the dot product $(a, c) \cdot (b, d)$ and the imaginary part looks like the determinant of $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$. This shows the inherent value of complex arithmetic, as both of these quantities have important interpretations in vector algebra. +The real part is the same as the dot product $(a, c) \cdot (b, d)$ and the imaginary part + looks like the determinant of $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$. +This shows the inherent value of complex arithmetic, as both of these quantities + have important interpretations in vector algebra. ### Onto the Circle -If $zz^* = a^2 + b^2 = 1$, then *z* lies on the complex unit circle. Since the norm is 1, a general complex number *w* has the same norm as the product *zw*. Thus, multiplication by *z* results in the rotation that maps 1 to *z*. But how do you get such a point on the unit circle? +If $zz^{*} = a^2 + b^2 = 1$, then *z* lies on the complex unit circle. +Since the norm is 1, a general complex number *w* has the same norm as its product with *z*, *zw*. +Specifically, if *w* is 1, multiplication by *z* can be seen as the rotation that maps 1 to *z*. +But how do you describe a point on the unit circle? -Simple. There are four complex numbers directly related to any complex number *w* that are guaranteed to share its norm: $w, -w, w^*$, and $-w^*$. Therefore, dividing any of these numbers by any other of these numbers results in a complex number on the unit circle. Division by -*w* is boring and just results in -1. However, division by $w^*$ turns out to be important: +Simple. +For any complex *w*, there are three other complex numbers that are guaranteed + to share its norm: $-w, w^{*}$, and $-w^{*}$. +Dividing any one of these numbers by another results in a complex number on the unit circle. +Dividing *w* by -*w* is boring, since that just results in -1. +However, division by $w^*$ turns out to be important: $$ \begin{align*} -z &= {w \over w^*} = -\left( {a + bi \over a -\ bi} \right) \left( {a + bi \over a + bi} \right) = 1 \\[8pt] -&= {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} + z &= {w \over w^*} = + \left( {c + di \over c - di} \right) + \left( {c + di \over c + di} \right) = 1 + \\[8pt] + &= {(c^2 - d^2) + (2cd)i \over c^2 + d^2} \end{align*} $$ -Dividing the numerator and denominator of this expression through by $a^2$ yields an expression in $t = b/a$, where t can be interpreted as the slope of the line joining 0 and *w*. +Dividing the numerator and denominator of this expression through by $c^2$ yields an expression + in $t = d/c$, where t can be interpreted as the slope of the line joining 0 and *w*. +Slopes range from $-\infty$ to $\infty$, which means that the variable *t* does as well. $$ \begin{align*} -{(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} -&= {1/a^2 \over 1/a^2} \cdot {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} \\ -&= {(1 -\ b^2/a^2) + (2b/a)i \over 1 + b^2/a^2} -= {(1 -\ t^2) + (2t)i \over 1 + t^2} \\ -&= {1 -\ t^2 \over 1 + t^2} + i{2t \over 1 + t^2} = z(t) + {(c^2 - d^2) + (2cd)i \over c^2 + d^2} + &= {1/c^2 \over 1/c^2} \cdot {(c^2 - d^2) + (2cd)i \over c^2 + d^2} + \\ + &= {(1 - d^2/c^2) + (2d/c)i \over 1 + d^2/c^2} + = {(1 - t^2) + (2t)i \over 1 + t^2} + \\ + &= {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2} = z(t) \end{align*} $$ -Dividing the numerator and denominator of this expression through by $a^2$ yields an expression in $t = b/a$, where *t* can be interpreted as the slope of the line joining 0 and *w*. +This gives us a point *z* on the unit circle. +Its reciprocal is: $$ -\begin{align*} -{(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} -&= {1/a^2 \over 1/a^2} \cdot {(a^2 -\ b^2) + (2ab)i \over a^2 + b^2} \\ -&= {(1 -\ b^2/a^2) + (2b/a)i \over 1 + b^2/a^2} -= {(1 -\ t^2) + (2t)i \over 1 + t^2} \\ -&= {1 -\ t^2 \over 1 + t^2} + i{2t \over 1 + t^2} = z(t) -\end{align*} +{1 \over z} = {z^{*} \over zz^{*}} = z^{*} $$ -Slopes range from $-\infty$ to $\infty$, which means that the variable *t* does as well. Moreover, if *z* is on the unit circle, then its reciprocal is: +Applying the same operation to *z* as we did to *w* gives us $z / z^{*} = z^2$, + or as an action on points, the rotation from 1 to $z^2$. -$$ -{1 \over z} = {z^* \over zz^*} = z^* -$$ - -In other words, for points already on the unit circle, $z / z^* = z^2$ describes the rotation from 1 to $z^2$. This demonstrates the decomposition of the rotation into two parts. For a general complex number *w* which may not lie on the unit circle, multiplication by this number dilates the complex plane as well rotating 1 toward the direction of *w*. Then, division by $w^*$ undoes the dilation and performs the rotation again. The proper description of the rotation described by $w/w^*$ is the combination of these half steps. +This demonstrates the decomposition of the rotation into two parts. +For a general complex number *w* which may not lie on the unit circle, multiplication by + this number dilates the complex plane as well rotating 1 toward the direction of *w*. +Then, division by $w^{*}$ undoes the dilation and performs the rotation again. +General (i.e., non-unit) rotations *must* occur in pairs. ### Two Halves, Twice Over -Plugging in some values for t reveals something else: $z(0) = 1 + 0i$, $z(1) = 0 + 1i$, and $z(-1) = 0 -\ 1i$. This shows that the semicircle in the right half-plane exists where $t \in (-1, 1)$. The other half of the circle comes from *t* with magnitude greater than 1. The point where $z(t) = -1$ does not properly exist unless it is allowed that $t = \pm \infty$. +Plugging in -1, 0, and 1 for *t* reveals some interesting behavior: + $z(0) = 1 + 0i$, $z(1) = 0 + 1i$, and $z(-1) = 0 - 1i$. +Assuming continuity, this shows that $t \in (-1, 1)$ plots the semicircle in the right half-plane. +The other half of the circle comes from *t* with magnitude greater than 1, + and the point where $z(t) = -1$ exists if we admit the extra point $t = \pm \infty$. -Is it possible to get the other half into the nicer interval? No problem, just double the rotation. +Is it possible to get the other half into a nicer interval? +No problem, just double the rotation. $$ \begin{align*} -z(t)^2 &= (\Re[z]^2 -\ \Im[z]^2) + i(2\Re[z]\Im[z]) \\[8pt] -&= {(1 -\ t^2)^2 -\ (2t)^2 \over (1 + t^2)^2} + i{2(1 -\ t^2)(2t) \over (1 + t^2)^2} \\[8pt] -&= {1 -\ 6t^2 + t^4 \over 1 + 2t^2 + t^4} -+i {4t -\ 4t^3 \over 1 + 2t^2 + t^4} \\ + z(t)^2 &= (\Re[z]^2 - \Im[z]^2) + i(2\Re[z]\Im[z]) + \\[8pt] + &= {(1 - t^2)^2 - (2t)^2 \over (1 + t^2)^2} + i{2(1 - t^2)(2t) \over (1 + t^2)^2} + \\[8pt] + &= {1 - 6t^2 + t^4 \over 1 + 2t^2 + t^4} + + i {4t - 4t^3 \over 1 + 2t^2 + t^4} \end{align*} $$ -Now the entire circle fits in the interval $t \in (-1, 1)$, and $z(-1)^2 = z(1)^2 = -1$. This action of squaring *z* means that as *t* ranges from $-\infty$ to $\infty$, the point $z^2$ loops around the circle twice, since the rotation *z* is applied twice. +Now $z(-1)^2 = z(1)^2 = -1$ and the entire circle fits in the interval \[-1, 1\]. +This action of squaring *z* means that as *t* ranges from $-\infty$ to $\infty$, the point + $z^2$ loops around the circle twice, since the unit rotation *z* is applied twice. -I'd now like to go on a brief tangent and compare these expressions to their transcendental trigonometric counterparts. Graphing the real and imaginary parts separately shows how much they resemble $\cos(\pi t)$ and $\sin(\pi t)$ around 0. -:::: {layout-ncol = "2"} -::: {} -![]() -::: +### Plotting Transcendence -::: {} -![]() -::: -:::: +Let's go on a brief tangent and compare these expressions to their + transcendental trigonometric counterparts. +Graphing the real and imaginary parts separately shows how much they resemble + $\cos(\pi t)$ and $\sin(\pi t)$ around 0[^1]. -Incidentally, an approximation of a function as the ratio of two polynomials is called a [Padé approximant](https://en.wikipedia.org/wiki/Pad%C3%A9_approximant). Specifically, the real part (which approximates cosine) has order \[4/4\] and the imaginary part (which approximates sine) has order \[3/4\]. Ideally, the zeroes of the real part should occur at $t = \pm 1/2$, since $\cos(\pm \pi/2) = 0$. Instead, they occur at +[^1]: An approximation of a function as the ratio of two polynomials is called a + [Padé approximant](https://en.wikipedia.org/wiki/Pad%C3%A9_approximant). + Specifically, the real part (which approximates cosine) has order \[4/4\] and the imaginary part + (which approximates sine) has order \[3/4\]. + +```{python} +#| echo: false +#| layout-ncol: 2 + +import matplotlib.pyplot as plt +import sympy +from sympy.abc import t + +z = (1 + sympy.I * t) / (1 - sympy.I * t) + +sympy.plot( + sympy.re(z**2), + sympy.im(z**2), + (t, -2, 2), + ylim=(-2, 2), + label=[ + r"$\Re \left [ \left ( \frac{1 + it}{1 - it} \right )^2 \right ]$", + r"$\Im \left [ \left ( \frac{1 + it}{1 - it} \right )^2 \right ]$", + ], + legend=True, + xlabel=False, + ylabel=False, +) + +sympy.plot( + sympy.re(z**2) - sympy.cos(sympy.pi * t), + sympy.im(z**2) - sympy.sin(sympy.pi * t), + (t, -2, 2), + ylim=(-2, 2), + label=[ + r"Cosine error", + r"Sine error", + ], + legend=True, + xlabel=False, + ylabel=False, +) +``` + +Ideally, the zeroes of the real part should occur at $t = \pm 1/2$, since $\cos(\pm \pi/2) = 0$. +Instead, they occur at $$ -0 = 1 -\ 6t^2 + t^4 = (t^2 -\ 2t -\ 1)(t^2 + 2t -\ 1) -\implies t = \pm {1 \over \delta_s} = \pm(\sqrt 2 -\ 1) \approx \pm0.414 +\begin{align*} + 0 &= 1 - 6t^2 + t^4 + = (t^2 - 2t - 1)(t^2 + 2t - 1) + \\ + &\implies t = \pm {1 \over \delta_s} = \pm(\sqrt 2 - 1) \approx \pm 0.414 +\end{align*} $$ -With a bit of polynomial interpolation, this can be rectified. A cubic interpolation is given by: +With a bit of polynomial interpolation, this can be rectified. +A cubic interpolation between the expected and actual values is given by: $$ \begin{gather*} -P(x) = ax^3 + bx^2 + cx + d \\ -\begin{pmatrix} -1 & 0 & 0 & 0 \\ -1 & 1/2 & 1/4 & 1/8 \\ -1 & -1/2 & 1/4 & -1/8 \\ -1 & 1 & 1 & 1 \\ -\end{pmatrix} \begin{pmatrix} -d \\ c \\ b \\ a -\end{pmatrix} = \begin{pmatrix} -0 \\ \sqrt 2 -\ 1 \\ -\sqrt 2 + 1 \\ 1 -\end{pmatrix} \\ -\implies \begin{pmatrix} -d \\ c \\ b \\ a - \end{pmatrix} = \begin{pmatrix} -0 \\ -3 + 8\sqrt 2/3 \\ 0 \\ 4 -\ 8\sqrt 2/3 \end{pmatrix} -\implies P(x) \approx 0.229 x^3 + 0.771x + P(x) = ax^3 + bx^2 + cx + d + \\ + \begin{pmatrix} + 1 & 0 & 0 & 0 \\ + 1 & 1/2 & 1/4 & 1/8 \\ + 1 & -1/2 & 1/4 & -1/8 \\ + 1 & 1 & 1 & 1 + \end{pmatrix} \begin{pmatrix} + d \\ + c \\ + b \\ + a + \end{pmatrix} = \begin{pmatrix} + 0 \\ + \sqrt 2 - 1 \\ + -\sqrt 2 + 1 \\ + 1 + \end{pmatrix} + \\ + \implies \begin{pmatrix} + d \\ + c \\ + b \\ + a + \end{pmatrix} = \begin{pmatrix} + 0 \\ + -3 + 8\sqrt 2/3 \\ + 0 \\ + 4 - 8\sqrt 2/3 + \end{pmatrix} + \implies P(x) \approx 0.229 x^3 + 0.771x \end{gather*} $$ -:::: {layout-ncol = "2"} ::: {} -![]() +```{python} +#| echo: false +#| layout-ncol: 2 + +interpolate = (-3 + 8 * sympy.sqrt(2)/3) * t + (4 - 8 * sympy.sqrt(2)/3) * t**3 + +sympy.plot( + sympy.re(z**2).subs(t, interpolate), + sympy.im(z**2).subs(t, interpolate), + (t, -2, 2), + ylim=(-2, 2), + label=[ + r"$\Re \left [ z(P(t))^2 \right ]$", + r"$\Im \left [ z(P(t))^2 \right ]$", + ], + legend=True, + xlabel=False, + ylabel=False, +) + +sympy.plot( + sympy.re(z**2).subs(t, interpolate) - sympy.cos(sympy.pi * t), + sympy.im(z**2).subs(t, interpolate) - sympy.sin(sympy.pi * t), + (t, -2, 2), + ylim=(-2, 2), + label=[ + r"Cosine error", + r"Sine error", + ], + legend=True, + xlabel=False, + ylabel=False, +) +``` + +The error over the interval \[-1, 1\] in these approximations is around 1% (RMS). ::: -::: {} -![]() -::: -The error over the interval [-1, 1] in these approximations is around 1% (RMS). -:::: +Notably, this interpolating polynomial is entirely odd, which gives it some symmetry about 0. +Along with being a very good approximation, it has the feature that a rotation by an + *n*^th^ of a turn is about $z(P(1/n))^2$. +The approximation be improved further by taking *z* to higher powers and deriving a higher-order + interpolating polynomial. -Notably, this interpolating polynomial is entirely odd, which gives it some symmetry about 0. Along with being a very good approximation, it has the feature that a rotation by an *n*^th^ of a turn is about $z(P(1/n))^2$. The approximation be improved further by taking *z* to higher powers and deriving another interpolating polynomial. +It is impossible to shrink this error to 0 because the derivative of the imaginary part at + $t = 1$ would be π. +But the imaginary part is a rational expression, so its derivative is also a rational expression. +Since π is transcendental, there must always be some error[^2]. -It is impossible to shrink this error to 0 because the derivative of the imaginary part at $t = 1$ would be $\pi$ with no error. But the imaginary part is a rational expression (technically with the interpolating polynomial, a ratio of two irrational numbers), so its derivative is also a rational expression. Since $\pi$ is transcendental, there must always be some error. +[^2]: Even with the interpolating polynomial, the quantity is a ratio of two algebraic numbers, + which is also algebraic and not transcendental. -Even though it is arguably easier to interpret, there probably isn't a definite benefit to this approximation. For example, +Even though it is arguably easier to calculate, there probably isn't a definite benefit to this approximation. +For example, -- Padé approximants can be calculated directly from the power series for sine and cosine. +- Other accurate approximants for sine and cosine can be calculated directly from their power series. - There are no inverse functions like `acos` or `asin` to complement these approximations. - - This isn't that bad, since I have refrained from describing rotations with an angle. Even so, the inverse functions have their use cases. + - This isn't that bad, since I have refrained from describing rotations with an angle + (i.e., the quantity returned by these functions). + Even so, the inverse functions have their use cases. - Trigonometric functions can be hardware-accelerated (at least in some FPUs). -- If no such hardware exists, approximations of `sin` and `cos` can be calculated by software beforehand and stored in a lookup table (which is also probably involved at some stage of the FPU). +- If no such hardware exists, approximations of `sin` and `cos` can be calculated by software beforehand + and stored in a lookup table (which is also probably involved at some stage of the FPU). -Elaborating on the latter two points, using a lookup table means evaluation happens in roughly constant time. A best-case analysis of the above approximation, given some value *t* is +Elaborating on the latter two points, using a lookup table means evaluation happens in roughly constant time. +A best-case analysis of the above approximation, given some value *t* is... -$$ -\begin{matrix} -\text{Expression} & \text{Additions} & \text{Multiplications} & \text{Divisions} \\ \hline -p = t \cdot (0.228763834 \cdot t \cdot t + 0.771236166) & -1 & 3 \\[4pt] -q = p \cdot p & -& 1 \\[4pt] -r = 1 + q & -1 \\[4pt] -c = {1 -\ q \over r} & -1 & & 1 \\[4pt] -s = {2 \cdot p \over r} & -2 & 1 & 1 \\[4pt] -real = c \cdot c -\ s \cdot s & -1 & 2 \\[4pt] -imag = 2 \cdot c \cdot s & -& 2 \\ -\hline -\Sigma & -6 & 9 & 2 \\ -\end{matrix} -$$ -Or 17 FLOPs. On a more optimistic note, a Monte Carlo test of the above approximation on my computer yields promising results when compared with the GCC `libmath` implementations of sin and cos (also tested with `cexp`). +| Expression | Additions | Multiplications | Divisions | +|-----------------------------------------------------------|-----------|-----------------|-----------| +| $p = t \cdot (0.228763834 \cdot t \cdot t + 0.771236166)$ | 1 | 3 | | +| $q = p \cdot p$ | | 1 | | +| $r = 1 + q$ | 1 | | | +| $c = { 1 - q \over r}$ | 1 | | 1 | +| $s = { p + p \over r}$ | 1 | | 1 | +| real = $c \cdot c - s \cdot s$ | 1 | 2 | | +| imag = $c \cdot s + c \cdot s$ | 1 | 1 | | +| Total | 6 | 7 | 2 | + +...or 15 FLOPs. + +On a more optimistic note, a Monte Carlo test of the above approximation on my computer yields + promising results when compared with GCC's `libm` implementations of `sin` and `cos` + (also tested with `cexp`). ```{} Timing for 100000 math.h sin and cos: 5409619ns @@ -234,172 +395,313 @@ Squared error in sines: For the source which I used to generate this output, see the repository linked below. -Even though results can be inaccurate, this exercise in the algebraic manipulation of complex numbers is fairly interesting since it requires no calculus to define, unlike sine, cosine, and their Padé approximants (and to a degree, $\pi$). +Even though results can be inaccurate, this exercise in the algebraic manipulation + of complex numbers is fairly interesting since it requires no calculus to define, + unlike sine, cosine, and their Padé approximants (and to a degree, π). From Circles to Spheres ----------------------- -The expression for complex points on the unit circle coincides with the stereographic projection of a circle. This method is achieved by selecting a point on the circle, fixing *t* along a line (such as the *y*-axis), and letting *t* range over all possible values. +The expression for complex points on the unit circle coincides with the stereographic projection of a circle. +This method is achieved by selecting a point on the circle, fixing *t* along a line + (such as the *y*-axis), and letting *t* range over all possible values. -![]() +![ + 2D stereographic projection +](./stereo_circle.png) -The equations of the line and circle appear in the diagram above. Through much algebra, expressions for *x* and *y* in *t* can be formed. +The equations of the line and circle appear in the diagram above. +Through much algebra, expressions for *x* and *y* as functions of *t* can be formed. + $$ \begin{align*} -y &= t(x + 1) \\ -1 &= x^2 + y^2 = x^2 + (tx + t)^2 \\ -&= x^2 + t^2x^2 + 2t^2 x + t^2 \\ -{1 \over t^2 + 1} -\ {t^2 \over t^2 + 1} &= -x^2 + {t^2 \over t^2 + 1} 2x \\ -{1 -\ t^2 \over t^2 + 1} + \left( {t^2 \over t^2 + 1} \right)^2 &= -\left(x + {t^2 \over t^2 + 1} \right)^2 + {t^2 \over t^2 + 1} \\ -{1 -\ t^4 \over (t^2 + 1)^2} + {t^4 \over (t^2 + 1)^2} &= -\left(x + {t^2 \over t^2 + 1} \right)^2 \\ -{1 \over (t^2 + 1)^2} &= -\left(x + {t^2 \over t^2 + 1} \right)^2 \\ -{1 \over t^2 + 1} &= -x + {t^2 \over t^2 + 1} \\ -x &= {1 -\ t^2 \over 1 + t^2} \\ -y &= t(x + 1) = -t\left( {1 -\ t^2 \over 1 + t^2} + {1 + t^2 \over 1 + t^2} \right) \\ -&= {2t \over 1 + t^2} + y &= t(x + 1) + \\ + 1 &= x^2 + y^2 = x^2 + (tx + t)^2 + \\ + &= x^2 + t^2x^2 + 2t^2 x + t^2 + \\[14pt] + {1 \over t^2 + 1} -\ {t^2 \over t^2 + 1} + &= x^2 + {t^2 \over t^2 + 1} 2x + \\ + {1 -\ t^2 \over t^2 + 1} + \left( {t^2 \over t^2 + 1} \right)^2 + &= \left(x + {t^2 \over t^2 + 1} \right)^2 + {t^2 \over t^2 + 1} + \\ + {1 -\ t^4 \over (t^2 + 1)^2} + {t^4 \over (t^2 + 1)^2} + &= \left(x + {t^2 \over t^2 + 1} \right)^2 + \\ + {1 \over (t^2 + 1)^2} + &= \left(x + {t^2 \over t^2 + 1} \right)^2 + \\ + {1 \over t^2 + 1} + &= x + {t^2 \over t^2 + 1} + \\ + x &= {1 -\ t^2 \over 1 + t^2} + \\ + y &= t(x + 1) + = t\left( {1 -\ t^2 \over 1 + t^2} + {1 + t^2 \over 1 + t^2} \right) + \\ + &= {2t \over 1 + t^2} \end{align*} $$ -These are exactly the same expressions which appear in the real and imaginary components of the ratio between $1 + ti$ and its conjugate. Compared to the algebra using complex numbers, this is quite a bit more work. Therefore, one might also ask whether, given a proper arithmetic setting, the projection of a sphere might be obtained in the same manner as the circle with respect to complex numbers. +These are exactly the same expressions which appear in the real and imaginary components + of the ratio between $1 + ti$ and its conjugate. +Compared to the algebra using complex numbers, this is quite a bit more work. +One might ask whether, given a proper arithmetic setting, this can be extended to three dimensions. +In other words, we want to find the projection of a sphere using some new number system + analogously to the circle with respect to complex numbers. -::: {} -![]() -What we are not doing: intersecting the sphere with a line through a pole and a point in the xy plane -::: +![ + What we are *not* doing: intersecting the sphere with a line through a pole and a point in the xy plane +](./stereo_sphere.png) ### Algebraic Projection -The simplest thing to do is try it out. Let's say an extended number $h = 1 + si + tj$ has a conjugate of $h^* = 1 -\ si -\ tj$. It's a bit of an assumption to be able to take the reciprocal of the latter (we don't know whether *i* and *j* create a field or division ring), but following our noses: +The simplest thing to do is try it out. +Let's say an extended number $h = 1 + si + tj$ has a conjugate of $h^{*} = 1 - si - tj$. +Then, following our noses[^3]: + +[^3]: We don't know whether *i* and *j* form a + [field](https://en.wikipedia.org/wiki/Field_%28mathematics%29) + or [division ring](https://en.wikipedia.org/wiki/Division_ring)), + so it's a bit of an assumption for division to be possible. + We'll ignore that. $$ \begin{gather*} -{h \over h^*} = {1 + si + tj \over 1 -\ si -\ tj} = -\left({1 + si + tj \over 1 -\ si -\ tj}\right) -\left({1 + si + tj \over 1 + si + tj}\right) = \\[8pt] { -1 + si + tj + si + s^2i^2 + stij + tj + stji + t^2j^2 -\over -1 -\ si -\ tj + si -\ s^2 i^2 -\ stij + tj -\ stji -\ t^2j^2 -} + {h \over h^{*}} = {1 + si + tj \over 1 - si - tj} + = \left({1 + si + tj \over 1 - si - tj}\right) + \left({1 + si + tj \over 1 + si + tj}\right) + \\[8pt] + = { + 1 + si + tj + si + s^2i^2 + stij + tj + stji + t^2j^2 + \over 1 - si - tj + si - s^2 i^2 - stij + tj - stji - t^2j^2 + } \end{gather*} $$ -While there is some cancellation in the denominator, both products are rather messy. To get more cancellation, we apply some nice algebraic properties between *i* and *j*. For the $stij$ terms to cancel, we let i and j be *anticommutative*, meaning that $ij = -ji$. So that the denominator is totally real (and therefore can be guaranteed to divide), we also apply the constraint that $i^2$ and $j^2$ are both real. Then the expression becomes +While there is some cancellation in the denominator, both products are rather messy. +To get more cancellation, we can add some nice algebraic properties between *i* and *j*. +If we let i and j be *anticommutative* (meaning that $ij = -ji$) then $stij$ cancels with $stji$. +So that the denominator is totally real (and therefore can be guaranteed to divide), + we can also assert that $i^2$ and $j^2$ are both real. +Then the expression becomes $$ -{1 + si + tj \over 1 -\ si -\ tj} = -{1 + s^2i^2 + t^2j^2 \over 1 -\ s^2 i^2 -\ t^2j^2} + -i{2s \over 1 -\ s^2 i^2 -\ t^2j^2} + -j{2t \over 1 -\ s^2 i^2 -\ t^2j^2} +{1 + si + tj \over 1 - si - tj} + = {1 + s^2i^2 + t^2j^2 \over 1 - s^2 i^2 - t^2 j^2} + + i{2s \over 1 - s^2 i^2 - t^2 j^2} + + j{2t \over 1 - s^2 i^2 - t^2 j^2} $$ -If it is chosen that $i^2 = j^2 = -1$, this produces the correct equations ([Wikipedia](https://en.wikipedia.org/wiki/Stereographic_projection#First_formulation)) parametrizing the unit sphere. One can also check that the squares of the real, *i*, and *j* components sum to 1. +If it is chosen that $i^2 = j^2 = -1$, this produces the correct equations parametrizing the unit sphere + (see [Wikipedia](https://en.wikipedia.org/wiki/Stereographic_projection#First_formulation)). + +$$ +{h \over h^{*}} = {1 - s^2 - t^2 \over 1 + s^2 + t^2} + + i{2s \over 1 + s^2 + t^2} + + j{2t \over 1 + s^2 + t^2} +$$ + +One can also check that this makes the squares of the real, *i*, and *j* components sum to 1. If both *i* and *j* anticommute, then their product also anticommutes with both *i* and *j*. $$ -\textcolor{red}{ij}j = -j\textcolor{red}{ij} ~,~ -i\textcolor{red}{ij} = -\textcolor{red}{ij}i +\textcolor{red}{ij}j = -j\textcolor{red}{ij} + ~,~ i\textcolor{red}{ij} = -\textcolor{red}{ij}i $$ -Calling this product *k* and noticing that $k^2 = (ij)(ij) = -ijji = i^2 = -1$, this completely characterizes the [quaternions](https://en.wikipedia.org/wiki/Quaternion). +Calling this product *k* and noticing that $k^2 = (ij)(ij) = -ijji = i^2 = -1$ + completely characterizes the [quaternions](https://en.wikipedia.org/wiki/Quaternion). -Choosing different values for $i^2$ and $j^2$ yield different shapes than a sphere. If you know a little group theory, you might know there are only two nonabelian (noncommutative) groups of order 8: the [quaternion group](https://en.wikipedia.org/wiki/Quaternion_group) and the [dihedral group of degree 4](https://en.wikipedia.org/wiki/Examples_of_groups#dihedral_group_of_order_8). In the latter group, *j* and *k* are both imaginary, but square to 1 (*i* still squares to -1). I'm being a bit careless with the meanings of "1" and "-1" here, but these objects come from the additive structure of the algebra, rather than the multiplicative. -Changing the sign of one (or both) of the imaginary squares in the expression $h/h^*$ above switches the multiplicative structure from quaternions to the dihedral group. In this group, picking $i^2 = -j^2 = -1$ parametrizes a hyperboloid of one sheet, and picking $i^2 = j^2 = 1$ parametrizes a hyperboloid of two sheets. +### Other projections + +Choosing different values for $i^2$ and $j^2$ yield different shapes than a sphere. +If you know a little group theory, you might know there are only two nonabelian + (noncommutative) groups of order 8: + +- the [quaternion group](https://en.wikipedia.org/wiki/Quaternion_group) +- and the [ + dihedral group of degree 4 + ](https://en.wikipedia.org/wiki/Examples_of_groups#dihedral_group_of_order_8). + +In the latter group, *j* and *k* are both imaginary, but square to 1 (*i* still squares to -1)[^4] . + +[^4]: I'm being a bit careless with the meanings of "1" and "-1" here. + Properly, these are the group identity and another group element which commutes with all others. + +Changing the sign of one (or both) of the imaginary squares in the expression $h / h^{*}$ above + switches the multiplicative structure from quaternions to the dihedral group. +In this group, picking $i^2 = -j^2 = -1$ parametrizes a hyperboloid of one sheet, + and picking $i^2 = j^2 = 1$ parametrizes a hyperboloid of two sheets. Quaternions and Rotation ------------------------ -Complex numbers are useful for describing 2D rotations and elegantly describe the stereographic projection of a circle. Consequently, since quaternions elegantly describe the stereographic projection of a sphere, they are useful for 3D rotations. +We've already established that complex numbers are useful for describing 2D rotations. +They also elegantly describe the stereographic projection of a circle. +Consequently, since quaternions elegantly describe the stereographic projection of a sphere, + they are useful for 3D rotations. -Since the imaginary units *i*, *j*, and *k* are anticommutative, a general quaternion does not commute with other quaternions like complex numbers do. This embodies a difficulty with 3D rotations in general: unlike 2D rotations, they do not commute. +Since the imaginary units *i*, *j*, and *k* are anticommutative, a general quaternion does not + commute with other quaternions like complex numbers do. +This embodies a difficulty with 3D rotations in general: unlike 2D rotations, they do not commute. -Only three of the four components in a quaternion are necessary to describe a point in 3D space. The three imaginary axes are symmetric in some way (i.e., they all square to -1), so we use the *ijk* subspace. Quaternions in this subspace are called *vectors*. Another reason for using the *ijk* subspace comes from considering a point $u = ai + bj + ck$ on the unit sphere ($a^2 + b^2 + c^2 = 1$). Then the square of this point is: +Only three of the four components in a quaternion are necessary to describe a point in 2D space. +The *ijk* subspace is ideal since the three imaginary axes are symmetric (i.e., they all square to -1). +Quaternions in this subspace are called *vectors*. +Another reason for using the *ijk* subspace comes from considering a point + $u = ai + bj + ck$ on the unit sphere ($a^2 + b^2 + c^2 = 1$). +Then the square of this point is: $$ \begin{align*} -u^2 &= (ai + bj + ck)(ai + bj + ck) \\ -&= a^2i^2 + abij + acik + abji + b^2j^2 + bcjk + acki + bckj + c^2k^2 \\ -&= (a^2 + b^2 + c^2)(-1) + ab(ij + ji) + ac(ik + ki) + bc(jk + kj) \\ -&= -1 + 0 + 0 + 0 + u^2 &= (ai + bj + ck)(ai + bj + ck) + \\ + &= a^2i^2 + abij + acik + abji + b^2j^2 + bcjk + acki + bckj + c^2k^2 + \\ + &= (a^2 + b^2 + c^2)(-1) + ab(ij + ji) + ac(ik + ki) + bc(jk + kj) + \\ + &= -1 + 0 + 0 + 0 \end{align*} $$ -This property means that *u* behaves similarly to a typical imaginary unit. If we form pseudo-complex numbers of the form $a + b u$, then ${1 + tu \over 1 -\ tu} = \alpha + \beta u$ specifies *some kind of rotation* in terms of the parameter t. Identically as with complex numbers, the inverse rotation is the conjugate $1 -\ tu \over 1 + tu$. +This property means that *u* behaves similarly to a typical imaginary unit. +If we form pseudo-complex numbers of the form $a + b u$, then ${1 + tu \over 1 - tu} = \alpha + \beta u$ + specifies *some kind* of rotation in terms of the parameter *t*. +As with complex numbers, the inverse rotation is the conjugate $1 - tu \over 1 + tu$. -Another useful feature of quaternion algebra deals with symmetry transformations of the imaginary units. If an imaginary unit (one of *i*, *j*, *k*) is left-multiplied by a quaternion $q$ and right-multiplied by its conjugate $q^*$, then the result is still imaginary. In other words, if p is a vector, then $qpq^*$ is also a vector since *i*, *j*, and *k* form a basis and multiplication distributes over addition. I will not demonstrate this fact, as it requires a large amount of algebra. +Another useful feature of quaternion algebra involves symmetry transformations of the imaginary units. +If an imaginary unit (one of *i*, *j*, *k*) is left-multiplied by a quaternion $q$ + and right-multiplied by its conjugate $q^{*}$, then the result is still imaginary. +In other words, if *p* is a vector, then $qpq^{*}$ is also a vector since + *i*, *j*, and *k* form a basis and multiplication distributes over addition. +I will not demonstrate this fact, as it requires a large amount of algebra. -These two features combine to characterize a transformation for a vector quaternion. For a vector *p*, if $q_u(t)$ is a "rotation" as above for a point *u* on the unit sphere, then another vector (the image of *p*) can be described by +These two features combine to characterize a transformation for a vector quaternion. +For a vector *p*, if $q_u(t)$ is a "rotation" as above for a point *u* on the unit sphere, + then another vector (the image of *p*) can be described by $$ \begin{gather*} -qpq^* = qpq^{-1} -= \left({1 + tu \over 1 -\ tu}\right) p \left({1 -\ tu \over 1 + tu}\right) -= (\alpha + \beta u)p(\alpha -\ \beta u) \\[4pt] -= (\alpha + \beta u)(\alpha p -\ \beta pu) -= \alpha^2 p -\ \alpha \beta pu + \alpha \beta up -\ \beta^2 upu + qpq^* = qpq^{-1} + = \left({1 + tu \over 1 - tu}\right) p \left({1 - tu \over 1 + tu}\right) + = (\alpha + \beta u)p(\alpha - \beta u) + \\[4pt] + = (\alpha + \beta u)(\alpha p - \beta pu) + = \alpha^2 p - \alpha \beta pu + \alpha \beta up - \beta^2 upu \end{gather*} $$ -Testing a Transformation ------------------------- +### Testing a Transformation -If this transformation is a rotation, then it should recreate the 2D case which can be described more simply by complex numbers. For example, if $u = k$ and $p = xi + yj$, then this can be expanded as: +3D space contains 2D subspaces. +If this transformation is a rotation, then a 2D subspace will be affected in a way + which can be described more simply by complex numbers. +For example, if $u = k$ and $p = xi + yj$, then this can be expanded as: $$ \begin{align*} -qpq^{-1} &= (\alpha + \beta k)(xi + yj)(\alpha -\ \beta k) \vphantom{\over} \\ -&= \alpha^2(xi + yj) -\ \alpha \beta (xi + yj)k + \alpha \beta k(xi + yj) -\ \beta^2 k(xi + yj)k -\vphantom{\over} \\ -&= \alpha^2(xi + yj) -\ \alpha \beta(-xj + yi) + \alpha \beta(xj -\ yi) -\ \beta^2(xi + yj) -\vphantom{\over} \\ -&= (\alpha^2 -\ \beta^2)(xi + yj) + 2\alpha \beta(xj - yi) -\vphantom{\over} \\ -&= [(\alpha^2 -\ \beta^2)x -\ 2\alpha \beta y]i + [(\alpha^2 -\ \beta^2)y + 2\alpha \beta x]j + qpq^{-1} &= (\alpha + \beta k)(xi + yj)(\alpha - \beta k) + \\[10pt] + &= \alpha^2(xi + yj) - \alpha \beta (xi + yj)k + \alpha \beta k(xi + yj) - \beta^2 k(xi + yj)k + \vphantom{\over} + \\[10pt] + &= \alpha^2(xi + yj) - \alpha \beta(-xj + yi) + \alpha \beta(xj - yi) - \beta^2(xi + yj) + \\[10pt] + &= (\alpha^2 - \beta^2)(xi + yj) + 2\alpha \beta(xj - yi) + \\[10pt] + &= [(\alpha^2 - \beta^2)x - 2\alpha \beta y]i + [(\alpha^2 - \beta^2)y + 2\alpha \beta x]j \end{align*} $$ -If *p* is rewritten as a vector (in the linear algebra sense), then this final expression can be rewritten as a linear transformation of *p*: +If *p* is rewritten as a vector (in the linear algebra sense), then this final expression + can be rewritten as a linear transformation of *p*: $$ -\begin{pmatrix} -(\alpha^2 -\ \beta^2)x -\ 2 \alpha \beta y \\ -(\alpha^2 -\ \beta^2)y + 2 \alpha \beta x -\end{pmatrix} = -\begin{pmatrix} -\alpha^2 -\ \beta^2 & -2\alpha \beta \\ -2\alpha \beta & \alpha^2 -\ \beta^2 -\end{pmatrix} -\begin{pmatrix} x \\ y \end{pmatrix} +\begin{align*} + \begin{pmatrix} + (\alpha^2 - \beta^2)x - 2 \alpha \beta y \\ + (\alpha^2 - \beta^2)y + 2 \alpha \beta x + \end{pmatrix} + &= \begin{pmatrix} + \alpha^2 - \beta^2 & -2\alpha \beta \\ + 2\alpha \beta & \alpha^2 - \beta^2 + \end{pmatrix} + \begin{pmatrix} + x \\ + y + \end{pmatrix} + \\ + &= \begin{pmatrix} + \alpha & -\beta \\ + \beta & \alpha + \end{pmatrix}^2 + \begin{pmatrix} + x \\ + y + \end{pmatrix} +\end{align*} $$ -The matrix which appears above is the square of the matrix $\begin{pmatrix}\alpha & -\beta \\ \beta & \alpha \end{pmatrix}$. This is isomorphic to the complex number $\alpha + \beta i$. Since $\alpha$ and $\beta$ were chosen so to lie on the (pseudo-) complex unit circle, this means that the vector (*x*, *y*) is rotated by $\alpha + \beta i$ twice. This also demonstrates that u specifies the axis of rotation, or equivalently, the plane of a great circle. This means that the "some kind of rotation" specified by $q = {1 + tu \over 1 -\ tu}$ is a rotation around the great circle normal to *u*. +The complex number $\alpha + \beta i$ is isomorphic to the matrix on the second line[^5]. +Since $\alpha$ and $\beta$ were chosen so to lie on "a" complex unit circle, + this means that the vector (*x*, *y*) is rotated by $\alpha + \beta i$ twice. +This also demonstrates that *u* specifies the axis of rotation, or equivalently, + the plane of a great circle. +This means that the "some kind of rotation" specified by *q* is a rotation around + the great circle normal to *u*. -The double rotation resembles the half-steps of rotation in the complex numbers, where $1 + ti$ performs a dilation, ${1 \over 1 -\ ti}$ reverses it, and together they describe a rotation. Since quaternions do not commute, sandwiching *p* between *q* and its conjugate makes the algebra simpler. The product of a vector with itself should be totally real, as shown above (and equal to the negative of the norm of *p*, the sum of the squares of its components). The rotated *p* shares the same norm as *p*, so it should equal $p^2$. Indeed, this is the case, as +[^5]: Observe this by comparing the entries of the square of the matrix and the square of the complex number. + +The double rotation resembles the half-steps in the complex numbers, + where $1 + ti$ performs a dilation, ${1 \over 1 - ti}$ reverses it, and together they describe a rotation. +The form $qpq^{-1}$ is similar to this -- the $q$ on the left performs some transformation which is (partially) + undone by the $q^{-1}$ on the right. +In this case, since *q*'s conjugate is its inverse and quaternions do not commute, the only thing to do without + the transformation being trivial is to sandwich *p* between them. + +As shown above, the product of a vector with itself should be totally real + and equal to the negative of the norm of *p*, the sum of the squares of its components. +The rotated *p* shares the same norm as *p*, so it should equal $p^2$. +Indeed, this is the case, as $$ (qpq^{-1})^2 = (qpq^{-1})(qpq^{-1}) = q p p q^{-1} = p^2 q q^{-1} = p^2 $$ -As a final remark, due to rotation through quaternions being doubled, the interpolating polynomial used above to smooth out double rotations can also be used here. That is, $q_u(P(t))pq_u^{-1}(P(t))$ rotates *p* by (approximately) the fraction of a turn specified by t through the great circle which intersects the plane normal to *u*. +As a final remark, due to rotation through quaternions being doubled, the interpolating polynomial + used above to smooth out double rotations can also be used here. +That is, $q_u(P(t))pq_u^{-1}(P(t))$ rotates *p* by (approximately) the fraction of a turn specified + by *t* through the great circle which intersects the plane normal to *u*. Closing ------- -It is difficult to find a treatment of rotations which does not hesitate to use the complex exponential $e^{ix} = \cos(x) + i \sin(x)$. The true criterion for rotations is simply that a point lies on the unit circle. Perhaps this contributes to why understanding the 3D situation with quaternions is challenging for many. Past the barrier to entry, I believe them to be rather intuitive. I have outlined some futher benefits of this approach in this post. +It is difficult to find a treatment of rotations which does not hesitate + to use the complex exponential $e^{ix} = \cos(x) + i \sin(x)$. +The true criterion for rotations is simply that a point lies on the unit circle. +Perhaps this contributes to why understanding the 3D situation with quaternions is challenging for many. +Past the barrier to entry, I believe them to be rather intuitive. +I have outlined some futher benefits of this approach in this post. -As a disclaimer, this post was inspired by [this video](https://www.youtube.com/watch?v=d4EgbgTm0Bg) by 3blue1brown on YouTube (at least subconsciously; I had not seen the video in years before checking that I wasn't just plagiarizing it). It *also* uses stereographic projections of the circle and sphere to describe rotations in the complex plane and quaternion vector space. However, I feel like it fails to provide an algebraic motivation for quaternions, or even stereography in the first place. Hopefully, my remarks on the algebraic approach can be used to augment the information in the video. +As a disclaimer, this post was (at least subconsciously) inspired by + [this video](https://www.youtube.com/watch?v=d4EgbgTm0Bg) by 3blue1brown on YouTube + (though I had not seen the video in years before checking that I wasn't just plagiarizing it). +It *also* uses stereographic projections of the circle and sphere to describe rotations + in the complex plane and quaternion vector space. +However, I feel like it fails to provide an algebraic motivation for quaternions, + or even stereography in the first place. +Hopefully, my remarks on the algebraic approach can be used to augment the information in the video. -Diagrams created with GeoGebra. Repository with approximations (as well as GeoGebra files) available [here](https://github.com/queue-miscreant/approx-trig). +Diagrams created with GeoGebra and Matplotlib. +Repository with approximations (as well as GeoGebra files) available [here](https://github.com/queue-miscreant/approx-trig). diff --git a/posts/stereo/1/stereo_circle.png b/posts/stereo/1/stereo_circle.png new file mode 100644 index 0000000000000000000000000000000000000000..fc967b26faa67a2caadd2dcf54214036718856f4 GIT binary patch literal 196771 zcmdqJ2UJsSw=Im_*IogosEDF;=^(`dYEVFW2kBA*(hU%>qadK5v``c&(xn6lkf0(> zN~DI4N(TWYgh;<@2g`fT8UGmf|HpsN9ry0>eJ>Gqc6s*mtTor1bL~f0RWEMea&QY1 z6VrCoB}EM;rVZWj|CP-f;Wvcsrx)SJde?I(t>Fck z?8*HtLf|?RyA5CH^ttnzE;^N{-8mWnP_)aHLxwE@oH;-O-aJUjTY-PuX z+9Nmf>AXSd&72(yi{I<;(wRD!ZKLfhExBC{)%T^g+NGLg*V3_Jo!s!j#<8|lLPR9N zJSUPsj9*zC=8dS88H%(JDI9c@;ka-1y0Njbj&eCV*-q?Wnoc_Wm5?TY+M}YPlB$!Q 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za|E*uly(Kn(g)yMgVyu_fa}VKj(~9Er288eA+Dr>8m^AqK|Y%Au;k_)g#4Y9Azk-x z&r$%k9O^WFK4oh71x?x{cWHz+=_Hu6H4@E08<|M650{T%Z$Q0Q-)VXC6{eEwMjUh} z9nK#-=lqrSE0Zm19Eu=aj<0uU-p2q$kmRK-T2$3Of~e5!RWJbrJi552g?7VD5Dz-} zu~js4zY5Bb!@mH5_JzMa?W_?RH7s2H{%&62vAhkhi?neh-Zfl2JR`qeE-YN zg+>lz1UPE(zghSbAkVC&#sCcE!;3TxNN{ckE5FM*XmyMVM`)<5NzHq@Df%hFh^BIw^RPsSjKK49=WJ=W zLS4ZR%qT1oVo%^Gr_Smh4NkB`8fI`RJR9;y1Gt0MF#1IM&pYnHw3x4hGO^=*b($9a zo&jwKgCGQ!76WiBX%C@kX@UShP*;F#4T2y7-pQS<`Bw{4c0W!SGE2kj1wTdq9*VX= zc5}NS5b1=LX4^qF>i=%Jc-}4@$%oS{@n(E(@%&+YweB^JAV*%2E1JYn!)Mu zPlF896vq$&P~3Ri z{}NQC|Bso^^8lO%TB>@|EBSBn()eu5=~S*6k5Mgp@v`QR{-tPZsK2_nSLz@A555Bs z#P>lVIu)|8f0lrJ%20rDIrHy-h1OnYdH~tplVq5(!9P8rA7v0!yMu9&{cVK-T>pY1ms|7X(Yz!9mAcl3gC+`ktJK^xe=SpvUkgGQxn9i;m|Y9SiH znZ!T;@q`CtVl%$o-GB|G4&#iu|W*e;VX}7W#)9`_Fm&;Vb@gxqrl^|D4Bv&VvSo{zD#r*v0=} c@!Hm|%we2cO!dQYI`DVloWa@LGj@0WA4-6c{Qv*} literal 0 HcmV?d00001 From 6f34d0ca0d53a4c521cec77fc29dbc2a771410b7 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 1 Jul 2025 04:44:54 -0500 Subject: [PATCH 03/14] revisions and images to stereo.2 (videos pending) --- posts/stereo/1/index.qmd | 12 +- posts/stereo/2/index.qmd | 635 +++++++++++++++++++++++++++------------ 2 files changed, 457 insertions(+), 190 deletions(-) diff --git a/posts/stereo/1/index.qmd b/posts/stereo/1/index.qmd index de9a4ea..bdeaecc 100644 --- a/posts/stereo/1/index.qmd +++ b/posts/stereo/1/index.qmd @@ -21,6 +21,14 @@ categories: } +```{python} +#| echo: false + +import sympy +from sympy.abc import t +``` + + Expressing rotation mathematically can be rather challenging, even in two dimensional space. Typically, the subject is approached from complicated-looking rotation matrices, with the occasional accompanying comment about complex numbers. @@ -203,10 +211,6 @@ Graphing the real and imaginary parts separately shows how much they resemble #| echo: false #| layout-ncol: 2 -import matplotlib.pyplot as plt -import sympy -from sympy.abc import t - z = (1 + sympy.I * t) / (1 - sympy.I * t) sympy.plot( diff --git a/posts/stereo/2/index.qmd b/posts/stereo/2/index.qmd index 56f2f13..a7cae19 100644 --- a/posts/stereo/2/index.qmd +++ b/posts/stereo/2/index.qmd @@ -1,44 +1,83 @@ --- +title: "Further Notes on Algebraic Stereography" +description: | + How do you rotate in 2D and 3D without standard trigonometry? format: html: html-math-method: katex +jupyter: python3 +date: "2021-10-10" +date-modified: "2025-06-30" +categories: + - algebra + - complex analysis + - polar roses + - generating function --- + -Further Notes on Algebraic Stereography -======================================= -In my previous post, I discussed the stereographic projection of a circle as it pertains to complex numbers, as well as its applications in 2D and 3D rotation. In an effort to document more interesting facts about this mathematical object (of which scarce information is immediately available online), I will now elaborate on more of its properties. +In my previous post, I discussed the stereographic projection of a circle as it pertains + to complex numbers, as well as its applications in 2D and 3D rotation. +In an effort to document more interesting facts about this mathematical object + (of which scarce information is immediately available online), + I will now elaborate on more of its properties. Chebyshev Polynomials --------------------- -[Previously](), I derived the [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials) with the archetypal complex exponential. These polynomials express the sines and cosines of a multiple of an angle from the sine and cosine of the base angle. +[Previously](/posts/chebyshev/1), I derived the + [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials) + with the archetypal complex exponential. +These polynomials express the sines and cosines of a multiple of an angle from + the sine and cosine of the base angle. +Where $T_n(t)$ are Chebyshev polynomials of the first kind and $U_n(t)$ are those of the second kind, $$ \begin{gather*} -\cos(n\theta) = T_n(\cos(\theta)) \\ -\sin(n\theta) = U_{n-1}(\cos(\theta)) \sin(\theta) + \cos(n \theta) = T_n(\cos(\theta)) + \\ + \sin(n \theta) = U_{n - 1}(\cos(\theta)) \sin(\theta) \end{gather*} $$ -Where $T_n(t)$ are Chebyshev polynomials of the first kind and $U_n(t)$ are those of the second kind. The complex exponential derivation begins by squaring and developing a second-order recurrence. +The complex exponential derivation begins by squaring and developing a second-order recurrence. $$ \begin{align*} -(e^{i\theta})^2 &= (\cos + i\sin)^2 \\ -&= \cos^2 + 2i\cos \cdot \sin -\ \sin^2 + (0 = \cos^2 + \sin^2 -\ 1) \\ -&= 2\cos^2 + 2i\cos \cdot \sin -\ 1 \\ -&= 2\cos \cdot (\cos + i\sin) -\ 1 \\ -&= 2\cos(\theta)e^{i\theta} -\ 1 \\ -(e^{i\theta})^{n+2} &= 2\cos(\theta)(e^{i\theta})^{n+1} -\ (e^{i\theta})^n + (e^{i\theta})^2 &= (\cos + i\sin)^2 + \\ + &= \cos^2 + 2i\cos \cdot \sin - \sin^2 + (0 = \cos^2 + \sin^2 - 1) + \\ + &= 2\cos^2 + 2i\cos \cdot \sin - 1 + \\ + &= 2\cos \cdot (\cos + i\sin) - 1 + \\ + &= 2\cos(\theta)e^{i\theta} - 1 + \\ + (e^{i\theta})^{n+2} &= 2\cos(\theta)(e^{i\theta})^{n+1} - (e^{i\theta})^n \end{align*} $$ -This recurrence relation can then be used to obtain the Chebyshev polynomials, and hence, the expressions using sine and cosine above. Presented this way with such a simple derivation, it appears as though these relationships are inherently trigonometric. However, these polynomials actually have *nothing* to do with sine and cosine. For one, they appear in graph theory, and for two, the importance of the complex exponential is overstated. +This recurrence relation can then be used to obtain the Chebyshev polynomials, and hence, + the expressions using sine and cosine above. +Presented this way with such a simple derivation, it appears as though these relationships + are inherently trigonometric. -$e^{i\theta}$ really just specifies a point on the complex unit circle. This property is used on the second line to coax the equation into a quadratic in $e^{i\theta}$. This is also the *only* property upon which the recurrence depends; all else is algebraic manipulation. +However, these polynomials actually have *nothing* to do with sine and cosine on their own. +For one, [they appear in graph theory](/posts/chebyshev/2), and for two, + the importance of the complex exponential is overstated. +$e^{i\theta}$ really just specifies a point on the complex unit circle. +This property is used on the second line to coax the equation into a quadratic in $e^{i\theta}$. +This is also the *only* property upon which the recurrence depends; all else is algebraic manipulation. ### Back to the Stereograph @@ -46,161 +85,240 @@ $e^{i\theta}$ really just specifies a point on the complex unit circle. This pro Knowing this, let's start over with the stereographic projection of the circle: $$ -o_1(t) = {1 + it \over 1 -\ it} -= {1 -\ t^2 \over 1 + t^2} + i {2t \over 1 + t^2} -= \text{c}_1 + i\text{s}_1 +o_1(t) = {1 + it \over 1 - it} + = {1 - t^2 \over 1 + t^2} + i {2t \over 1 + t^2} + = \text{c}_1 + i\text{s}_1 $$ -The subscript "1" is because as *t* ranges over $(-\infty, \infty)$, a single loop around the circle is made. Thanks to the properties of complex numbers, it is impossible to grow the norm by exponentiating, and therefore impossible for points to stray from the unit circle. Thus, more loops can be made around the circle by raising this expression to higher powers: +The subscript "1" is because as *t* ranges over $(-\infty, \infty)$, the function loops once + around the unit circle. +Taking this to higher powers keeps points on the circle since all points on the circle + have a norm of 1. +It also makes more loops around the circle, which we can denote by larger subscripts: $$ \begin{align*} -o_n &= (o_1)^n -= \left( {1 + it \over 1 -\ it} \right)^n \\ -\text{c}_n + i\text{s}_n -&= (\text{c}_1 + i\text{s}_1)^n + o_n &= (o_1)^n + = \left( {1 + it \over 1 - it} \right)^n + \\ + \text{c}_n + i\text{s}_n + &= (\text{c}_1 + i\text{s}_1)^n \end{align*} $$ -This mirrors raising the complex exponential to a power, i.e., increasing the number of loops *it* makes in the interval $(-\pi, \pi)$. The final line is analogous to de Moivre's formula, but in a form where everything is a ratio of polynomials in *t*. This also means that the Chebyshev polynomials can be obtained directly from these rational expressions: +This mirrors raising the complex exponential to a power + (which loops over the range $(-\pi, \pi)$ instead). +The final line is analogous to de Moivre's formula, but in a form where everything is + a ratio of polynomials in *t*. +This means that the Chebyshev polynomials can be obtained directly from these rational expressions: $$ \begin{align*} -o_2 = (o_1)^2 &= (\text{c}_1 + i\text{s}_1)^2 \\ -&= \text{c}_1^2 + 2i\text{c}_1\text{s}_1 -\ \text{s}_1^2 + -(0 = \text{c}_1^2 + \text{s}_1^2 -\ 1) \\ -&= 2\text{c}_1^2 + 2i\text{c}_1\text{s}_1 -\ 1 \\ -&= 2\text{c}_1(\text{c}_1 + i\text{s}_1) -\ 1 \\ -&= 2\text{c}_1 o_1 -\ 1 \\ -o_2 \cdot (o_1)^n &= 2\text{c}_1 o_1 \cdot (o_1)^n -\ (o_1)^n \\ -o_{n+2} &= 2\text{c}_1 o_{n+1} -\ o_n \\ + o_2 = (o_1)^2 &= (\text{c}_1 + i\text{s}_1)^2 + \\ + &= \text{c}_1^2 + 2i\text{c}_1\text{s}_1 - \text{s}_1^2 + + (0 = \text{c}_1^2 + \text{s}_1^2 - 1) + \\ + &= 2\text{c}_1^2 + 2i\text{c}_1\text{s}_1 - 1 + \\ + &= 2\text{c}_1(\text{c}_1 + i\text{s}_1) - 1 + \\ + &= 2\text{c}_1 o_1 - 1 + \\ + o_2 \cdot (o_1)^n &= 2\text{c}_1 o_1 \cdot (o_1)^n - (o_1)^n + \\ + o_{n+2} &= 2\text{c}_1 o_{n+1} - o_n \end{align*} $$ -This matches the earlier recurrence relation with the complex exponential and therefore the recurrence relation of the Chebyshev polynomials. This also means that the the rational functions obey the same relationship as sine and cosine +This matches the earlier recurrence relation with the complex exponential and therefore + the recurrence relation of the Chebyshev polynomials. +It also means that the the rational functions obey the same relationship as sine and cosine: $$ \begin{matrix} -\begin{gather} -\text{c}_n = T_n(\text{c}_1) \\ -\text{s}_n = U_{n-1}(\text{c}_1) \text{s}_1 -\end{gather} & -\text{where } -\text{c}_1 = {1 -\ t^2 \over 1 + t^2}, & -\text{s}_1 = {2t \over 1 + t^2} + \begin{gather*} + \text{c}_n = T_n(\text{c}_1) + \\ + \text{s}_n = U_{n-1}(\text{c}_1) \text{s}_1 + \end{gather*} + & \text{where } + \text{c}_1 = {1 - t^2 \over 1 + t^2}, & + \text{s}_1 = {2t \over 1 + t^2} \end{matrix} $$ -Thus it is shown that the Chebyshev polynomials are tied to (coordinates on) circles, rather than explicitly to trigonometry. It is a bit strange that these polynomials are in terms of rational functions, but no stranger than them being in terms of *ir* rational functions like sine and cosine. +Thus, the Chebyshev polynomials are tied to (coordinates on) circles, + rather than explicitly to the trigonometric functions. +It is a bit strange that these polynomials are in terms of rational functions, but no stranger + than them being in terms of *ir*rational functions like sine and cosine. Calculus -------- -Since these functions behave similarly to sine and cosine, one might wonder about the nature of these expressions in the context of calculus. For comparison, the complex exponential (as it is a parallel construction) has a simple derivative: +Since these functions behave similarly to sine and cosine, one might wonder about + the nature of these expressions in the context of calculus. + +For comparison, the complex exponential (as it is a parallel construction) has a simple derivative[^1]. +Since the exponential function is its own derivative, the expression acquires + an imaginary coefficient through the chain rule. + +[^1]: This is forgoing the fact that complex derivatives require more care than their real counterparts. + It matters slightly less in this case since this function is complex-valued, but has a real parameter. $$ \begin{align*} -e^{it} &= \cos(t) + i\sin(t) \\ -{d \over dt} e^{it} &= {d \over dt} \cos(t) + {d \over dt} i\sin(t) \\ -i e^{it} &= -\sin(t) + i\cos(t) \\ -i[\cos(t) + i\sin(t)] &\stackrel{\checkmark}{=} -\sin(t) + i\cos(t) + e^{it} &= \cos(t) + i\sin(t) + \\ + {d \over dt} e^{it} + &= {d \over dt} \cos(t) + {d \over dt} i\sin(t) + \\ + i e^{it} &= -\sin(t) + i\cos(t) + \\ + i[\cos(t) + i\sin(t)] + &\stackrel{\checkmark}{=} -\sin(t) + i\cos(t) \end{align*} $$ -By applying the chain rule, one can easily differentiate the complex exponential. Since the exponential function is its own derivative, the expression acquires an imaginary coefficient through the chain rule. Meanwhile, the complex stereograph has derivative +Meanwhile, the complex stereograph has derivative $$ \begin{align*} -{d \over dt} o_1(t) &= {d \over dt} {1 + it \over 1 -\ it} -= {i(1 -\ it) + i(1 + it) \over (1 -\ it)^2} \\ -&= {2i \over (1 -\ it)^2} -= {2i(1 + it)^2 \over (1 + t^2)^2} -= {2i(1 -\ t^2 + 2it) \over (1 + t^2)^2} \\ -&= {-4t \over (1 + t^2)^2} + i {2(1 -\ t^2) \over (1 + t^2)^2} \\ -&= {-2 \over 1 + t^2}s_1 + i {2 \over 1 + t^2}c_1 \\ -&= -(1 + c_1)s_1 + i(1 + c_1)c_1 \\ -&= i(1 + c_1)o_1 + {d \over dt} o_1(t) &= {d \over dt} {1 + it \over 1 - it} + = {i(1 - it) + i(1 + it) \over (1 - it)^2} + \\ + &= {2i \over (1 - it)^2} + = {2i(1 + it)^2 \over (1 + t^2)^2} + = {2i(1 - t^2 + 2it) \over (1 + t^2)^2} + \\ + &= {-4t \over (1 + t^2)^2} + i {2(1 - t^2) \over (1 + t^2)^2} + \\ + &= {-2 \over 1 + t^2}s_1 + i {2 \over 1 + t^2}c_1 + \\ + &= -(1 + c_1)s_1 + i(1 + c_1)c_1 + \\ + &= i(1 + c_1)o_1 \end{align*} $$ -Just like the complex exponential, an imaginary coefficient falls out. However, the expression also accrues a $1 + c_1$ term, almost like an adjustment factor for its failure to be the complex exponential. Sine and cosine obey a simpler relationship with respect to the derivative, and thus need no adjustment. +Just like the complex exponential, an imaginary coefficient falls out. +However, the expression also accrues a $1 + c_1$ term, almost like an adjustment factor + for its failure to be the complex exponential. +Sine and cosine obey a simpler relationship with respect to the derivative, + and thus need no adjustment. -Complex Analysis ----------------- +### Complex Analysis -Since $o_n$ is a curve which loops around the unit circle *n* times, that possibly suits it to showing a simple result from complex analysis. Integrating along a contour which wraps around a sufficiently nice function's pole (i.e., where its magnitude grows without bound) yields a familiar value. This is easiest to see with $f(z) = 1 / z$: +Since $o_n$ is a curve which loops around the unit circle *n* times, that possibly suits it + to showing a simple result from complex analysis. +Integrating along a contour which wraps around a sufficiently nice function's pole + (i.e., where its magnitude grows without bound) yields a familiar value. +This is easiest to see with $f(z) = 1 / z$: $$ -\oint_\Gamma {1 \over z} dz = -\int_a^b {\gamma'(t) \over \gamma(t)} dt -= 2\pi i +\oint_\Gamma {1 \over z} dz + = \int_a^b {\gamma'(t) \over \gamma(t)} dt + = 2\pi i $$ -In this example, $\Gamma$ loops once counterclockwise around the pole at *z* = 0; more loops will scale this by a factor according to the number of loops. Normally this is demonstrated with the complex exponential, but will $o_1$ work just as well? If $\Gamma$ is the unit circle, the integral is: +In this example, Γ is a counterclockwise curve parametrized by γ which loops once around + the pole at *z* = 0. +More loops will scale this by a factor according to the number of loops. + +Normally this equality is demonstrated with the complex exponential, but will $o_1$ work just as well? +If Γ is the unit circle, the integral is: $$ -\oint_\Gamma {1 \over z} dz = -\int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt -= \int_{-\infty}^\infty i(1 + c_1(t)) dt -= 2i\int_{-\infty}^\infty {1 \over 1 + t^2} dt +\oint_\Gamma {1 \over z} dz + = \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt + = \int_{-\infty}^\infty i(1 + c_1(t)) dt + = 2i\int_{-\infty}^\infty {1 \over 1 + t^2} dt $$ -If one has studied their integral identities, the indefinite version of the final integral will be obvious as $\arctan(t)$, which has horizontal asymptotes of $\pi / 2$ and $-\pi / 2$. Therefore, the value of the integral is indeed $2\pi i$. +If one has studied their integral identities, the indefinite version of the final integral + will be obvious as $\arctan(t)$, which has horizontal asymptotes of $\pi / 2$ and $-\pi / 2$. +Therefore, the value of the integral is indeed $2\pi i$. -If there are *n* loops, then naturally there are *n* of these $2\pi i$s. Fortunately, powers of *o* are more loops around the circle, and the chain and power rules can be used to show: +If there are *n* loops, then naturally there are *n* of these $2\pi i$s. +Since powers of *o* are more loops around the circle, the chain and power rules show: $$ \begin{gather*} -{d \over dt} (o_1)^n = n(o_1)^{n-1} {d \over dt} o_1 \\ \\ -\oint_\Gamma {1 \over z} dz = -\int_{-\infty}^\infty {n o_1(t)^{n-1} o_1'(t) \over o_1(t)^n} dt -= n \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt -= 2 \pi i n + {d \over dt} (o_1)^n = n(o_1)^{n-1} {d \over dt} o_1 + \\[14pt] + \oint_\Gamma {1 \over z} dz + = \int_{-\infty}^\infty {n o_1(t)^{n-1} o_1'(t) \over o_1(t)^n} dt + = n \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt + = 2 \pi i n \end{gather*} $$ -It is certainly possible to perform these contour integrals along straight lines; in fact, integrating along lines from 1 to *i* to -1 to -*i* similarly deals with an indefinite integral involving arctangent. However, the best one can do to construct more loops with lines is to count each line multiple times, which isn't extraordinarily convincing. +It is certainly possible to perform these contour integrals along straight lines; + in fact, integrating along lines from 1 to *i* to -1 to -*i* deals with a + similar integral involving arctangent. +However, the best one can do to construct more loops with lines is to count each line + multiple times, which isn't extraordinarily convincing. -Perhaps the use of $\infty$ in the integral bounds is also unconvincing, but the integral can be shifted back into the realm of plausibility by considering simpler bounds on $o_2$: +Perhaps the use of $\infty$ in the integral bounds is also unconvincing. +The integral can be shifted back into the realm of plausibility by considering simpler bounds on $o_2$: $$ \begin{align*} -\oint_\Gamma {1 \over z} dz &= -\int_{-1}^1 {2 o_1(t) o_1'(t) \over o_1(t)^2} dt \\ -&= 2 \int_{-1}^1 {o_1'(t) \over o_1(t)} dt \\ -&= 2(2i\arctan(1) -\ 2i\arctan(-1)) \\ -&= 2\pi i + \oint_\Gamma {1 \over z} dz + &= \int_{-1}^1 {2 o_1(t) o_1'(t) \over o_1(t)^2} dt + \\ + &= 2 \int_{-1}^1 {o_1'(t) \over o_1(t)} dt + \\ + &= 2(2i\arctan(1) - 2i\arctan(-1)) + \\ + &= 2\pi i \end{align*} $$ -This has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating, one obtains the series form of the arctangent. This series converges for $-1 \le t \le 1$, which happens to match the bounds of integration. The convergence of this series is fairly important, since it is tied to formulas for $\pi$, in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80). +This has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating, + one obtains the series form of the arctangent. +This series converges for $-1 \le t \le 1$, which happens to match the bounds of integration. +The convergence of this series is fairly important, since it is tied to formulas for π, + in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80). -Were one to integrate with the complex exponential, the bounds $(0, 2\pi)$ would be invoked, since at this point a full loop has been made. But think to yourself -- how do you know that $2\pi$ radians is equivalent to 0 radians? How do you know the period of the complex exponential? The result using stereography relies on neither of these prior results and is directly pinned to a formula for $\pi$ instead an apparent detour through the number e. +Were one to integrate with the complex exponential, we would instead use the bounds $(0, 2\pi)$, + since at this point a full loop has been made. +But think to yourself -- how do you know the period of the complex exponential? +How do you know that 2π radians is equivalent to 0 radians? +The result using stereography relies on neither of these prior results and is directly pinned + to a formula for π instead an apparent detour through the number *e*. Polar Curves ------------ -Polar coordinates are useful for expressing for which the distance from the origin is a function of the angle with respect to the positive *x*-axis. They can also be readily converted to parametric forms: +Polar coordinates are useful for expressing for which the distance from the origin is + a function of the angle with respect to the positive *x*-axis. +They can also be readily converted to parametric forms: $$ \begin{gather*} -r(\theta) &\Longleftrightarrow& -\begin{matrix} -x(\theta) = r \cos(\theta) \\ -y(\theta) = r \sin(\theta) -\end{matrix} + r(\theta) &\Longleftrightarrow& + \begin{matrix} + x(\theta) = r \cos(\theta) \\ + y(\theta) = r \sin(\theta) + \end{matrix} \end{gather*} $$ -Polar curves frequently loop in on themselves, and so choosing appropriate bounds for $\theta$ (usually as multiples of $\pi$) is necessary for plotting. Evidently, this is due to the use of sine and cosine in the above parametrization. Fortunately, $s_n$ and $c_n$ (as shown by the calculus above) potentially have much simpler bounds. So what happens when one substitutes the rational functions in place of the trig ones? +Polar curves frequently loop in on themselves, and so it is necessary to choose appropriate bounds + for θ (usually as multiples of π) when plotting. +Evidently, this is due to the use of sine and cosine in the above parametrization. +Fortunately, $s_n$ and $c_n$ (as shown by the calculus above) have much simpler bounds. +So what happens when one substitutes the rational functions in place of the trig ones? ### Polar Roses -[Polar roses](https://en.wikipedia.org/wiki/Rose_(mathematics)) are beautiful shapes which have a simple form when expressed in polar coordinates. +[Polar roses](https://en.wikipedia.org/wiki/Rose_(mathematics)) are beautiful shapes which have + a simple form when expressed in polar coordinates. $$ r(\theta) = \cos \left( {p \over q} \cdot \theta \right) @@ -208,204 +326,349 @@ $$ The ratio $p/q$ in least terms uniquely determines the shape of the curve. -If this weren't this post, you might assume this curve is transcendental since it uses cosine, but you probably know better at this point. The Chebyshev examples above demonstrate the resemblance between $c_n$ and $\cos(n\theta)$. The subscript of $c$ is easiest to work with as an integer, so let $q = 1$. +If you weren't reading this post, you might assume this curve is transcendental since it uses cosine, + but you probably know better at this point. +The Chebyshev examples above demonstrate the resemblance between $c_n$ and $\cos(n\theta)$. +The subscript of $c$ is easiest to work with as an integer, so let $q = 1$. -:::: {layout-ncol = "2"} -::: {} -![]() -::: - -::: {} $$ x(t) = c_p(t) c_1(t) \qquad y(t) = c_p(t) s_1(t) $$ -will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$. Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0). -::: -:::: +will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$. -$q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder whether the other polar curves can be obtained by allowing it to vary as well. And you'd be right. +```{python} +#| echo: false +#| fig-cap: | +#| p/1 polar roses as rational curves +#| Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0). -:::: {layout-ncol = "2"} -::: {} -![]() -::: +# TODO: video + +import sympy +from sympy.abc import t + +o = (1 + sympy.I*t) / (1 - sympy.I*t) +``` + +$q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder + whether the other polar curves can be obtained by allowing it to vary as well. +And you'd be right. -::: {} $$ x(t) = c_p(t) c_q(t) \qquad y(t) = c_p(t) s_q(t) $$ will plot a $p/q$ polar rose as t ranges over $(-\infty, \infty)$. -::: -:::: -Just as with the prior calculus examples, doubling all subscripts of *c* and s will only require t to range over $(-1, 1)$, which removes the ugly bite mark. Perhaps it is also slightly less satisfying, since the fraction $p/q$ directly appears in the typical polar incarnation with cosine. On the other hand, it exposes an important property of these curves: they are all rational. +```{python} +#| echo: false +#| fig-cap: p/q polar roses as rational curves -This approach lends additional precision to a prospective pseudo-polar coordinate system. In the next examples, I will be using the following notation for compactness: +# TODO: video +``` + + +Just as with the prior calculus examples, doubling all subscripts of *c* and *s* will + only require *t* to range over $(-1, 1)$, which removes the ugly bite mark. +Perhaps it is also slightly less satisfying, since the fraction $p/q$ directly appears in the + typical polar incarnation with cosine. +On the other hand, it exposes an important property of these curves: they are all rational. + +This approach lends additional precision to a prospective pseudo-polar coordinate system. +In the next few examples, I will be using the following notation for compactness: $$ -\begin{gather*} -R_n(t) = f(t) &\Longleftrightarrow& -\begin{matrix} -x(t) = f(t) c_n(t) \\ -y(t) = f(t) s_n(t) -\end{matrix} + \begin{gather*} + R_n(t) = f(t) &\Longleftrightarrow& + \begin{matrix} + x(t) = f(t) c_n(t) \\ + y(t) = f(t) s_n(t) + \end{matrix} \end{gather*} $$ ### Conic Sections -The polar equation for a conic section (with a particular unit length occurring somewhere) in terms of its eccentricity $\varepsilon$ is: +The polar equation for a conic section (with a particular unit length occurring somewhere) + in terms of its eccentricity $\varepsilon$ is: $$ -r(\theta) = {1 \over 1 -\ \varepsilon \cos(\theta)} +r(\theta) = {1 \over 1 - \varepsilon \cos(\theta)} $$ Correspondingly, the rational polar form can be expressed as $$ -R_1(t) = {1 \over 1 -\ \varepsilon c_1} +R_1(t) = {1 \over 1 - \varepsilon c_1} $$ -Since polynomial arithmetic is (slightly) easier to work with than trigonometric identities, it is a matter of pencil-and-paper algebra to recover the implicit form from a parametric one. +Since polynomial arithmetic is easier to work with than trigonometric identities, + it is a matter of pencil-and-paper algebra to recover the implicit form from a parametric one. #### Parabola ($|\varepsilon| = 1$) -The conic section with the simplest implicit equation is the parabola. Since $c_n$ is a simple ratio of polynomials in *t*, it is much simpler to recover the implicit equation. For $\varepsilon = 1$, +The conic section with the simplest implicit equation is the parabola. +Since $c_n$ is a simple ratio of polynomials in *t*, it is much simpler to recover the implicit equation. +For $\varepsilon = 1$, -:::: {layout-ncol = "2"} -::: {} -![]() -::: +:::: {layout-ncol="2"} +```{python} +#| echo: false +#| fig-cap: $x = {c_1 \over 1 - c_1} \quad y = {s_1 \over 1 - c_1}$ + +sympy.plot_parametric( + 1 / (1 - sympy.re(o)) * sympy.re(o), + 1 / (1 - sympy.re(o)) * sympy.im(o), + xlim=(-5,5), + ylim=(-5,5), +) +``` ::: {} $$ \begin{align*} -1 -\ c_1 &= 1 -\ {1 -\ t^2 \over 1 + t^2} = {2 t^2 \over 1 + t_1} \\ -y &= {s_1 \over 1 -\ c_1} = {2t \over 1 + t^2} {1 + t^2 \over 2 t_1} = {1 \over t} \\ -x &= {c_1 \over 1 -\ c_1} = {1 -\ t^2 \over 1 + t^2} \cdot {1 + t^2 \over 2 t^2} = {1 -\ t^2 \over 2t^2} \\ -&= {1 \over 2t^2} -\ {1 \over 2} = {y^2 \over 2} -\ {1 \over 2} + 1 - c_1 &= 1 - {1 - t^2 \over 1 + t^2} + = {2 t^2 \over 1 + t^2} + \\ + y &= {s_1 \over 1 - c_1} + = {2t \over 1 + t^2} {1 + t^2 \over 2 t^2} + = {1 \over t} + \\ + x &= {c_1 \over 1 - c_1} + = {1 - t^2 \over 1 + t^2} \cdot {1 + t^2 \over 2 t^2} + = {1 - t^2 \over 2t^2} + \\ + &= {1 \over 2t^2} - {1 \over 2} + = {y^2 \over 2} - {1 \over 2} \end{align*} $$ ::: :::: -*x* is a quadratic polynomial in *y*, so trivially the figure formed is a parabola. Technically it is missing the point where $y = 0 (t = \infty)$, and this is not a circumstance where using a higher $c_n$ would help. It is however, similar to the situation where we allow $o_1(\infty) = -1$, and an argument can be made to waive away any concerns one might have. +*x* is a quadratic polynomial in *y*, so trivially the figure formed is a parabola. +Technically it is missing the point where $y = 0 ~ (t = \infty)$, and this is not a circumstance + where using a higher $c_n$ would help. +It is however, similar to the situation where we allow $o_1(\infty) = -1$, and an argument + can be made to waive away any concerns one might have. #### Ellipse ($|\varepsilon| < 1$) -Ellipses are next. The simplest fraction between zero and one is 1/2, so for $\varepsilon = 1/2$, +Ellipses are next. +The simplest fraction between zero and one is 1/2, so for $\varepsilon = 1/2$, :::: {layout-ncol = "2"} -::: {} -![]() -::: +```{python} +#| echo: false +#| fig-cap: $x = {c_1 \over 1 - c_1 / 2} \quad y = {s_1 \over 1 - c_1 / 2}$ + +sympy.plot_parametric( + 1 / (1 - (1/2)*sympy.re(o)) * sympy.re(o), + 1 / (1 - (1/2)*sympy.re(o)) * sympy.im(o), + xlim=(-5,5), + ylim=(-5,5), +) +``` ::: {} $$ \begin{align*} -1 -\ {1 \over 2}c_1 &= 1 -\ {1 \over 2} \cdot {1 -\ t^2 \over 1 + t^2} = {3 t^2 + 1 \over 2 + 2t^2} \\ -y &= {s_1 \over 1 -\ {1 \over 2}c_1} = {4t \over 3t^2 + 1} \\ -x &= {c_1 \over 1 -\ {1 \over 2}c_1} = {2 -\ 2t^2 \over 3t^2 + 1} \\ \\ + 1 - {1 \over 2}c_1 &= 1 - {1 \over 2} \cdot {1 - t^2 \over 1 + t^2} + = {3 t^2 + 1 \over 2 + 2t^2} + \\ + y &= {s_1 \over 1 - {1 \over 2}c_1} + = {4t \over 3t^2 + 1} + \\ + x &= {c_1 \over 1 - {1 \over 2}c_1} + = {2 - 2t^2 \over 3t^2 + 1} \end{align*} $$ ::: :::: -There isn't an obvious way to combine products of *x* and *y* into a single equation. However, we do know that such an equation must have total degree 2. Any monomial of degree 2 must has a squared denominator, but must be able to add with the lower-degree terms. Therefore, the numerator of the sum of these degree 2 terms has the (degree 1) denominator as a factor. +There isn't an obvious way to combine products of *x* and *y* into a single equation. +The general form of a conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, so + we know that the implicit equation for the curve almost certainly involves $x^2$ and $y^2$. -Since the coefficient of $t^4$ in $x^2$ is 4, it must be multiplied by a multiple of 3 (the coefficient of $t^2$ in the denominator) for any factorization to occur. +$$ +x^2 = {4 - 8t^2 + 4t^4 \over (3t^2 + 1)^2} \qquad + y^2 = {16t^2 \over (3t^2 + 1)^2} +$$ + +Squaring produces some $t^4$ terms which cannot exist outside of these terms and *xy*. +A linear combination of $x^2$ and $y^2$ never includes any cubic terms in the numerator + which would appear in *xy*, so $B = 0$. +Since all remaining terms are linear in *x* and *y*, their denominator must appear as a factor + in the numerator of $Ax^2 + Cy^2$, whatever *A* and *C* are. + +Since the coefficient of $t^4$ in $x^2$ is 4, *A* must be multiple of 3. +Through trial and error, $A = 3, C = 4$ gives: $$ \begin{align*} -x^2 &= {4 -\ 8t^2 + 4t^4 \over (3t^2 + 1)^2} \qquad -y^2 = {16t^2 \over (3t^2 + 1)^2} \\ \\ -3x^2 + 4y^2 &= -{12 -\ 24t^2 + 12t^4 + 64t^2 \over (3t^2 + 1)^2} \\ -&= {12 -\ 40t^2 + 12t^4 \over (3t^2 + 1)^2} \\ -&= {(4t^2 + 12) (3t^2 + 1) \over (3t^2 + 1)^2} \\ -&= {4t^2 + 12 \over 3t^2 + 1} + 3x^2 + 4y^2 + &= {12 - 24t^2 + 12t^4 + 64t^2 \over (3t^2 + 1)^2} + \\ + &= {12 - 40t^2 + 12t^4 \over (3t^2 + 1)^2} + \\ + &= {(4t^2 + 12) (3t^2 + 1) \over (3t^2 + 1)^2} + \\ + &= {4t^2 + 12 \over 3t^2 + 1} \end{align*} $$ -The linear term *t* occurs nowhere in the result, so this is clearly some combination of *x* and a constant. By the previous line of thought, the constant term must be a multiple of 4, and picking the smallest option finally results in the implicit form: +Since the numerator of *y* has a *t*, this is clearly some combination of *x* and a constant. +By the previous line of thought, the constant term must be a multiple of 4, and picking the smallest + option finally results in the implicit form: $$ \begin{align*} -4 &= {4(3t^2 + 1) \over 3t^2 + 1} = {12t^2 + 4) \over 3t^2 + 1} \\ -{4t^2 + 12 \over 3t^2 + 1} -\ 4 &= {8 -\ 8t^2 \over 3t^2 + 1} = 4x \\ \\ -3x^2 + 4y^2 &= 4x + 4 \\ -3x^2 + 4y^2 -\ 4x -\ 4 &= 0 + 4 &= {4(3t^2 + 1) \over 3t^2 + 1} + = {12t^2 + 4 \over 3t^2 + 1} + \\ + {4t^2 + 12 \over 3t^2 + 1} - 4 + &= {8 - 8t^2 \over 3t^2 + 1} = 4x + \\[14pt] + 3x^2 + 4y^2 &= 4x + 4 + \\ + 3x^2 + 4y^2 - 4x - 4 &= 0 \end{align*} $$ -Notably, the coefficients of x and y are 3 and 4; simultaneously, $o_1(\varepsilon) = o_1(1/2) = {3 \over 5} + i{4 \over 5}$. This binds together three concepts: the simplest case of the Pythagorean theorem, the 3-4-5 right triangle; the coefficients of the implicit form; and the role of eccentricity with respect to stereography. +Notably, the coefficients of *x* and *y* are 3 and 4. +Simultaneously, $o_1(\varepsilon) = o_1(1/2) = {3 \over 5} + i{4 \over 5}$. +This binds together three concepts: the simplest case of the Pythagorean theorem, + the 3-4-5 right triangle; the coefficients of the implicit form; and the role of eccentricity + with respect to stereography. #### Hyperbola ($|\varepsilon| > 1$) -As evidenced by the bound on the eccentricity above, hyperbolae are in some way the inverses of ellipses. Since $o_1(2)$ is a reflection of $o_1(1/2)$, you might think the implicit equation for $\varepsilon = 2$ to be the same, but with a flipped sign or two. Unfortunately, you'd be wrong. +As evidenced by the bound on the eccentricity above, hyperbolae are in some way the inverses of ellipses. +Since $o_1(2)$ is a reflection of $o_1(1/2)$, you might think the implicit equation for + $\varepsilon = 2$ to be the same, but with a flipped sign or two. +Unfortunately, you'd be wrong. -:::: {layout-ncol = "2"} -::: {} -![]() -::: +:::: {layout-ncol="2"} +```{python} +#| echo: false +#| fig-cap: $x = {c_1 \over 1 - 2c_1} \quad y = {s_1 \over 1 - 2c_1}$ + +# Ignore discontinuity at sqrt(1/3) +disc = (1/3)**(1/2) +sympy.plot_parametric( + ( + 1 / (1 - 2*sympy.re(o)) * sympy.re(o), + 1 / (1 - 2*sympy.re(o)) * sympy.im(o), + (t, -disc + 1e-5, disc - 1e-5) + ), + ( + 1 / (1 - 2*sympy.re(o)) * sympy.re(o), + 1 / (1 - 2*sympy.re(o)) * sympy.im(o), + (t, -5, -disc - 1e-5) + ), + ( + 1 / (1 - 2*sympy.re(o)) * sympy.re(o), + 1 / (1 - 2*sympy.re(o)) * sympy.im(o), + (t, disc + 1e-5, 5) + ), + xlim=(-5,5), + ylim=(-5,5), + line_color = "blue", +) +``` ::: {} $$ -\begin{align*} -x &= {c_1 \over 1 -\ 2c_1} = {1 -\ t^2 \over 3t^2 -\ 1} \\ -y &= {s_1 \over 1 -\ 2c_1} = {2t \over 3t^2 -\ 1} \\ \\ -3x^2 -\ y^2 &= {3 -\ 6t^2 + 3t^4 -\ 4t^2 \over 3t^2 -\ 1} \\ -&= {3 -\ 10t^2 + 3t^4 \over 3t^2 -\ 1} \\ -&= {(t^2 -\ 3)(3t^2 -\ 1) \over (3t^2 -\ 1)^2 } \\ -&= {t^2 -\ 3 \over 3t^2 -\ 1 } \\ -&= {3t^2 -\ 1 + t^2 -\ 3 \over 3t^2 -\ 1 } -\ 1 \\ -&= {4t^2 -\ 4 \over 3t^2 -\ 1 } -\ 1 -= -4x -\ 1 -\end{align*} -$$ - -$$ -3x^2 -\ y^2 + 4x + 1 = 0 +\begin{gather*} + \begin{align*} + x &= {c_1 \over 1 - 2c_1} = {1 - t^2 \over 3t^2 - 1} + \\ + y &= {s_1 \over 1 - 2c_1} = {2t \over 3t^2 - 1} + \\[14pt] + 3x^2 - y^2 &= {3 - 6t^2 + 3t^4 - 4t^2 \over 3t^2 - 1} + \\ + &= {(t^2 - 3)(3t^2 - 1) \over (3t^2 - 1)^2 } + \\ + &= {t^2 - 3 \over 3t^2 - 1 } + = ... = -4x - 1 + \end{align*} + \\[14pt] + 3x^2 - y^2 + 4x + 1 = 0 +\end{gather*} $$ ::: :::: -At the very least, the occurrences of 1 in the place of 4 have a simple explanation: 1 = 4 - 3. This reinforces the previous example's link to the Pythagorean triple 3-4-5. +At the very least, the occurrences of 1 in the place of 4 have a simple explanation: 1 = 4 - 3. ### Archimedean Spiral -Arguably the simplest (non-circular) polar curve is $r(\theta) = \theta$. This is also the unit [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral). Since the curve is defined by a constant turning, this is a natural application of the properties of sine and cosine. The closest equivalent in rational polar coordinates is $R_1(t) = t$. But this can be converted to an implicit form: +Arguably the simplest (non-circular) polar curve is $r(\theta) = \theta$, the unit + [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral). +Since the curve is defined by a constant turning, this is a natural application of the properties + of sine and cosine. +The closest equivalent in rational polar coordinates is $R_1(t) = t$. +But this can be converted to an implicit form: $$ \begin{gather*} -x = tc_1 \qquad y = ts_1 \\ \\ -x^2 + y^2 = t^2(c_1^2 + s_1^2) = t^2\\ -y = {2t^2 \over 1 + t^2} = {2(x^2 + y^2) \over 1 + (x^2 + y^2)} \\ \\ -(1 + x^2 + y^2)y = 2(x^2 + y^2) + x = tc_1 \qquad y = ts_1 + \\[14pt] + x^2 + y^2 = t^2(c_1^2 + s_1^2) = t^2 + \\ + y = {2t^2 \over 1 + t^2} = {2(x^2 + y^2) \over 1 + (x^2 + y^2)} + \\[14pt] + (1 + x^2 + y^2)y = 2(x^2 + y^2) \end{gather*} $$ -The curve produced by this equation is a [right strophoid](https://mathworld.wolfram.com/RightStrophoid.html) with a node at (0, 1) and asymptote $y = 2$. This form suggests something interesting about this curve: it approximates the Archimedean spiral (specifically the one with polar equation $r(\theta) = \theta/2$). Indeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate the (unit) spiral for larger *n*, as can be seen in the following video. +The curve produced by this equation is a + [right strophoid](https://mathworld.wolfram.com/RightStrophoid.html) + with a node at (0, 1) and asymptote $y = 2$. +This form suggests something interesting about this curve: it approximates the Archimedean spiral + (specifically the one with polar equation $r(\theta) = \theta/2$). +Indeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate the (unit) spiral + for larger *n*, as can be seen in the following video. -![]() +```{python} +#| echo: false +#| fig-cap: $x = {c_1 \over 1 - 2c_1} \quad y = {s_1 \over 1 - 2c_1}$ -Since R necessarily defines a rational curve, the curves will never be equal, just as any stretching of $c_n$ will never exactly become cosine. +# TODO: video +``` + +Since R necessarily defines a rational curve, the curves will never be equal, + just as any stretching of $c_n$ will never exactly become cosine. Closing ------- -There is nothing wrong with using sine, cosine, the exponential function, or any of the entailing math, especially in a calculus setting. The benefit to using these functions is their constant "velocity" around the circle as their parameter varies continuously. Indeed, velocity, change, and continuity are some of the concepts which calculus deals with in the first place. Further, nearly every modern scientific calculator in the world features buttons for trigonometric functions. +Sine, cosine, and the exponential function, are useful in a calculus setting precisely + because of their constant "velocity" around the circle. +Also, nearly every modern scientific calculator in the world features buttons + for trigonometric functions, so there seems to be no reason *not* to use them. -We can however be misled by their apparent omnipresence. Stereographic projection has been around for *millennia*; just because Newton invented calculus 3.5 centuries ago does not mean every formula needs to be rewritten in its language. For example (and as previously mentioned), the Chebyshev polynomials are tied to the multiplication of two complex numbers whose norm cannot grow. This explanation requires only an understanding of complex number arithmetic, and not of trigonometry and dividing angles. Many other instances of sine and cosine merely rely on a number (or ratio) of loops around a circle, rather than their properties in calculus, and for these instances it will obviously do to "stay rational". +We can however be misled by their apparent omnipresence. +Stereographic projection has been around for *millennia*, and not every formula needs to be rewritten + in its language. +For example (and as previously mentioned), defining the Chebyshev polynomials really only requires + understanding the multiplication of two complex numbers whose norm cannot grow, + not trigonometry and dividing angles. +Many other instances of sine and cosine merely rely on a number (or ratio) of loops around a circle. +When velocity does not factor, it will obviously do to "stay rational". -One of my favorite things to plot as a kid were polar roses, so it was somewhat of a shock to see that they are, in fact, rational curves. On the other hand, their rationality follows immediately from the rationality of the circle (which itself follows from the existence of Pythagorean triples). If I were more experienced with manipulating Chebyshev polynomials or willing to set up a linear system in (way too) many terms, I might have considered attempting to find an implicit form for them as well. However, only the latter approach would involve polynomial arithmetic and therefore be relevant to the contents of this post. +One of my favorite things to plot as a kid were polar roses, so I was somewhat intrigued + to see that they are, in fact, rational curves. +On the other hand, their rationality follows immediately from the rationality of the circle + (which itself follows from the existence of Pythagorean triples). +If I were more experienced with manipulating Chebyshev polynomials or willing to set up a + linear system in (way too) many terms, I might have considered attempting to find + an implicit form for them as well. Diagrams created with Sympy and Matplotlib. From e3f4bb83c8b2e44d5dcd8b1d3484c30a5f6300dc Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 1 Jul 2025 04:55:45 -0500 Subject: [PATCH 04/14] add old stereograph_notes script --- posts/stereo/2/stereograph_notes.py | 93 +++++++++++++++++++++++++++++ 1 file changed, 93 insertions(+) create mode 100644 posts/stereo/2/stereograph_notes.py diff --git a/posts/stereo/2/stereograph_notes.py b/posts/stereo/2/stereograph_notes.py new file mode 100644 index 0000000..13b02b1 --- /dev/null +++ b/posts/stereo/2/stereograph_notes.py @@ -0,0 +1,93 @@ +from sympy import symbols, I, im, re, sympify, plot_parametric, sin, cos +import numpy as np +import matplotlib.pyplot as plt +from matplotlib import animation + +class LazyList: + def __init__(self, generator): + self._list = [] + self._generator = generator + + def __getitem__(self, idx): + final = idx + if isinstance(idx, slice): + final = idx.stop + while final >= len(self._list): + self._list.append(next(self._generator)) + return self._list[idx] + + def __repr__(self): + if len(self._list): + return repr(self._list)[:-1] + ", ...]" + return "[...]" + +t = symbols('t', real=True) + +e = (1 + I*t) / (1 - I*t) + +def generate_es(): + ret = sympify(1) + while True: + yield ret + ret = (ret * e).expand().cancel().simplify() + +es = LazyList(generate_es()) +cs = LazyList(map(lambda x: re(x).simplify(), es)) +ss = LazyList(map(lambda x: im(x).simplify(), es)) + + +def make_animation(fig, frames, **kwargs): + #fig.tight_layout() + #why isn't this how the function is exposed by default + return lambda func: animation.FuncAnimation(fig, func, frames, init_func=lambda: None, **kwargs) + +def bindfig(fig): + def ret(**kwargs): + for i, j in kwargs.items(): + if i == "figsize": + if j is not None: + fig.set_figwidth(j[0]) + fig.set_figheight(j[1]) + continue + fig.__dict__["set_" + i](j) + return fig + return ret + +def anim_curves(stuff=[], interval=200): + #zero[0,0] = 1 + fig = plt.gcf() + #plt.colorbar() + + #temp = plt.figure + #I hate doing it, but there's no other way to get the figure before it's plotted + plt.figure = bindfig(fig) + + @make_animation(fig, len(stuff), interval=interval) + def ret(fr): + fig.clf() + idx1, idx2 = stuff[fr] + + plot_parametric(idx1, idx2, xlim=(-2,2), ylim=(-2,2), backend='matplotlib') + #plt.title(f"$x = c_{{{idx1}}}c_{{{idx2}}}; y = c_{{{idx1}}}s_{{{idx2}}}$") + plt.tight_layout() + + return ret + +plot_roses = lambda roses: anim_curves([(cs[i]*cs[j], cs[i]*ss[j]) for i,j in roses], interval=500) +#plot_roses([(1,1),(1,2),(2,1),(3,1),(3,2),(2,3),(1,3),(1,4),(3,4),(4,3),(4,1)]).save() + +def archimedes_frames(): + writer = animation.writers['ffmpeg'](fps=15, metadata={'artist': 'Me'}) + #fig = plt.gcf() + + for i in range(10): + #for i in range(1): + i += 1 + p = plot_parametric(t*cos(t), t*sin(t), xlim=(-4,4), ylim=(-4,4), label="Archimedean Spiral", show=False) + p.save(f"frame{i}.png") + p.extend( + plot_parametric(2*i*t*cs[i], 2*i*t*ss[i], line_color="black", label=f"$R_{{{i}}}(t) = {2*i}t$", + show=False) + ) + p.legend = True + p.save(f"frame%02d.png" % i) From a15fca65996c6f02a8d020f6c1bf43a59776ffad Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Thu, 3 Jul 2025 03:53:49 -0500 Subject: [PATCH 05/14] refactored notes into markdown, add gitignore --- posts/polycount/sand-1/todo.txt | 2 + posts/stereo/2/.gitignore | 1 + posts/stereo/2/anim.py | 67 +++++++++++++++++ posts/stereo/2/index.qmd | 108 +++++++++++++++++++++++++--- posts/stereo/2/stereograph_notes.py | 93 ------------------------ 5 files changed, 170 insertions(+), 101 deletions(-) create mode 100644 posts/stereo/2/.gitignore create mode 100644 posts/stereo/2/anim.py delete mode 100644 posts/stereo/2/stereograph_notes.py diff --git a/posts/polycount/sand-1/todo.txt b/posts/polycount/sand-1/todo.txt index b9bd8d8..a9b5086 100644 --- a/posts/polycount/sand-1/todo.txt +++ b/posts/polycount/sand-1/todo.txt @@ -3,3 +3,5 @@ mp4s saved do not get copied or linked over to the rendering folder center figures make sure to object-fit: scale + +use SympyAnimationWrapper from stereo/2 diff --git a/posts/stereo/2/.gitignore b/posts/stereo/2/.gitignore new file mode 100644 index 0000000..2641667 --- /dev/null +++ b/posts/stereo/2/.gitignore @@ -0,0 +1 @@ +*.mp4 diff --git a/posts/stereo/2/anim.py b/posts/stereo/2/anim.py new file mode 100644 index 0000000..07b77f7 --- /dev/null +++ b/posts/stereo/2/anim.py @@ -0,0 +1,67 @@ +import logging +import time +from pathlib import Path +from threading import Thread +from typing import Callable + +from matplotlib import animation, pyplot as plt + + +def animate( + fig, frames: list[int] | None, **kwargs +) -> Callable[..., animation.FuncAnimation]: + """Decorator-style wrapper for FuncAnimation.""" + # Why isn't this how the function is exposed by default? + return lambda func: animation.FuncAnimation(fig, func, frames, **kwargs) + +# writer = animation.writers["ffmpeg"](fps=15, metadata={"artist": "Me"}) + +def bindfig(fig): + """ + Create a function which is compatible with the signature of `pyplot.figure`, + but alters the already-existing figure `fig` instead. + """ + def ret(**kwargs): + for i, j in kwargs.items(): + if i == "figsize": + if j is not None: + fig.set_figwidth(j[0]) + fig.set_figheight(j[1]) + continue + fig.__dict__["set_" + i](j) + return fig + return ret + +class SympyAnimationWrapper: + """ + Context manager for binding a figurebinfoinfoinfoinfoinfoinfo + """ + def __init__(self, filename: Path | str): + self._filename = filename if isinstance(filename, Path) else Path(filename) + self._current_figure = plt.gcf() + + self._figure_temp = plt.figure + + def __enter__(self): + plt.figure = bindfig(self._current_figure) + return self.animate + + def __exit__(self, *_): + plt.figure = self._figure_temp + + def animate(self, *args, **kwargs): + def wrapper(func): + ret = animate(self._current_figure, *args, **kwargs)(func) + current_save = ret.save + def save(*args, **kwargs): + if self._filename.exists(): + logging.error("Ignoring saving animation '%s': file already exists", self._filename) + return + thread = Thread(target=lambda: time.sleep(1) or plt.close()) + thread.run() + current_save(self._filename, *args, **kwargs) + ret.save = save + + return ret + + return wrapper diff --git a/posts/stereo/2/index.qmd b/posts/stereo/2/index.qmd index a7cae19..c73201a 100644 --- a/posts/stereo/2/index.qmd +++ b/posts/stereo/2/index.qmd @@ -23,6 +23,32 @@ categories: } +```{python} +#| echo: false + +from dataclasses import dataclass + +import matplotlib.pyplot as plt +import sympy +from sympy.abc import t + +from anim import SympyAnimationWrapper + +# rational circle function +o = (1 + sympy.I*t) / (1 - sympy.I*t) + +# cache some values for plotting later +def generate_os(amt): + ret = sympy.sympify(1) + for _ in range(amt): + yield ret + ret = (ret * o).expand().cancel().simplify() + +os = list(generate_os(10)) +cs = [sympy.re(i).simplify() for i in os] +ss = [sympy.im(i).simplify() for i in os] +``` + In my previous post, I discussed the stereographic projection of a circle as it pertains to complex numbers, as well as its applications in 2D and 3D rotation. @@ -340,15 +366,46 @@ will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$. ```{python} #| echo: false #| fig-cap: | -#| p/1 polar roses as rational curves +#| p/1 polar roses as rational curves. #| Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0). -# TODO: video +@dataclass +class Rose: + x: sympy.Expr + y: sympy.Expr + title: str | None = None -import sympy -from sympy.abc import t + @classmethod + def from_rational(cls, p: int, q: int): + return cls( + cs[p]*cs[q], + cs[p]*ss[q], + title=f"$x = c_{{{p}}}c_{{{q}}}; y = c_{{{p}}}s_{{{q}}}$", + ) -o = (1 + sympy.I*t) / (1 - sympy.I*t) + +def animate_roses(filename: str, arguments: list[Rose], interval=200): + with SympyAnimationWrapper(filename) as animate: + @animate(list(range(len(arguments))), interval=interval) + def ret(fr): + plt.clf() + argument = arguments[fr] + + sympy.plot_parametric( + argument.x, + argument.y, + xlim=(-2,2), + ylim=(-2,2), + title=argument.title, + backend="matplotlib", + ) + + ret.save() # type: ignore + +animate_roses( + "polar_roses_1.mp4", + [ Rose.from_rational(i, 1) for i in range(1, 11) ], +) ``` $q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder @@ -365,7 +422,13 @@ will plot a $p/q$ polar rose as t ranges over $(-\infty, \infty)$. #| echo: false #| fig-cap: p/q polar roses as rational curves -# TODO: video +animate_roses( + "polar_roses_2.mp4", + [ + Rose.from_rational(i,j) + for i,j in [(1,1),(1,2),(2,1),(3,1),(3,2),(2,3),(1,3),(1,4),(3,4),(4,3),(4,1)] + ], +) ``` @@ -637,9 +700,38 @@ Indeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate t ```{python} #| echo: false -#| fig-cap: $x = {c_1 \over 1 - 2c_1} \quad y = {s_1 \over 1 - 2c_1}$ +#| fig-cap: Approximations to the Archimedean spiral -# TODO: video +with SympyAnimationWrapper("approximate_archimedes.mp4") as animate: + @animate(list(range(10)), interval=500) + def ret(fr): + plt.clf() + + p = sympy.plot_parametric( + t*sympy.cos(t), + t*sympy.sin(t), + xlim=(-4,4), + ylim=(-4,4), + label="Archimedean Spiral", + backend="matplotlib", + show=False + ) + i = fr + 1 + p.extend( + sympy.plot_parametric( + 2*i*t*cs[i], + 2*i*t*ss[i], + (t, -5, 5), + line_color="black", + label=f"$R_{{{i}}}(t) = {2*i}t$", + backend="matplotlib", + show=False + ) + ) + p.show() + plt.legend() + + ret.save() # type: ignore ``` Since R necessarily defines a rational curve, the curves will never be equal, diff --git a/posts/stereo/2/stereograph_notes.py b/posts/stereo/2/stereograph_notes.py deleted file mode 100644 index 13b02b1..0000000 --- a/posts/stereo/2/stereograph_notes.py +++ /dev/null @@ -1,93 +0,0 @@ -from sympy import symbols, I, im, re, sympify, plot_parametric, sin, cos -import numpy as np -import matplotlib.pyplot as plt -from matplotlib import animation - -class LazyList: - def __init__(self, generator): - self._list = [] - self._generator = generator - - def __getitem__(self, idx): - final = idx - if isinstance(idx, slice): - final = idx.stop - while final >= len(self._list): - self._list.append(next(self._generator)) - return self._list[idx] - - def __repr__(self): - if len(self._list): - return repr(self._list)[:-1] + ", ...]" - return "[...]" - -t = symbols('t', real=True) - -e = (1 + I*t) / (1 - I*t) - -def generate_es(): - ret = sympify(1) - while True: - yield ret - ret = (ret * e).expand().cancel().simplify() - -es = LazyList(generate_es()) -cs = LazyList(map(lambda x: re(x).simplify(), es)) -ss = LazyList(map(lambda x: im(x).simplify(), es)) - - -def make_animation(fig, frames, **kwargs): - #fig.tight_layout() - #why isn't this how the function is exposed by default - return lambda func: animation.FuncAnimation(fig, func, frames, init_func=lambda: None, **kwargs) - -def bindfig(fig): - def ret(**kwargs): - for i, j in kwargs.items(): - if i == "figsize": - if j is not None: - fig.set_figwidth(j[0]) - fig.set_figheight(j[1]) - continue - fig.__dict__["set_" + i](j) - return fig - return ret - -def anim_curves(stuff=[], interval=200): - #zero[0,0] = 1 - fig = plt.gcf() - #plt.colorbar() - - #temp = plt.figure - #I hate doing it, but there's no other way to get the figure before it's plotted - plt.figure = bindfig(fig) - - @make_animation(fig, len(stuff), interval=interval) - def ret(fr): - fig.clf() - idx1, idx2 = stuff[fr] - - plot_parametric(idx1, idx2, xlim=(-2,2), ylim=(-2,2), backend='matplotlib') - #plt.title(f"$x = c_{{{idx1}}}c_{{{idx2}}}; y = c_{{{idx1}}}s_{{{idx2}}}$") - plt.tight_layout() - - return ret - -plot_roses = lambda roses: anim_curves([(cs[i]*cs[j], cs[i]*ss[j]) for i,j in roses], interval=500) -#plot_roses([(1,1),(1,2),(2,1),(3,1),(3,2),(2,3),(1,3),(1,4),(3,4),(4,3),(4,1)]).save() - -def archimedes_frames(): - writer = animation.writers['ffmpeg'](fps=15, metadata={'artist': 'Me'}) - #fig = plt.gcf() - - for i in range(10): - #for i in range(1): - i += 1 - p = plot_parametric(t*cos(t), t*sin(t), xlim=(-4,4), ylim=(-4,4), label="Archimedean Spiral", show=False) - p.save(f"frame{i}.png") - p.extend( - plot_parametric(2*i*t*cs[i], 2*i*t*ss[i], line_color="black", label=f"$R_{{{i}}}(t) = {2*i}t$", - show=False) - ) - p.legend = True - p.save(f"frame%02d.png" % i) From 3799b5c0772686bc9bef1a8da6ee8da47cabd604 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Mon, 21 Jul 2025 21:34:18 -0500 Subject: [PATCH 06/14] merge branch 'rewritten' into algebraic-sterography (squash) --- .../1/index/execute-results/html.json | 12 + .../1/index/execute-results/html.json | 12 + .../2/index/execute-results/html.json | 12 + _quarto.yml | 2 +- index.qmd | 6 +- logo-favicon.png | Bin 0 -> 2960 bytes logo.png | Bin 408 -> 0 bytes posts/chebyshev/1/index.qmd | 2 +- 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posts/permutations/appendix/h3/icosahedral_embedding.png create mode 100644 posts/permutations/appendix/h3/rhomicosidodecahedral_graph.png create mode 100644 posts/permutations/appendix/h3/truncated_icosidodecahedral_graph.png create mode 100644 posts/permutations/appendix/index.qmd create mode 100644 posts/permutations/appendix/tetrahedron_wikipedia.png create mode 100644 posts/permutations/index.qmd diff --git a/_freeze/posts/chebyshev/1/index/execute-results/html.json b/_freeze/posts/chebyshev/1/index/execute-results/html.json new file mode 100644 index 0000000..3914367 --- /dev/null +++ b/_freeze/posts/chebyshev/1/index/execute-results/html.json @@ -0,0 +1,12 @@ +{ + "hash": "f0d1956695c45b1ca4b803d88c0a4bf5", + "result": { + "engine": "jupyter", + "markdown": "---\ntitle: \"Generating Polynomials, Part 1: Regular Constructibility\"\ndescription: |\n What kinds of regular polygons are constructible with compass and straightedge?\nformat:\n html:\n html-math-method: katex\ndate: \"2021-08-18\"\ndate-modified: \"2025-06-17\"\ncategories:\n - geometry\n - generating functions\n - algebra\n - python\n---\n\n\n\n\n\n[Recently](/posts/misc/platonic-volume), I used coordinate-free geometry to derive\n the volumes of the Platonic solids, a problem which was very accessible to the ancient Greeks.\nOn the other hand, they found certain problems regarding which figures can be constructed via\n compass and straightedge to be very difficult. For example, they struggled with problems\n like [doubling the cube](https://en.wikipedia.org/wiki/Doubling_the_cube)\n or [squaring the circle](https://en.wikipedia.org/wiki/Squaring_the_circle),\n which are known (through circa 19th century mathematics) to be impossible.\nHowever, before even extending planar geometry by a third dimension or\n calculating the areas of circles, a simpler problem becomes apparent.\nNamely, what kinds of regular polygons are constructible?\n\n\nRegular Geometry and a Complex Series\n-------------------------------------\n\nWhen constructing a regular polygon, one wants a ratio between the length of a edge\n and the distance from a vertex to the center of the figure.\n\n![\n Regular triangle, square, and pentagons inscribed in a unit circle.\n Note the right triangle formed by the apothem, half of an edge, and circumradius.\n](./central_angle_figures.png){.wide}\n\nIn a convex polygon, the total central angle is always one full turn, or 2π radians.\nThe central angle of a regular *n*-gon is ${2\\pi \\over n}$ radians,\n and the green angle above (which we'll call *θ*) is half of that.\nThis means that the ratio we're looking for is $\\sin(\\theta) = \\sin(\\pi / n)$.\nWe can multiply by *n* inside the function on both sides to give\n $\\sin(n\\theta) = \\sin(\\pi) = 0$.\nTherefore, constructing a polygon is actually equivalent to solving this equation,\n and we can rephrase the question as how to express $\\sin(n\\theta)$ (and $\\cos(n\\theta)$).\n\n\n### Complex Recursion\n\nThanks to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula)\n and [de Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula),\n the expressions we're looking for can be phrased in terms of the complex exponential.\n\n$$\n\\begin{align*}\n e^{i\\theta}\n &= \\text{cis}(\\theta) = \\cos(\\theta) + i\\sin(\\theta)\n & \\text{ Euler's formula}\n \\\\\n \\text{cis}(n \\theta) = e^{i(n\\theta)}\n &= e^{(i\\theta)n} = {(e^{i\\theta})}^n = \\text{cis}(\\theta)^n\n \\\\\n \\cos(n \\theta) + i\\sin(n \\theta)\n &= (\\cos(\\theta) + i\\sin(\\theta))^n\n & \\text{ de Moivre's formula}\n\\end{align*}\n$$\n\nDe Moivre's formula for $n = 2$ gives\n\n$$\n\\begin{align*}\n \\text{cis}(\\theta)^2\n &= (\\text{c} + i\\text{s})^2\n \\\\\n &= \\text{c}^2 + 2i\\text{cs} - \\text{s}^2 + (0 = \\text{c}^2 + \\text{s}^2 - 1)\n \\\\\n &= 2\\text{c}^2 + 2i\\text{cs} - 1\n \\\\\n &= 2\\text{c}(\\text{c} + i\\text{s}) - 1\n \\\\\n &= 2\\cos(\\theta)\\text{cis}(\\theta) - 1\n\\end{align*}\n$$\n\nThis can easily be massaged into a recurrence relation.\n\n$$\n\\begin{align*}\n \\text{cis}(\\theta)^2\n &= 2\\cos(\\theta)\\text{cis}(\\theta) - 1\n \\\\\n \\text{cis}(\\theta)^{n+2}\n &= 2\\cos(\\theta)\\text{cis}(\\theta)^{n+1} - \\text{cis}(\\theta)^n\n \\\\\n \\text{cis}((n+2)\\theta)\n &= 2\\cos(\\theta)\\text{cis}((n+1)\\theta) - \\text{cis}(n\\theta)\n\\end{align*}\n$$\n\nRecurrence relations like this one are powerful.\nThrough some fairly straightforward summatory manipulations,\n the sequence can be interpreted as the coefficients in a Taylor series,\n giving a [generating function](https://en.wikipedia.org/wiki/Generating_function).\nCall this function *F*. Then,\n\n$$\n\\begin{align*}\n \\sum_{n=0}^\\infty \\text{cis}((n+2)\\theta)x^n\n &= 2\\cos(\\theta) \\sum_{n=0}^\\infty \\text{cis}((n+1)\\theta) x^n\n - \\sum_{n=0}^\\infty \\text{cis}(n\\theta) x^n\n \\\\\n {F(x; \\text{cis}(\\theta)) - 1 - x\\text{cis}(\\theta) \\over x^2}\n &= 2\\cos(\\theta) {F(x; \\text{cis}(\\theta)) - 1 \\over x}\n - F(x; \\text{cis}(\\theta))\n \\\\[10pt]\n F - 1 - x\\text{cis}(\\theta)\n &= 2\\cos(\\theta) x (F - 1)\n - x^2 F\n \\\\\n F - 2\\cos(\\theta) x F + x^2 F\n &= 1 + x(\\text{cis}(\\theta) - 2\\cos(\\theta))\n \\\\[10pt]\n F(x; \\text{cis}(\\theta))\n &= {1 + x(\\text{cis}(\\theta) - 2\\cos(\\theta)) \\over\n 1 - 2\\cos(\\theta)x + x^2}\n\\end{align*}\n$$\n\nSince $\\text{cis}$ is a complex function, we can separate *F* into real and imaginary parts.\nConveniently, these correspond to $\\cos(n\\theta)$ and $\\sin(n\\theta)$, respectively.\n\n$$\n\\begin{align*}\n \\Re[ F(x; \\text{cis}(\\theta)) ]\n &= {1 + x(\\cos(\\theta) - 2\\cos(\\theta)) \\over 1 - 2\\cos(\\theta)x + x^2}\n \\\\\n &= {1 - x\\cos(\\theta) \\over 1 - 2\\cos(\\theta)x + x^2} = A(x; \\cos(\\theta))\n \\\\\n \\Im[ F(x; \\text{cis}(\\theta)) ]\n &= {x \\sin(\\theta) \\over 1 - 2\\cos(\\theta)x + x^2} = B(x; \\cos(\\theta))\\sin(\\theta)\n\\end{align*}\n$$\n\nIn this form, it becomes obvious that the even though the generating function *F* was originally\n parametrized by $\\text{cis}(\\theta)$, *A* and *B* are parametrized only by $\\cos(\\theta)$.\nExtracting the coefficients of *x* yields an expression for $\\cos(n\\theta)$ and $\\sin(n\\theta)$\n in terms of $\\cos(\\theta)$ (and in the latter case, a common factor of $\\sin(\\theta)$).\n\nIf $\\cos(\\theta)$ in *A* and *B* is replaced with the parameter *z*, then all trigonometric functions\n are removed from the equation, and we are left with only polynomials[^1].\nThese polynomials are [*Chebyshev polynomials*](https://en.wikipedia.org/wiki/Chebyshev_polynomial)\n *of the first (A) and second (B) kind*.\nIn actuality, the polynomials of the second kind are typically offset by 1\n (the x in the numerator of *B* is omitted).\nHowever, retaining this term makes indexing consistent between *A* and *B*\n (and will make things clearer later).\n\n[^1]:\n This can actually be observed as early as the recurrence relation.\n\n $$\n \\begin{align*}\n \\text{cis}(\\theta)^{n+2}\n &= 2\\cos(\\theta)\\text{cis}(\\theta)^{n+1} - \\text{cis}(\\theta)^n\n \\\\\n a_{n+2}\n &= 2 z a_{n+1} - a_n\n \\\\\n \\Re[ a_0 ]\n &= 1,~~ \\Im[ a_0 ] = 0\n \\\\\n \\Re[ a_1 ]\n &= z,~~ \\Im[ a_1 ] = 1 \\cdot \\sin(\\theta)\n \\end{align*}\n $$\n\n\nWe were primarily interested in $\\sin(n\\theta)$, so let's tabulate\n the first few polynomials of the second kind (at $z / 2$).\n\n::: {#tbl-chebyshevu .cell .plain tbl-cap='[OEIS A049310](http://oeis.org/A049310)' execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=2}\n*n* $[x^n]B(x; z / 2) = U_{n - 1}(z / 2)$ Factored\n----- --------------------------------------------- -------------------------------------------------------------------------------------------------\n0 $0$ $0$\n1 $1$ $1$\n2 $z$ $z$\n3 $z^{2} - 1$ $\\left(z - 1\\right) \\left(z + 1\\right)$\n4 $z^{3} - 2 z$ $z \\left(z^{2} - 2\\right)$\n5 $z^{4} - 3 z^{2} + 1$ $\\left(z^{2} - z - 1\\right) \\left(z^{2} + z - 1\\right)$\n6 $z^{5} - 4 z^{3} + 3 z$ $z \\left(z - 1\\right) \\left(z + 1\\right) \\left(z^{2} - 3\\right)$\n7 $z^{6} - 5 z^{4} + 6 z^{2} - 1$ $\\left(z^{3} - z^{2} - 2 z + 1\\right) \\left(z^{3} + z^{2} - 2 z - 1\\right)$\n8 $z^{7} - 6 z^{5} + 10 z^{3} - 4 z$ $z \\left(z^{2} - 2\\right) \\left(z^{4} - 4 z^{2} + 2\\right)$\n9 $z^{8} - 7 z^{6} + 15 z^{4} - 10 z^{2} + 1$ $\\left(z - 1\\right) \\left(z + 1\\right) \\left(z^{3} - 3 z - 1\\right) \\left(z^{3} - 3 z + 1\\right)$\n10 $z^{9} - 8 z^{7} + 21 z^{5} - 20 z^{3} + 5 z$ $z \\left(z^{2} - z - 1\\right) \\left(z^{2} + z - 1\\right) \\left(z^{4} - 5 z^{2} + 5\\right)$\n:::\n:::\n\n\nEvaluating the polynomials at $z / 2$ cancels the 2 in the denominator (and recurrence),\n making these expressions much simpler.\nThis evaluation has an interpretation in terms of the previous diagram --\n recall we used *half* the length of a side as a leg of the right triangle.\nFor a unit circumradius, the side length itself is then $2\\sin( {\\pi / n} )$.\nTo compensate for this doubling, the Chebyshev polynomial must be evaluated at half its normal argument.\n\n\n### Back on the Plane\n\nThe constructibility criterion is deeply connected to the Chebyshev polynomials.\nIn compass and straightedge constructions, one only has access to linear forms (lines)\n and quadratic forms (circles).\nThis means that a figure is constructible if and only if the root can be expressed using\n normal arithmetic (which is linear) and square roots (which are quadratic).\n\n\n#### Pentagons\n\nLet's look at a regular pentagon.\nThe relevant polynomial is\n\n$$\n[x^5]B ( x; z / 2 )\n = z^4 - 3z^2 + 1\n = (z^2 - z - 1) (z^2 + z - 1)\n$$\n\nAccording to how we derived this series, when $z = 2\\cos(\\theta)$, the roots of this polynomial\n correspond to when $\\sin(5\\theta) / \\sin(\\theta) = 0$.\nThis relation itself is true when $\\theta = \\pi / 5$, since $\\sin(5 \\pi / 5) = 0$.\n\nOne of the factors must therefore be the minimal polynomial of $2\\cos(\\pi / 5 )$.\nThe former happens to be correct correct, since $2\\cos( \\pi / 5 ) = \\varphi$, the golden ratio.\nNote that the second factor is the first evaluated at -*z*.\n\n\n#### Heptagons\n\nAn example of where constructability fails is for $2\\cos( \\pi / 7 )$.\n\n$$\n\\begin{align*}\n [x^7]B ( x; z / 2 )\n &= z^6 - 5 z^4 + 6 z^2 - 1\n \\\\\n &= ( z^3 - z^2 - 2 z + 1 ) ( z^3 + z^2 - 2 z - 1 )\n\\end{align*}\n$$\n\nWhichever is the minimal polynomial (the former), it is a cubic, and constructing\n a regular heptagon is equivalent to solving it for *z*.\nBut there are no (nondegenerate) cubics that one can produce via compass and straightedge,\n and all constructions necessarily fail.\n\n\n#### Decagons\n\nOne might think the same of $2\\cos(\\pi /10 )$\n\n$$\n\\begin{align*}\n [x^{10}]B ( x; z / 2 )\n &= z^9 - 8 z^7 + 21 z^5 - 20 z^3 + 5 z\n \\\\\n &= z ( z^2 - z - 1 )( z^2 + z - 1 )( z^4 - 5 z^2 + 5 )\n\\end{align*}\n$$\n\nThis expression also contains the polynomials for $2\\cos( \\pi / 5 )$.\nThis is because a regular decagon would contain two disjoint regular pentagons,\n produced by connecting every other vertex.\n\n![\n  \n](./decagon_divisible.png)\n\nThe polynomial which actually corresponds to $2\\cos( \\pi / 10 )$ is the quartic,\n which seems to suggest that it will require a fourth root and somehow decagons are not constructible.\nHowever, it can be solved by completing the square...\n\n$$\n\\begin{align*}\n z^4 - 5z^2 &= -5\n \\\\\n z^4 - 5z^2 + (5/2)^2 &= -5 + (5/2)^2\n \\\\\n ( z^2 - 5/2)^2 &= {25 - 20 \\over 4}\n \\\\\n ( z^2 - 5/2) &= {\\sqrt 5 \\over 2}\n \\\\\n z^2 &= {5 \\over 2} + {\\sqrt 5 \\over 2}\n \\\\\n z &= \\sqrt{ {5 + \\sqrt 5 \\over 2} }\n\\end{align*}\n$$\n\n...and we can breathe a sigh of relief.\n\n\nThe Triangle behind Regular Polygons\n------------------------------------\n\nPreferring *z* to be halved in $B(x; z/2)$ makes something else more evident.\nObserve these four rows of the Chebyshev polynomials\n\n::: {#273cfec0 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=3}\n*n* $[x^n]B(x; z / 2)$ *k* $[z^{k}][x^n]B(x; z / 2)$\n----- ------------------------------- ----- ---------------------------\n4 $z^{3} - 2 z$ 3 1\n5 $z^{4} - 3 z^{2} + 1$ 2 -3\n6 $z^{5} - 4 z^{3} + 3 z$ 1 3\n7 $z^{6} - 5 z^{4} + 6 z^{2} - 1$ 0 -1\n:::\n:::\n\n\nThe last column looks like an alternating row of Pascal's triangle\n (namely, ${n - \\lfloor {k / 2} \\rfloor - 1 \\choose k}(-1)^k$).\nThis resemblance can be made more apparent by listing the coefficients of the polynomials in a table.\n\n::: {#92d420a6 .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=4}\n n $z^9$ $z^8$ $z^7$ $z^6$ $z^5$ $z^4$ $z^3$ $z^2$ $z$ $1$\n--- ------------------------------ ------------------------------------ ------------------------------------- ----------------------------------- ----------------------------------- ----------------------------------- ------------------------------------ ------------------------------------- ------------------------------------- -------------------------------------\n 1 1\n 2 1 0\n 3 1 0 -1\n 4 1 0 -2 0\n 5 1 0 -3 0 1\n 6 1 0 -4 0 3 0\n 7 1 0 -5 0 6 0 -1\n 8 1 0 -6 0 10 0 -4 0\n 9 1 0 -7 0 15 0 -10 0 1\n 10 1 0 -8 0 21 0 -20 0 5 0\n:::\n:::\n\n\nThough they alternate in sign, the rows of Pascal's triangle appear along diagonals,\n which I have marked in rainbow.\nMeanwhile, alternating versions of the naturals (1, 2, 3, 4...),\n the triangular numbers (1, 3, 6, 10...),\n the tetrahedral numbers (1, 4, 10, 20...), etc.\n are present along the columns, albeit spaced out by 0's.\n\nThe relationship of the Chebyshev polynomials to the triangle is easier to see if\n the coefficient extraction of $B(x; z / 2)$ is reversed.\nIn other words, we extract *z* before extracting *x*.\n\n$$\n\\begin{align*}\n B(x; z / 2) &= {x \\over 1 - zx + x^2}\n = {x \\over 1 + x^2 - zx}\n = {x \\over 1 + x^2}\n \\cdot {1 \\over {1 + x^2 \\over 1 + x^2} - z{x \\over 1 + x^2}}\n \\\\[10pt]\n [z^n]B(x; z / 2) &= {x \\over 1 + x^2} [z^n] {1 \\over 1 - z{x \\over 1 + x^2}}\n = {x \\over 1 + x^2} \\left( {x \\over 1 + x^2} \\right)^n\n \\\\\n &= \\left( {x \\over 1 + x^2} \\right)^{n+1}\n = x^{n+1} (1 + x^2)^{-n - 1}\n \\\\\n &= x^{n+1} \\sum_{k=0}^\\infty {-n - 1 \\choose k}(x^2)^k\n \\quad \\text{Binomial theorem}\n\\end{align*}\n$$\n\nWhile the use of the binomial theorem is more than enough to justify\n the appearance of Pascal's triangle (along with explaining the 0's),\n I'll simplify further to explicitly show the alternating signs.\n\n$$\n\\begin{align*}\n {(-n - 1)_k} &= (-n - 1)(-n - 2) \\cdots (-n - k)\n \\\\\n &= (-1)^k (n + k)(n + k - 1) \\cdots (n + 1)\n \\\\\n &= (-1)^k (n + k)_k\n \\\\\n \\implies {-n - 1 \\choose k}\n &= {n + k \\choose k}(-1)^k\n \\\\[10pt]\n [z^n]B(x; z / 2)\n &= x^{n+1} \\sum_{k=0}^\\infty {n + k \\choose k} (-1)^k x^{2k}\n\\end{align*}\n$$\n\nSquinting hard enough, the binomial coefficient is similar to the earlier\n which gave the third row of Pascal's triangle.\nIf k is fixed, then this expression actually generates the antidiagonal entries\n of the coefficient table, which are the columns with uniform sign.\nThe alternation instead occurs between antidiagonals (one is all positive,\n the next is 0's, the next is all negative, etc.).\nThe initial $x^{n+1}$ lags these sequences so that they reproduce the triangle.\n\n\n### Imagined Transmutation\n\nThe generating function of the Chebyshev polynomials resembles other two term recurrences.\nFor example, the Fibonacci numbers have generating function\n\n$$\n\\sum_{n = 0}^\\infty \\text{Fib}_n x^n = {1 \\over 1 - x - x^2}\n$$\n\nThis resemblance can be made explicit with a simple algebraic manipulation.\n\n$$\n\\begin{align*}\n B(ix; -iz / 2)\n &= {1 \\over 1 -\\ (-i z)(ix) + (ix)^2}\n = {1 \\over 1 -\\ (-i^2) z x + (i^2)(x^2)}\n \\\\\n &= {1 \\over 1 -\\ z x -\\ x^2}\n\\end{align*}\n$$\n\nIf $z = 1$, these two generating functions are equal.\nThe same can be said for $z = 2$ with the generating function of the Pell numbers,\n and so on for higher recurrences (corresponding to metallic means) for higher integral *z*.\n\nIn terms of the Chebyshev polynomials, this series manipulation removes the alternation in\n the coefficients of $U_n$, restoring Pascal's triangle to its nonalternating form.\nRelated to the previous point, it is possible to find the Fibonacci numbers (Pell numbers, etc.)\n in Pascal's triangle, which you can read more about\n [here](http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Pascal/Koshy2011.pdf).\n\n\nManipulating the Series\n-----------------------\n\nLook back to the table of $U_{n - 1}(z / 2)$ (@tbl-chebyshevu).\nWhen I brought up $U_{10 - 1}(z / 2)$ and decagons, I pointed out their relationship to pentagons\n as an explanation for why $U_{5 -\\ 1}(z / 2)$ appears as a factor.\nConveniently, $U_{2 -\\ 1}(z / 2) = z$ is also a factor, and 2 is likewise a factor of 10.\n\nThis pattern is present throughout the table; $n = 6$ contains factors for\n $n = 2 \\text{ and } 3$ and the prime numbers have no smaller factors.\nIf this observation is legitimate, call the newest term $f_n(z)$\n and denote $p_n(z) = U_{n -\\ 1}( z / 2 )$.\n\n\n### Factorization Attempts\n\nThe relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,\n can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).\n\n$$\n\\begin{align*}\n p_n(z) &= \\prod_{d|n} f_n(z)\n \\\\\n \\log( p_n(z) )\n &= \\log \\left( \\prod_{d|n} f_d(z) \\right)\n = \\sum_{d|n} \\log( f_d(z) )\n \\\\\n \\log( f_n(z) ) &= \\sum_{d|n} { \\mu \\left({n \\over d} \\right)}\n \\log( p_d(z) )\n \\\\\n f_n(z) &= \\prod_{d|n} p_d(z)^{ \\mu (n / d) }\n \\\\[10pt]\n f_6(z) = g_6(z)\n &= p_6(z)^{\\mu(1)}\n p_3(z)^{\\mu(2)}\n p_2(z)^{\\mu(3)}\n \\\\\n &= {p_6(z) \\over p_3(z) p_2(z)}\n\\end{align*}\n$$\n\nUnfortunately, it's difficult to apply this technique across our whole series.\nMöbius inversion over series typically uses more advanced generating functions such as\n [Dirichlet series](https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series)\n or [Lambert series](https://en.wikipedia.org/wiki/Lambert_series).\nHowever, naively reaching for these fails for two reasons:\n\n- We built our series of polynomials on a recurrence relation, and these series\n are opaque to such manipulations.\n- To do a proper Möbius inversion, we need these kinds of series over the *logarithm*\n of each polynomial (*B* is a series over the polynomials themselves).\n\nIgnoring these (and if you're in the mood for awful-looking math) you may note\n the Lambert equivalence[^2]:\n\n[^2]:\n This equivalence applies to other polynomial series obeying the same factorization rule\n such as the [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial).\n\n$$\n\\begin{align*}\n \\log( p_n(z) )\n &= \\sum_{d|n} \\log( f_d(z) )\n \\\\\n \\sum_{n = 1}^\\infty \\log( p_n ) x^n\n &= \\sum_{n = 1}^\\infty \\sum_{d|n} \\log( f_d ) x^n\n \\\\\n &= \\sum_{k = 1}^\\infty \\sum_{m = 1}^\\infty \\log( f_m ) x^{m k}\n \\\\\n &= \\sum_{m = 1}^\\infty \\log( f_m ) \\sum_{k = 1}^\\infty (x^m)^k\n \\\\\n &= \\sum_{m = 1}^\\infty \\log( f_m ) {x^m \\over 1 - x^m}\n\\end{align*}\n$$\n\nEither way, the number-theoretic properties of this sequence are difficult to ascertain\n without advanced techniques.\nIf research has been done, it is not easily available in the OEIS.\n\n\n### Total Degrees\n\nIt can be also be observed that the new term is symmetric ($f(z) = f(-z)$), and is therefore\n either irreducible or the product of polynomial and its reflection (potentially negated).\nFor example,\n\n$$\np_9(z) = \\left\\{\n\\begin{matrix}\n (z - 1)(z + 1)\n & \\cdot\n & (z^3 - 3z - 1)(z^3 - 3z + 1)\n \\\\\n \\shortparallel && \\shortparallel\n \\\\\n f_3(z)\n & \\cdot\n & f_9(z)\n \\\\\n \\shortparallel && \\shortparallel\n \\\\\n g_3(z) \\cdot g_3(-z)\n & \\cdot\n & g_9(z) \\cdot -g_9(-z)\n\\end{matrix}\n\\right.\n$$\n\nThese factor polynomials $g_n$ are the minimal polynomials of $2\\cos( \\pi / n )$.\n\nMultiplying these minimal polynomials by their reflection can be observed in the Chebyshev polynomials\n for $n = 3, 5, 7, 9$, strongly implying that it occurs on the odd terms.\nAssuming this is true, we have\n\n$$\nf_n(z) = \\begin{cases}\n g_n(z) & \\text{$n$ is even}\n \\\\\n g_n(z)g_n(-z)\n & \\text{$n$ is odd and ${\\deg(f_n) \\over 2}$ is even}\n \\\\\n -g_n(z)g_n(-z)\n & \\text{$n$ is odd and ${\\deg(f_n) \\over 2}$ is odd}\n\\end{cases}\n$$\n\nWithout resorting to any advanced techniques, the degrees of $f_n$ are\n not too difficult to work out.\nThe degree of $p_n(z)$ is $n -\\ 1$, which is also the degree of $f_n(z)$ if *n* is prime.\nIf *n* is composite, then the degree of $f_n(z)$ is $n -\\ 1$ minus the degrees\n of the divisors of $n -\\ 1$.\nThis leaves behind how many numbers less than *n* are coprime to *n*.\nTherefore $\\deg(f_n) = \\phi(n)$, the\n [Euler totient function](https://en.wikipedia.org/wiki/Euler_totient_function) of the index.\n\nThe totient function can be used to examine the parity of *n*.\nIf *n* is odd, it is coprime to 2 and all even numbers.\nThe introduced factor of 2 to 2*n* removes the evens from the totient, but this is compensated by\n the addition of the odd multiples of old numbers coprime to *n* and new primes.\nThis means that $\\phi(2n) = \\phi(n)$ for odd *n* (other than 1).\n\nThe same argument can be used for even *n*: there are as many odd numbers from 0 to *n* as there are\n from *n* to 2*n*, and there are an equal number of numbers coprime to 2*n* in either interval.\nTherefore, $\\phi(2n) = 2\\phi(n)$ for even *n*.\n\nThis collapses all cases of the conditional factorization of $f_n$ into one,\n and the degrees of $g_n$ are\n\n$$\n\\begin{align*}\n \\deg( g_n(z) )\n &= \\begin{cases}\n \\deg( f_n(z) )\n = \\phi(n)\n & n \\text{ is even} & \\implies \\phi(n) = \\phi(2n) / 2\n \\\\\n \\deg( f_n(z) ) / 2\n = \\phi(n) / 2\n & n \\text{ is odd} & \\implies \\phi(n) / 2 = \\phi(2n) / 2\n \\end{cases}\n \\\\\n &= \\varphi(2n) / 2\n\\end{align*}\n$$\n\nThough they were present in the earlier Chebyshev table,\n the $g_n$ themselves are presented again, along with the expression for their degree\n\n::: {#835a5369 .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=5}\nn $\\varphi(2n)/2$ $g_n(z)$ Coefficient list, rising powers\n--- --------------------------------------- ------------------------- ---------------------------------------\n2 1 $z$ [0, 1]\n3 1 $z - 1$ [-1, 1]\n4 2 $z^{2} - 2$ [-2, 0, 1]\n5 2 $z^{2} - z - 1$ [-1, -1, 1]\n6 2 $z^{2} - 3$ [-3, 0, 1]\n7 3 $z^{3} - z^{2} - 2 z + 1$ [1, -2, -1, 1]\n8 4 $z^{4} - 4 z^{2} + 2$ [2, 0, -4, 0, 1]\n9 3 $z^{3} - 3 z - 1$ [-1, -3, 0, 1]\n9 3 $z^{3} - 3 z + 1$ [1, -3, 0, 1]\n10 4 $z^{4} - 5 z^{2} + 5$ [5, 0, -5, 0, 1]\n- [OEIS A055034](http://oeis.org/A055034) - [OEIS A187360](http://oeis.org/A187360)\n:::\n:::\n\n\nClosing\n-------\n\nMy initial jumping off point for writing this article was completely different.\nHowever, in the process of writing, its share of the article shrank and shrank until its\n introduction was only vaguely related to what preceded it.\nBut alas, the introduction via geometric constructions flows better coming off my\n [post about the Platonic solids](/posts/misc/platonic-volume).\nAlso, it reads better if I rely less on \"if you search for this sequence of numbers\"\n and more on how to interpret the definition.\n\nConsider reading [the follow-up](../2) to this post if you're interested in another way\n one can obtain the Chebyshev polynomials.\n\nDiagrams created with GeoGebra.\n\n\n\n", + "supporting": [ + "index_files" + ], + "filters": [], + "includes": {} + } +} \ No newline at end of file diff --git a/_freeze/posts/permutations/1/index/execute-results/html.json b/_freeze/posts/permutations/1/index/execute-results/html.json new file mode 100644 index 0000000..a363e57 --- /dev/null +++ b/_freeze/posts/permutations/1/index/execute-results/html.json @@ -0,0 +1,12 @@ +{ + "hash": "895ef8d2c498a83dc789e762bbc35fd2", + "result": { + "engine": "jupyter", + "markdown": "---\ntitle: \"A Game of Permutations, Part 1\"\ndescription: |\n Some basic, interesting connections between graph theory and permutation groups.\nformat:\n html:\n html-math-method: katex\njupyter: haskell\ndate: \"2022-01-18\"\ndate-modified: \"2025-07-04\"\ncategories:\n - graph theory\n - group theory\n - sorting algorithms\n---\n\n\n\nIn the time since [my last post](../chebyshev/2) discussing graphs,\n I have been spurred on to continue playing with them, with a slight focus on abstract algebra.\nThis post will primarily focus on some fundamental concepts before launching into some constructions which will make the journey down this road more manageable.\nHowever, I will still assume you are already familiar with what both a\n [group](https://mathworld.wolfram.com/Group.html) and a\n [graph](https://mathworld.wolfram.com/Graph.html) are.\n\n\nThe Symmetric Group\n-------------------\n\nSome of the most important finite groups are the symmetric groups of degree *n* ($S_n$).\nThey are the groups formed by permutations of lists containing *n* items.\nThere is a group element for each permutation, so the order of the group is the\n same as the number of permutations, *n!*.\n\n

\n On lists versus sets \n\n The typical definition of the symmetric group uses sets, which are fundamentally unordered.\n This is potentially confusing, since it can appear as though an object after applying\n a permutation is the same object as before.\n\n For example, let *p* be the permutation swapping \"1\" and \"3\".\n Then,\n\n $$\n \\begin{align*}\n p(\\{ 1, 2, 3 \\}) = \\{ p(1), p(2), p(3) \\} = \\{ 3, 2, 1 \\}\n \\\\\n p([ 1, 2, 3 ]) = [ p(1), p(2), p(3) ] = [ 3, 2, 1 ]\n \\end{align*}\n $$\n\n After applying *p*, the set is unchanged, since all of the elements are the same.\n On the other hand, the lists differ in the first element and cannot be equal.\n\n Sets are still useful as a container.\n For example, the elements of a group are unordered.\n To keep vocabulary simple, I will do my best to refer to objects in a group as\n \"group elements\" and the objects in a list as \"items\".\n
\n\nThere are many ways to denote elements of the symmetric group.\nI will take the liberty of explaining some common notations, each of which are useful in different ways.\nMore information about them can be found [elsewhere](https://mathworld.wolfram.com/Permutation.html)\n [online](https://en.wikipedia.org/wiki/Permutation#Notations)\n as well as any adequate group theory text.\n\n\n### Naive List Notation\n\nArguably the simplest way to do things is to denote the permutation by the result\n of applying it to a canonical list.\nTake the element of $S_3$ which can be described as the action\n \"assign the first item to the second index, the second to the third, and the third to the first\".\nOur canonical list in this case is $[1, 2, 3]$, matching the degree of the group.\nThis results in $[3, 1, 2]$, since \"1\" is in now in position two, and similarly for the other items.\n\nUnfortunately, this choice is too result-oriented.\nThis choice makes it difficult to compose group elements in a meaningful way, since all\n of the information about the permutation is in the position of items in the list,\n rather than the items of the list themselves.\nFor example, under the same rule, $[c, b, a]$ is mapped to $[a, c, b]$.\n\n\n### True List Notations (Two- and One-line Notation)\n\nInstead, let's go back to the definition of the element.\nAll we have to do is list out every index on one line, then the destination\n of every index on the next.\nThis is known as *two-line notation*.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n p(1) & p(2) & p(3)\n\\end{pmatrix}\n= \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}\n$$\n\nFor simplicity, the first row is kept as a list whose items match their indices.\n\nThis notation makes it easier to identify the inverse of a group element.\nAll we have to do is sort the columns of the permutation by the second row,\n then swap the two rows.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}^{-1}\n= \\begin{pmatrix}\n 3 & 1 & 2 \\\\\n 1 & 2 & 3\n\\end{pmatrix}^{-1}\n= \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 3 & 1 & 2\n\\end{pmatrix}\n$$\n\nNote that the second row is now the same as the result from the naive notation.\n\nSince the first row will be the same in all cases, we can omit it, which results in\n *one-line notation*.\nThe *n*th item in the list now describes the position in which *n* can be found after\n the permutation.\n\n$$\n\\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n\\end{pmatrix}\n\\equiv [\\![2, 3, 1]\\!]\n$$\n\nDouble brackets are used to distinguish this as a permutation and not an ordinary list.\n\nThese notations make it straightforward to encode symmetric group elements on a computer.\nAfter all, we only have to read the items of a list by the indices in another.\nHere's a compact definition in Haskell:\n\n::: {#2e6870e0 .cell execution_count=2}\n``` {.haskell .cell-code}\n-- convenient wrapper type for below\nnewtype Permutation = P { unP :: [Int] }\n\napply :: Permutation -> [a] -> [a]\napply = flip (\\xs ->\n map ( -- for each item of the permutation, map it to...\n (xs !!) -- the nth item of the first list\n . (+(-1)) -- (indexed starting with 1)\n ) . unP) -- (after undoing the type wrapping)\n-- written in a non-point free form\napply' (P xs) ys = map ( \\n -> ys !! (n-1) ) xs\n\nprint $ P [2,3,1] `apply` [1,2,3]\n```\n\n::: {.cell-output .cell-output-display}\n```\n[2,3,1]\n```\n:::\n:::\n\n\nNote that this means `P [2,3,1]` is actually equivalent to $[\\![2, 3, 1]\\!]^{-1}$,\n since we don't get $[3, 1, 2]$.\n\nWhile these notations are fairly explicit and easy to describe to a computer,\n it's easy to misinterpret an element as its inverse.\nThere is also some redundancy: $[\\![2, 1]\\!]$ and $[\\![2, 1, 3]\\!]$ both describe a group element\n which swaps the first two items of a list.\nOn one hand, the $S_n$ each belongs to is explicit, but on the other,\n every element of a smaller symmetric group also belongs to a larger one.\nThe verbosity of these notations also makes composing group elements difficult[^1].\n\n[^1]: Composition is relatively easy to describe in two-line notation.\n Recall that we reordered columns when finding an inverse.\n We can do so to match rows of two elements, then compose a new element\n by looking at the first and last rows.\n For example, with group inverses:\n $$\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n \\end{pmatrix}\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 2 & 3 & 1\n \\end{pmatrix}^{-1}\n = \\begin{matrix}\n \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n \\cancel{2} & \\cancel{3} & \\cancel{1}\n \\end{pmatrix}\n \\\\\n \\begin{pmatrix}\n \\cancel{2} & \\cancel{3} & \\cancel{1} \\\\\n 1 & 2 & 3\n \\end{pmatrix}\n \\end{matrix}\n = \\begin{pmatrix}\n 1 & 2 & 3 \\\\\n 1 & 2 & 3\n \\end{pmatrix}\n $$\n\n\n### Cycle Notation\n\n*Cycle notation* addresses all of these issues, but gets rid of the transparency with respect to lists.\nLet's try phrasing the element we've been describing differently.\n\n> assign the first item to the second index, the second to the third, and the third to the first\n\nWe start at index 1 and follow it to index 2, and from there follow it to index 3.\nContinuing from index 3, we return to index 1, and from then we'd loop forever.\nThis describes a *cycle*, denoted as $(1 ~ 2 ~ 3)$.\n\nCycle notation is much more delicate than list notation, since the notation is nonunique:\n\n- Naturally, the elements of a cycle may be cycled to produce an equivalent one.\n - $(1 ~ 2 ~ 3) = (3 ~ 1 ~ 2) = (2 ~ 3 ~ 1)$\n- Cycles which have no common elements (i.e., are disjoint) commute,\n since they act on separate parts of the list.\n - $(1 ~ 2 ~ 3)(4 ~ 5) = (4 ~ 5)(1 ~ 2 ~ 3)$\n\n\n#### Cycle Algebra\n\nThe true benefit of cycles is that they are easy to manipulate algebraically.\nFor some reason, [Wikipedia](https://en.wikipedia.org/wiki/Permutation#Cycle_notation)\n does not elaborate on the composition rules for cycles,\n and the text which I read as an introduction to group theory simply listed it as an exercise.\nWhile playing around with them and deriving these rules oneself *is* a good idea,\n I will list the most important here:\n\n- Cycles can be inverted by reversing their order.\n - $(1 ~ 2 ~ 3)^{-1} = (3 ~ 2 ~ 1) = (1 ~ 3 ~ 2)$\n- Cycles may be composed if the last element in the first is the first index on the right.\n Inversely, cycles may also be decomposed by partitioning on an index and duplicating.\n - $(1 ~ 2 ~ 3) = (1 ~ 2)(2 ~ 3)$\n- If an index in a cycle is repeated twice, it may be omitted from the cycle.\n - $(1 ~ 2 ~ 3)(1 ~ 3) = (1 ~ 2 ~ 3)(3 ~ 1) = (1 ~ 2 ~ 3 ~ 1) = (1 ~ 1 ~ 2 ~ 3) = (2 ~ 3)$\n\nGoing back to $(1 ~ 2 ~ 3)$, if we apply this permutation to the list $[1, 2, 3]$:\n\n$$\n(1 ~ 2 ~ 3) \\left( \\vphantom{0^{0^0}} [1, 2, 3] \\right)\n = (1 ~ 2)(2 ~ 3) \\left( \\vphantom{0^{0^0}} [1, 2, 3] \\right)\n = (1 ~ 2) \\left( \\vphantom{0^{0^0}} [1, 3, 2] \\right)\n = [3, 1, 2]\n$$\n\nWhich is exactly what we expected with our naive notation.\n\n\nGenerators, Permutation Groups, and Sorting\n-------------------------------------------\n\nIf we have a group *G*, then we can select a set of elements\n $\\langle g_1, g_2, g_3, ... \\rangle$ as *generators*.\nIf we form all possible products -- not only the pairwise ones $g_1 g_2$,\n but also $g_1 g_2 g_3$ and all powers of any $g_n$ -- then the products form a subgroup of *G*.\nNaturally, such a set is called a *generating set*.\n\nSymmetric groups are of primary interest because of their subgroups, also known as permutation groups.\n[Cayley's theorem](https://en.wikipedia.org/wiki/Cayley%27s_theorem),\n a fundamental result of group theory, states that all finite groups\n are isomorphic to one of these subgroups.\nThis means that we can encode effectively any group using elements of the symmetric group.\n\nFor example, consider the generating set $\\langle (1 ~ 2) \\rangle$, which contains a single element\n that swaps the first two items of a list.\nIts square is the identity, meaning that its inverse is itself.\nThe identity and $(1 ~ 2)$ are the only two elements of the generated group,\n which is isomorphic to $C_2$, the cyclic group of order 2.\nSimilarly, the 3-cycle in the generating set $\\langle (1 ~ 2 ~ 3) \\rangle$ generates $C_3$,\n the cyclic group of order 3.\n\nHowever, the generating set $\\langle (1 ~ 2), (1 ~ 2 ~ 3)\\rangle$, which contains both of these permutations, generates the entirety of $S_3$.\nIn fact, [\n every symmetric group can be generated by two elements\n ](https://groupprops.subwiki.org/wiki/Symmetric_group_on_a_finite_set_is_2-generated):\n a permutation which cycles all elements once, and the permutation which swaps the first two elements.\n\n$$\nS_n = \\langle (1 ~ 2), (1 ~ 2 ~ 3 ~ 4 ~ ... ~ n) \\rangle\n$$\n\n\n### Sorting and 2-Cycles\n\nThe proof linked to above, that every symmetric group can be generated by two elements,\n uses the (somewhat obvious) result that one can produce any permutation of a list by picking two items,\n swapping them, and repeating until the list is in the desired order.\n\nThis is reminiscent of how sorting algorithms are able to sort a list by only comparing and swapping items.\nAs mentioned earlier when finding inverses using in two-line notation, sorting is almost the inverse of permuting.\nHowever, not all 2-cycles are necessary to generate the whole symmetric group.\n\n\n### Bubble Sort\n\nConsider [bubble sort](https://en.wikipedia.org/wiki/Bubble_sort).\nIn this algorithm, we swap two items when the latter is less than the former,\n looping over the list until it is sorted.\nUntil the list is sorted, the algorithm finds all such adjacent inversions.\nIn the worst case, it will swap every pair of adjacent items, some possibly multiple times.\nThis corresponds to the generating set\n $\\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~\\ ~ n) \\rangle$.\n\n![\n Bubble sort ordering a reverse-sorted list\n](./bubble_sort.gif){.narrow}\n\n\n### Selection Sort\n\nAnother method, [selection sort](https://en.wikipedia.org/wiki/Selection_sort),\n searches through the list for the smallest item and swaps it to the beginning.\nIf this is the final item of the list, this results in the permutation $(1 ~ n)$.\nSupposing that process is continued with the second item and we also want it\n to swap with the final item, then the next swap corresponds to $(2 ~ n)$.\nContinuing until the last item, this gives the generating set\n $\\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~\\ ~ n) \\rangle$.\n\n![\n Selection sort ordering a particular list, using only swaps with the final item\n](./selection_sort.gif){.narrow}\n\nThis behavior for selection sort is uncommon, and this animation omits the selection of a swap candidate.\nThe animation below shows a more destructive selection sort, in which the\n candidate least item is placed at the end of the list (position 5).\nOnce the algorithm hits the end of the list, the candidate is swapped to the least unsorted position,\n and the algorithm continues on the rest of the list.\n\n![\n \"Destructive\" selection sort.\n The actual list being sorted consists of the initial four items, with the final item as temporary storage.\n](./destructive_selection_sort.gif){.narrow}\n\n\nSwap Diagrams\n-------------\n\nGiven a set of 2-cycles, it would be nice to know at a glance if the entire group is generated.\nIn cycle notation, a 2-cycle is an unordered pair of natural numbers which swap items of an *n*-list.\nSimilarly, the edges of an undirected graph on *n* vertices (labelled from 1 to *n*)\n may be interpreted as an unordered pair of the vertices it connects.\n\nIf we treat the two objects as the same, then we can convert between graphs and sets of 2-cycles.\nGoing from the latter to the former, we start on an empty graph on n vertices\n (labelled from 1 to *n*).\nThen, we connect two vertices with an edge when the set includes the permutation swapping\n the indices labelled by the vertices.\n\nReturning to the generating sets we identified with sorting algorithms,\n we identify with each a graph family.\n\n- Bubble sort: $\\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~~ n) \\rangle$\n - The path graphs ($P_n$), which are precisely as they sound:\n an unbranching path formed by *n* vertices.\n- \"Selection\" sort: $\\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~~ n) \\rangle$\n - The star graphs ($\\bigstar_n$, as $S_n$ means the symmetric group),\n one vertex connected to all others.\n- Every 2-cycle in $S_n$\n - The complete graphs ($K_n$), which connect every vertex to every other vertex.\n\n![ ](./example_graphs.png)\n\nThis interpretation of these objects doesn't have proper name, but I think the name \"swap diagram\" fits.\nThey allow us to answer at least one question about the generating set from a graph theory perspective.\n\n\n### Connected Graphs\n\nA graph is connected if a path exists between all pairs of vertices.\nThe simplest possible path is simply a single edge, which we already know to be an available 2-cycle.\n\nThe next simplest case is a path consisting of two edges.\nSome cycle algebra shows that we can produce a third cycle which corresponds\n to an edge connecting the two distant vertices.\n\n:::: {layout-ncol=\"2\"}\n![ ](./induced_edge.png)\n\n::: {}\n$$\n\\begin{align*}\n &(m ~ n) (n ~ o) (m ~ n) \\\\\n &= (m ~ n ~ o) (m ~ n) \\\\\n &= (n ~ o ~ m) (m ~ n) \\\\\n &= (n ~ o ~ m ~ n) \\\\\n &= (o ~ m) = (m ~ o)\n\\end{align*}\n$$\nNote that this is just the conjugation of $(n ~ o)$ by $(m ~ n)$\n:::\n::::\n\nIn other words, if we have have two adjacent edges, the new edge corresponds to\n a product of elements from the generating set.\nGraph theory has a name for this operation: when we produce *all* new edges by linking vertices\n that were separated by a distance of 2, the result is called the *square of that graph*.\nIn fact, higher [graph powers](https://en.wikipedia.org/wiki/Graph_power) will reflect connections\n induced by more conjugations of adjacent edges.\n\n![\n As you might be able to guess, this implies that $(1 ~ 4) = (3 ~ 4)(2 ~ 3)(1 ~ 2)(2 ~ 3)(3 ~ 4)$.\n Also, $\\bigstar_n^2 \\cong K_n$.\n](./P4_powers.png)\n\nIf our graph is connected, then repeating this operation will tend toward a complete graph.\nComplete graphs contain every possible edge, and so correspond to all possible 2-cycles,\n which trivially generate the symmetric group.\nConversely, if a graph has *n* vertices, then for it to be connected, it must have at least $n - 1$ edges.\nThus, a generating set of 2-cycles must have at least $n - 1$ items to generate the symmetric group.\n\nPicking a different vertex labelling will correspond to a different generating set.\nFor example, in the image of $P_4$ above, if the edge connecting vertices 1 and 2\n is replaced with an edge connecting 1 and 4, then the resulting graph\n is still $P_4$, even though it describes a different generating set.\nWe can ignore these extra cases entirely -- either way, the graph power argument shows\n that a connected graph corresponds to a set generating the whole symmetric group.\n\n\n### Disconnected Graphs\n\nA disconnected graph is the\n [disjoint union](https://en.wikipedia.org/wiki/Disjoint_union_of_graphs) of connected graphs.\nUnder graph powers, we know that each connected graph tends toward a complete graph,\n meaning a disconnected graph as a whole tends toward a disjoint union of complete graphs,\n or [cluster graph](https://en.wikipedia.org/wiki/Cluster_graph).\nBut what groups do cluster graphs correspond to?\n\nThe simplest case to consider is what happens when the graph is $P_2 \\oplus P_2$.\nIf there is an edge connecting vertices 1 and 2 and an edge connecting vertices 3 and 4,\n it corresponds to the generating set $\\langle (1 ~ 2), (3 ~ 4) \\rangle$.\nThis is a pair of disjoint cycles, and the group they generate is\n\n$$\n\\{e, (1 ~ 2), (3 ~ 4), (1 ~ 2)(3 ~ 4) \\}\n \\cong S_2 \\times S_2\n \\cong C_2 \\times C_2\n$$\n\nOne way to look at this is by considering paths on each component:\n we can either cross an edge on the first component (corresponding to $(1 ~ 2)$),\n the second component (corresponding to $(3 ~ 4)$),\n or both at the same time.\nThis independence means that one group's structure is duplicated over the other's,\n or more succinctly, gives the direct product.\nIn general, if we denote *γ* as the map which \"runs\" the swap diagram and produces the group, then\n\n$$\n\\gamma( A \\oplus B ) = S_{|A|} \\times S_{|B|},\n ~ A, B \\text{ connected}\n$$\n\nwhere $|A|$ is the number of vertices in *A*.\n\n*γ* has the interesting property of mapping a sum-like object onto a product-like object.\nIf we express a disconnected graph *U* as the disjoint union of its connected components $V_i$, then\n\n$$\n\\begin{gather*}\n U = \\bigsqcup_i V_i\n \\\\\n \\gamma( U ) = \\gamma \\left( \\bigsqcup_i V_i \\right) = \\prod_i S_{|V_i|}\n\\end{gather*}\n$$\n\nThis describes *γ* for every simple graph.\nIt also shows that we're rather limited in the kinds of groups which can be expressed by a swap diagram.\n\n\nClosing\n-------\n\nThis concludes the dry introduction to some investigations of mine into symmetric groups.\nWhile I could have omitted the sections about permutation notation and generators,\n I wanted to be thorough and tie it to concepts which were useful to my understanding.\nThe notion of a graph encoding a generating set in particular will be fairly important going forward.\n\nOriginally, this post was half of a single, sprawling, meandering article.\nI hope I've improved the organization by keeping the digression about sorting algorithms\n to this initial article.\nThe [next post](../2) will cover some interesting structures which fill Euclidean space\n and incredibly large graphs.\n\nSorting and graph diagrams made with GeoGebra.\n\n", + "supporting": [ + "index_files" + ], + "filters": [], + "includes": {} + } +} \ No newline at end of file diff --git a/_freeze/posts/permutations/2/index/execute-results/html.json b/_freeze/posts/permutations/2/index/execute-results/html.json new file mode 100644 index 0000000..295151f --- /dev/null +++ b/_freeze/posts/permutations/2/index/execute-results/html.json @@ -0,0 +1,12 @@ +{ + "hash": "cfab1947f023fe38d9c86abc8bb49fc3", + "result": { + "engine": "jupyter", + "markdown": "---\ntitle: \"A Game of Permutations, Part 2\"\ndescription: |\n Notes on an operation which makes some very large graphs.\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2022-01-18\"\ndate-modified: \"2025-07-06\"\ncategories:\n - graph theory\n - group theory\n---\n\n\n\n\n\nThis post assumes you have read (or at least skimmed over parts of) the [first post](../1),\n which talks about graphs and the symmetric group.\nThis post will contain some more \"empirical\" results, since I'm not an expert on graph theory.\nHowever, one hardly needs to be an expert to learn or to make computations, observations, and predictions.\n\nWe left off talking about producing a group from a graph, so we begin now by considering how to do the reverse.\n\n\nCayley Graphs\n-------------\n\nFor a given generating set, we can assign every element in the group it generates to a vertex in a graph.\nStarting with each of the generators, we draw edges from one vertex to another when\n the product of the initial vertex and a generator (in that order) is the product vertex.\nThis process continues until there are no more arrows to draw.\nThe resulting figure is known as a [Cayley graph](https://mathworld.wolfram.com/CayleyGraph.html)[^1].\n\n[^1]: The construction implies a labelling of edges by its generating set and a labelling\n of vertices by the generated elements.\n It is also common to describe an unlabelled graph as \"Cayley\" if it could\n be generated by this procedure.\n\nCayley graphs depend on the generating set used, so they can take a wide variety of shapes.\nHere are a few examples of Cayley graphs made from elements of $S_4$:\n\n![\n Left: $\\{(1 ~ 3 ~ 2 ~ 4), (3 ~ 4), (1 ~ 4 ~ 2 ~ 3)\\}$, cube graph
\n Middle: $\\{(1 ~ 2 ~ 3), (2 ~ 3 ~ 4)\\}$, cuboctahedral graph
\n Right: $\\{(2 ~ 3), (3 ~ 4), (2 ~ 3 ~ 4)\\}$, octahedral graph
\n Generating sets obtained from the previous MathWorld article.\n](./s4_cayley_graphs.png)\n\nOwing to the way in which they are defined, Cayley graphs have a few useful properties as graphs.\nAt every vertex, we have as many outward edges as we do generators in the generating set,\n so the outward (and in fact, inward) degree of each vertex is the same.\nIn other words, it is a regular graph.\nMore than that, it is [vertex-transitive](https://mathworld.wolfram.com/Vertex-TransitiveGraph.html),\n since labelling a single vertex's outward edges will label that of the entire graph.\n\nIn general, the Cayley graph is a directed graph.\nHowever, if for every member of the generating set, we also include its inverse,\n every directed edge will be matched by an edge in the opposite direction,\n and the Cayley graph may be considered undirected.\n\n\nGraphs to Graphs\n----------------\n\nAll 2-cycles are their own inverse, so generating sets which include only them\n produce undirected Cayley graphs.\nSince this kind of generating set can itself be thought of as a graph,\n we may consider an operation on graphs that maps a swap diagram to its Cayley graph.\n\n![\n An example of this operation, denoted as $\\exp$.\n The proper Cayley graphs for $(1 ~ 2)$ and $(3 ~ 4)$ are not shown;\n they are isomorphic to the corresponding swap graphs, but have different vertex labels.\n](./exp_p2_oplus_p2.png)\n\nIt seems to be the case that $\\exp( A \\oplus B ) = \\exp( A ) \\times \\exp( B )$,\n where $\\oplus$ signifies the disjoint uinion and $\\times$ signifies the\n [Cartesian (box) product of graphs](https://en.wikipedia.org/wiki/Cartesian_product_of_graphs)[^2].\nUnlike *γ* from the previous post, both the input and output of this operation are graphs.\nBecause of this and the sum/product relationship, I've taken to calling this operation the\n \"graph exponential\"[^3].\n\n[^2]: Graphs have many product structures, such as the tensor product and strong product.\n The Cartesian product is (categorically) more natural when paired with disjoint unions.\n\n[^3]: Originally, I called this operation the \"graph factorial\", since it involves permutations\n and the number of vertices in the resulting graph grows factorially.\n\nThis operation is my own invention, so I am unsure whether or not\n it constitutes anything useful.\nIn fact, the possible graphs grow so rapidly that computing anything about the exponential\n of order 8 graphs starts to overwhelm a single computer.\nIt is, however, interesting, as I will hopefully be able to convince.\n\nA random graph will not generally correspond to an interesting generating set,\n and therefore, will also generally have an uninteresting exponential graph.\nHence, I will continue with the examples used previously: paths, stars, and complete graphs.\nThey are among the simplest graphs one can consider, and as we will see shortly,\n have exponentials which appear to have natural correspondences to other graph families.\n\n\nSome Small Exponential Graphs\n-----------------------------\n\nBecause of [the difficulty in determining graph isomorphism](\n https://en.wikipedia.org/wiki/Graph_isomorphism_problem\n ), it is challenging for a computer to find a graph in an encyclopedia.\nComputers think of graphs as a list of vertices and their outward edges,\n but this implementation faces inherent labelling issues.\nThese persist even if the graph is described as a list of (un)ordered pairs,\n an adjacency matrix, or an incidence matrix,\n the latter two of which have very large memory footprints[^4].\n\n[^4]: I was able to locate a project named the\n [Encyclopedia of Finite Graphs](https://github.com/thoppe/Encyclopedia-of-Finite-Graphs),\n but it is only able to build a database simple connected graphs which\n can be queried by invariants (and is outdated since it uses Python 2).\n\nHowever, as visual objects, humans can compare graphs fairly easily\n -- the name means \"drawing\" after all.\nExponentials of order 3 and order 4 graphs are neither so small as to be uninteresting\n nor so big as to be unparsable by humans.\n\n\n### Order 3\n\n![ ](./exp_order_3.png){.narrow}\n\nAt this stage, we only really have two graphs to consider, since $P_3 = \\bigstar_3$.\nImmediately, one can see that $\\exp( P_3 ) = \\exp( \\bigstar_3 ) = C_6$,\n the 6-cycle graph (or hexagonal graph).\nIt is also apparent that $\\exp( K_3 )$ is the utility graph, $K_{3,3}$.\n\n![\n Graph exponential of a disconnected graph.
\n Note that the red element commutes with the green and blue ones.\n This produces two types of squares in the exponential graph.\n Meanwhile, the green and blue elements have an order 3 product, and produce the hexagon.\n](./exp_p3_oplus_p2.png)\n\nHere, we can again demonstrate the sum rule of the graph exponential with $\\exp( P_3 \\oplus P_2 )$.\nSimplifying, since we know $\\exp( P_3 ) = C_6$, the result is $C_6 \\times P_2 = \\text{Prism}_6$,\n the hexagonal prism graph.\n\n\n### Order 4 (and beyond)\n\n::: {layout=\"[[1,1],[1]]\"}\n![$\\exp( P_4 )$](./exp_p4_networkx.png)\n\n![$\\exp( \\bigstar_4 )$](./exp_star4_networkx.png)\n\n![](./exp_order_4.png)\n:::\n\nWith some effort, $\\exp( P_4 )$ can be imagined as a projection of a 3D object,\n the [truncated octahedron](https://en.wikipedia.org/wiki/Truncated_octahedron).\nBecause of its correspondence to a 3D solid, this graph is planar.\nBoth the hexagon and this solid belong to a class of polytopes called\n [*permutohedra*](https://en.wikipedia.org/wiki/Permutohedron), which are figures\n that are also formed by permutations of the coordinate (1, 2, 3, ..., *n*) in Euclidean space[^5].\nTechnically, there is a distinction between the Cayley graphs and permutohedra\n since their labellings differ.\nBoth have edges generated by swaps, but in the latter case, the connected vertices are expected to be\n separated by a certain distance.\nMore information about the distinction can be found at this article on [Wikimedia](\n https://commons.wikimedia.org/wiki/Category:Permutohedron_of_order_4_%28raytraced%29#Permutohedron_vs._Cayley_graph\n )[^6].\n\n[^5]: In fact, these figures are able to completely tessellate the $n-1$ dimensional subspace of\n $\\mathbb{R}^n$ where the coordinates sum to the $n-1$th triangular number.\n Note also that the previous graph in the sequence of $\\exp(P_n)$, the hexagonal graph,\n is visible in the truncated octahedron.\n This corresponds to the projection $(x,y,z,w) \\mapsto (x,y,z)$ over\n the coordinates of the permutohedra.\n\n[^6]: Actually, if one considers a *right* Cayley graph, where each generator is right-multiplied\n to the permutation at a node rather than left-multiplied, then a true correspondence is obtained,\n at least for order 4.\n\nMeanwhile, $\\exp( \\bigstar_4 )$ is more difficult to identify, at least without rearranging its vertices.\nIt turns out to be isomorphic to the [Nauru graph](https://mathworld.wolfram.com/NauruGraph.html),\n a graph with many strange properties.\nNotably, whereas the graph isomorphic to the permutohedron is obviously a spherical polyhedron,\n the Nauru graph can be topologically embedded on a torus.\nThe Nauru graph also belongs to the family of\n [permutation star graphs](https://mathworld.wolfram.com/PermutationStarGraph.html)\n $PS_n$ (*n* = 4), which also includes the hexagonal graph (*n* = 3).\nThe MathWorld article confirms some kind of correspondence, stating graphs of this form\n are generated by pairwise swaps.\n\nMy attempts at finding a graph isomorphic to $\\exp( K_4 )$ have thus far ended in failure.\nIt is certainly *not* isomorphic to $K_{4,4}$, since this graph has 8 vertices,\n as opposed to 24 in $\\exp( K_4 )$.\n\n\nGraph Invariants\n----------------\n\nWhile I have managed to identify the families to which some of these graphs belong,\n I am rather fond of computing (and conjecturing) sequences from objects.\nNot only is it much easier to consult something like [the OEIS](https://oeis.org/) for these quantities,\n but after finding a matching sequence, there are ample articles to consult for more information.\nBy linking to their respective entries, I hope you'll consider reading more there.\n\nEven though I have obtained these values empirically, I am certain that the sequences for\n $\\exp( P_n )$ and $\\exp( \\bigstar_n )$ match the corresponding OEIS entries.\nI also have great confidence in the sequences I found for $\\exp( K_n )$.\n\n\n### Edge Counts\n\nDespite knowing how many vertices there are (*n*!, the order of the symmetric group),\n we don't necessarily know how many edges there are.\n\n::: {#a412d7a0 .cell .plain execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=7}\n*n* $\\#E(\\exp( P_n ))$ $\\#E(\\exp( \\bigstar_n ))$ $\\#E(\\exp( K_n ))$\n-------- ------------------------------------------------------------------------- --------------------------- --------------------------------------\n3 6 6 9\n4 36 36 72\n5 240 240 600\n6 1800 1800 5400\n7 15120 15120 52920\nSequence Second column of Lah numbers
[OEIS A001286](http://oeis.org/A001286) Same as previous [OEIS 001809](http://oeis.org/A001809)\nRule $L(n,2) = n!{(n-1)(n-2) \\over 4}$ $n!{n(n-1) \\over 4}$\n:::\n:::\n\n\n### Radius and Distance Classes\n\nThe radius of a graph is the smallest possible distance which separates two maximally-separated vertices.\nDue to vertex transitivity, the greatest distance between two vertices is the same for every vertex.\n\n::: {#8a1cc334 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=8}\n*n* $r(\\exp( P_n ))$ $r(\\exp( \\bigstar_n ))$ $r(\\exp( K_n ))$\n-------- --------------------------------------------------------------- -------------------------------------------------------------------------------- ------------------\n3 4 4 3\n4 7 5 4\n5 11 7 5\n6 16 8 6\n7 22 10 7\nSequence Triangular numbers
[OEIS A000217](http://oeis.org/A000217) Integers not congruent to 2 (mod 3)
[OEIS A032766](http://oeis.org/A032766) *n* - 1\nRule $\\Delta_{n-1} = {n(n-1) \\over 2}$ $\\lfloor {n-1 \\over 2} \\rfloor + n -\\ 1$\n:::\n:::\n\n\nThese quantities are somewhat reductive.\nIf a vertex is distinguished, the remaining vertices may be partitioned into classes by their\n distance from it.\nIncluding the vertex itself (which is distance 0 away), there will be $r + 1$ such classes,\n where *r* is the radius.\nThese classes are the same for every vertex due to transitivity.\n\nIn the case of these graphs, they are a partition of *n*!.\n\n::: {#b3698cfa .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=9}\n*n* $\\text{dists}(\\exp( P_n ))$ $\\text{dists}(\\exp( \\bigstar_n ))$ $\\text{dists}(\\exp( K_n ))$\n-------- -------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------\n3 [1, 2, 2, 1] [1, 2, 2, 1] [1, 3, 2]\n4 [1, 3, 5, 6, 5, 3, 1] [1, 3, 6, 9, 5] [1, 6, 11, 6]\n5 [1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1] [1, 4, 12, 30, 44, 26, 3] [1, 10, 35, 50, 24]\n6 [1, 5, 14, 29, 49, 71, 90, 101
101, 90, 71, 49, 29, 14, 5, 1] [1, 5, 20, 70, 170, 250, 169, 35] [1, 15, 85, 225, 274, 120]\n7 [1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573
573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1] [1, 6, 30, 135, 460, 1110, 1689, 1254, 340, 15] [1, 21, 175, 735, 1624, 1764, 720]\nSequence Mahonian numbers
[OEIS A008302](http://oeis.org/A008302) Whitney numbers of the second kind (star poset)
[OEIS A007799](http://oeis.org/A007799) Stirling numbers of the first kind
[OEIS A132393](http://oeis.org/A132393)\n:::\n:::\n\n\nI am certain that the appearance of the Stirling numbers here is legitimate,\n since these numbers count the number of permutations of *n* objects with *k* disjoint cycles.\nObviously, the identity element is distance 1 from all 2-cycles since they are all in the generating set;\n likewise, all 3-cycles are distance 2 from the identity (but distance 1 from the 2-cycles),\n and so on until the entire graph has been mapped.\nThe shapes induced by these classes were used to create the diagrams\n of $\\exp( K_3 )$ and $\\exp( K_4 )$ above.\n\n\n#### Spectrum\n\nThe eigenvalues of the adjacency matrix of a graph can be interesting\n and sometimes help in identifying a graph.\nUnfortunately, eigenvalues are not necessarily integers, and therefore not easily\n found in the OEIS (though they are always real for graphs).\n\n::: {#1ce3c98e .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=10}\n *n* $\\text{Spec}(\\exp( P_n ))$ $\\text{Spec}(\\exp( \\bigstar_n )$ $\\text{Spec}(\\exp( K_n ))$\n----- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n 3 $$\\begin{gather*}(\\pm1)^{2\\phantom{0}} \\\\ (\\pm2)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(\\pm1)^{2\\phantom{0}} \\\\ (\\pm2)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm3)^{1\\phantom{0}}\\end{gather*}$$\n 4 $$\\begin{gather*}(\\pm1)^{3} (\\pm3)^{1} \\\\ \\left(x^{2} - 3\\right)^{2} \\\\ \\left(x^{2} + 2 x - 1\\right)^{3} \\\\ \\left(x^{2} - 2 x - 1\\right)^{3} \\\\ \\left(x^{2} + 2 x - 1\\right)^{3} \\\\ \\left(x^{2} - 2 x - 1\\right)^{3}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm1)^{3\\phantom{0}} \\\\ (\\pm2)^{6\\phantom{0}} \\\\ (\\pm3)^{1\\phantom{0}}\\end{gather*}$$ $$\\begin{gather*}(0)^{4\\phantom{0}} \\\\ (\\pm2)^{9\\phantom{0}} \\\\ (\\pm6)^{1\\phantom{0}}\\end{gather*}$$\n 5 $$\\begin{gather*}(0)^{12} (\\pm1)^{6} (\\pm4)^{1} \\\\ \\left(x^{2} - 5\\right)^{6} \\\\ \\left(x^{2} - 5 x + 5\\right)^{4} \\\\ \\left(x^{2} + 5 x + 5\\right)^{4} \\\\ \\left(x^{2} - 3 x + 1\\right)^{4} \\\\ \\left(x^{2} + 3 x + 1\\right)^{4} \\\\ \\left(x^{2} + 2 x - 1\\right)^{5} \\\\ \\left(x^{2} - 2 x - 1\\right)^{5} \\\\ \\left(x^{3} + 2 x^{2} - 5 x - 4\\right)^{5} \\\\ \\left(x^{3} - 2 x^{2} - 5 x + 4\\right)^{5}\\end{gather*}$$ $$\\begin{gather*}(0)^{30\\phantom{00}} \\\\ (\\pm1)^{4\\phantom{00}} \\\\ (\\pm2)^{28\\phantom{00}} \\\\ (\\pm3)^{12\\phantom{00}} \\\\ (\\pm4)^{1\\phantom{00}}\\end{gather*}$$ $$\\begin{gather*}(0)^{36\\phantom{00}} \\\\ (\\pm2)^{25\\phantom{00}} \\\\ (\\pm5)^{16\\phantom{00}} \\\\ (\\pm10)^{1\\phantom{00}}\\end{gather*}$$\n 6 $$\\begin{gather*}(0)^{20} (\\pm1)^{25} (\\pm2)^{15}\\\\(\\pm3)^{5} (\\pm4)^{5} (\\pm5)^{1} \\\\ \\left(x^{2} - 3\\right)^{20}\\end{gather*}$$Not shown: 558 other roots $$\\begin{gather*}(0)^{168\\phantom{000}} \\\\ (\\pm1)^{30\\phantom{000}} \\\\ (\\pm2)^{120\\phantom{000}} \\\\ (\\pm3)^{105\\phantom{000}} \\\\ (\\pm4)^{20\\phantom{000}} \\\\ (\\pm5)^{1\\phantom{000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{256\\phantom{000}} \\\\ (\\pm3)^{125\\phantom{000}} \\\\ (\\pm5)^{81\\phantom{000}} \\\\ (\\pm9)^{25\\phantom{000}} \\\\ (\\pm15)^{1\\phantom{000}}\\end{gather*}$$\n 7 $$\\begin{gather*}(0)^{35\\phantom{00}} \\\\ (\\pm1)^{20\\phantom{00}} \\\\ (\\pm2)^{45\\phantom{00}} \\\\ (\\pm6)^{1\\phantom{00}}\\end{gather*}$$Not shown: 4873 other roots $$\\begin{gather*}(0)^{840\\phantom{000}} \\\\ (\\pm1)^{468\\phantom{000}} \\\\ (\\pm2)^{495\\phantom{000}} \\\\ (\\pm3)^{830\\phantom{000}} \\\\ (\\pm4)^{276\\phantom{000}} \\\\ (\\pm5)^{30\\phantom{000}} \\\\ (\\pm6)^{1\\phantom{000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{400\\phantom{0000}} \\\\ (\\pm1)^{441\\phantom{0000}} \\\\ (\\pm3)^{1225\\phantom{0000}} \\\\ (\\pm6)^{196\\phantom{0000}} \\\\ (\\pm7)^{225\\phantom{0000}} \\\\ (\\pm9)^{196\\phantom{0000}} \\\\ (\\pm14)^{36\\phantom{0000}} \\\\ (\\pm21)^{1\\phantom{0000}}\\end{gather*}$$\n 8 Not shown: all 40320 roots $$\\begin{gather*}(0)^{3960\\phantom{0000}} \\\\ (\\pm1)^{5691\\phantom{0000}} \\\\ (\\pm2)^{2198\\phantom{0000}} \\\\ (\\pm3)^{6321\\phantom{0000}} \\\\ (\\pm4)^{3332\\phantom{0000}} \\\\ (\\pm5)^{595\\phantom{0000}} \\\\ (\\pm6)^{42\\phantom{0000}} \\\\ (\\pm7)^{1\\phantom{0000}}\\end{gather*}$$ $$\\begin{gather*}(0)^{9864\\phantom{0000}} \\\\ (\\pm2)^{3136\\phantom{0000}} \\\\ (\\pm4)^{6125\\phantom{0000}} \\\\ (\\pm7)^{4096\\phantom{0000}} \\\\ (\\pm8)^{196\\phantom{0000}} \\\\ (\\pm10)^{784\\phantom{0000}} \\\\ (\\pm12)^{441\\phantom{0000}} \\\\ (\\pm20)^{49\\phantom{0000}} \\\\ (\\pm28)^{1\\phantom{0000}}\\end{gather*}$$\n:::\n:::\n\n\nFrom what I have been able to identify, the spectrum of an exponential graph is symmetric about 0,\n by which I mean that the characteristic polynomial is even.\nThis has been the case for all graphs I have tried testing, even outside these graph families.\n\nSince all eigenvalues of $\\exp( \\bigstar_n )$ calculated are integers,\n it appears they are integral graphs, a fact of which I am reasonably sure\n because of the aforementioned correspondence to permutation star graphs.\nAdditionally, the eigenvalues are very conveniently the integers up to $n-1$ and down to $-n+1$.\nUnfortunately, despite the ease of reading the eigenvalues themselves,\n there isn't an OEIS entry for the multiplicities.\nI was able to identify the multiplicity of the 0 eigenvalue with\n [OEIS A217213](http://oeis.org/A217213), which counts orderings on Dyck paths.\nIf this is truly the sequence being generated, it means there is a 1:1 correspondence between\n these orderings and a basis of the nullspace of the adjacency matrix.\n\nIt seems to be the case that $\\exp( K_n )$ are also integral graphs.\nPerplexingly, the multiplicities for each of the eigenvalues appear to mostly be perfect powers.\nThis is the case until n = 8, which ruins the pattern because neither of\n $9864 = 2^3 \\cdot 3^2 \\cdot 137$ or $6125 = 5^3 \\cdot 7^2$ are perfect powers.\nI find both this[^7] and the fact that such a large prime appears among the factorization of the former\n rather creepy since all other primes which appear here are small -- 2, 3, 5, and 7.\n\n[^7]: Some physicists are fond of 137 for its closeness to the reciprocal\n of the fine structure constant (a bit of mostly-harmless numerology).\n\n\n### Notes about Spectral Computation\n\nFor *n* = 3 through 6, exactly computing the spectrum\n (or more accurately, the characteristic polynomial)\n is possible, although it takes upwards of 10 minutes for *n* = 6 on my machine using `sympy`.\nThe spectra of *n* = 7, 8 are marked with an asterisk because they were computed numerically,\n which still took nearly 8 hours in the case of the latter.\nIn fact, these graphs grow so quickly that it becomes nearly impossible to compute\n the spectrum without an explicit formula.\n\nFor *n* = 8, even storing the adjacency matrix in memory is a problem.\nAssuming the use of single-precision floating point, this behemoth of a matrix is\n $(8!)^2 \\cdot 4 \\text{ bytes} = 6.5\\text{GB}$.\nThis doesn't even factor in additional space requirements for eigenvalue algorithms,\n and is the reason I certainly won't be attempting to compute the spectrum for *n* = 9.\n\n\nGallery of Adjacency Matrices\n-----------------------------\n\nThe patterns in adjacency matrices depend on an enumeration of $S_n$ so that the vertices\n can be labelled from 1 to *n*!.\nWhile this is a fascinating topic unto itself, this post is already long enough,\n and I feel comfortable with just sharing the pictures.\n\n\n### [Plain Changes](https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm)\n\n::: {layout-ncol=\"3\"}\n![$P_5$](plain-changes/plain_path_5.png)\n\n![$\\bigstar_5$](plain-changes/plain_star_5.png)\n\n![$K_5$](plain-changes/plain_k_5.png)\n\n![$P_6$](plain-changes/plain_path_6.png)\n\n![$\\bigstar_6$](plain-changes/plain_star_6.png)\n\n![$K_6$](plain-changes/plain_k_6.png)\n:::\n\n\n### [Heap's Algorithm](https://en.wikipedia.org/wiki/Heap%27s_algorithm)\n\n:::: {}\n::: {layout-ncol=\"3\"}\n![$P_5$](heap/heap_path_5.png)\n\n![$\\bigstar_5$](heap/heap_star_5.png)\n\n![$K_5$](heap/heap_k_5.png)\n\n![$P_6$](heap/heap_path_6.png)\n\n![$\\bigstar_6$](heap/heap_star_6.png)\n\n![$K_6$](heap/heap_k_6.png)\n:::\n\nNote: GHC's `Data.List.permutations` is slightly different from Heap's algorithm as displayed on Wikipedia\n::::\n\n\nClosing\n-------\n\nAs previously stated, I am only mostly sure of the validity of the exponential law for graphs.\nIt *seems* too good to be true, but testing it directly on some graphs by comparing the spectra\n of the exponential of the sum against the product of the exponentials shows that they are at least cospectral.\nTry it yourself, preferably with a better tool than [the ones I made in Haskell](https://github.com/queue-miscreant/SymmetricGraph).\n\nFrom the articles I was able to find, permutation star graphs have applications to parallel computing,\n which is somewhat ironic considering how little care I had for the topic when writing this article.\nIf I needed ruthless efficiency, I probably could have used a library with GPU algorithms\n (or taken a stab at writing a shader myself).\nHowever, I *was* able to use this as a learning experience regarding mutable objects in Haskell.\nWith only immutable objects (and enough garbage to create an island in the Pacific),\n I was running out of memory even with 16GB of RAM and 16GB of swap.\nIntroducing mutability not only brought improvements in space, but also a great deal of speedup,\n enough to make rendering adjacency matrix images of order 8 graphs just barely doable within\n a reasonable time span.\n\nSaid images are cursed.\nRemember, as raw bitmaps, these files are on the order of *gigabytes* big.\nOn a much weaker computer than I used to render the images, merely opening my file explorer\n to the folder containing *the folder containing* the images\n caused its all-too-eager thumbnailer to run.\nThis started consuming all of my system resources, crashed all of my browser's tabs,\n distorted audio, and finally locked up the computer.\nDespite this, PNG is a wonderful format that is able to compress them down to just 4MB,\n which demonstrates just how sparse these matrices are.\n\nDespite everything I was able to find about permutation star graphs and permutohedra,\n I was surprised that there is no information about the Cayley graphs generated by *all*\n 2-cycles (or at least information which is easy to find).\nThis is especially disappointing considering the phantom pattern which gets destroyed by 137,\n and I would love to know more about why this happens in the first place.\n\nGraph diagrams made with GeoGebra and NetworkX (GraphViz).\n\n\n### Additional Links\n\n- [Whitney Numbers of the Second Kind for the Star Poset (\n Paper from ScienceDirect\n )](https://www.sciencedirect.com/science/article/pii/S0195669813801278)\n - This article includes a section about representing a list of generators as a graph,\n making me wonder if someone has tried defining this operation before\n- [Whitney Numbers (Josh Cooper's Mathpages)](https://people.math.sc.edu/cooper/graph.html)\n- [\n The Many Faces of the Nauru Graph (Blogpost by David Eppstein)\n ](https://11011110.github.io/blog/2007/12/12/many-faces-of.html)\n\n", + "supporting": [ + "index_files" + ], + "filters": [], + "includes": {} + } +} \ No newline at end of file diff --git a/_quarto.yml b/_quarto.yml index a13a08f..976dea5 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -2,7 +2,7 @@ project: type: website website: - favicon: ./logo.png + favicon: ./logo-favicon.png title: 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a slight focus on abstract algebra. +This post will primarily focus on some fundamental concepts before launching into some constructions which will make the journey down this road more manageable. +However, I will still assume you are already familiar with what both a + [group](https://mathworld.wolfram.com/Group.html) and a + [graph](https://mathworld.wolfram.com/Graph.html) are. + + +The Symmetric Group +------------------- + +Some of the most important finite groups are the symmetric groups of degree *n* ($S_n$). +They are the groups formed by permutations of lists containing *n* items. +There is a group element for each permutation, so the order of the group is the + same as the number of permutations, *n!*. + +
+ On lists versus sets + + The typical definition of the symmetric group uses sets, which are fundamentally unordered. + This is potentially confusing, since it can appear as though an object after applying + a permutation is the same object as before. + + For example, let *p* be the permutation swapping "1" and "3". + Then, + + $$ + \begin{align*} + p(\{ 1, 2, 3 \}) = \{ p(1), p(2), p(3) \} = \{ 3, 2, 1 \} + \\ + p([ 1, 2, 3 ]) = [ p(1), p(2), p(3) ] = [ 3, 2, 1 ] + \end{align*} + $$ + + After applying *p*, the set is unchanged, since all of the elements are the same. + On the other hand, the lists differ in the first element and cannot be equal. + + Sets are still useful as a container. + For example, the elements of a group are unordered. + To keep vocabulary simple, I will do my best to refer to objects in a group as + "group elements" and the objects in a list as "items". +
+ +There are many ways to denote elements of the symmetric group. +I will take the liberty of explaining some common notations, each of which are useful in different ways. +More information about them can be found [elsewhere](https://mathworld.wolfram.com/Permutation.html) + [online](https://en.wikipedia.org/wiki/Permutation#Notations) + as well as any adequate group theory text. + + +### Naive List Notation + +Arguably the simplest way to do things is to denote the permutation by the result + of applying it to a canonical list. +Take the element of $S_3$ which can be described as the action + "assign the first item to the second index, the second to the third, and the third to the first". +Our canonical list in this case is $[1, 2, 3]$, matching the degree of the group. +This results in $[3, 1, 2]$, since "1" is in now in position two, and similarly for the other items. + +Unfortunately, this choice is too result-oriented. +This choice makes it difficult to compose group elements in a meaningful way, since all + of the information about the permutation is in the position of items in the list, + rather than the items of the list themselves. +For example, under the same rule, $[c, b, a]$ is mapped to $[a, c, b]$. + + +### True List Notations (Two- and One-line Notation) + +Instead, let's go back to the definition of the element. +All we have to do is list out every index on one line, then the destination + of every index on the next. +This is known as *two-line notation*. + +$$ +\begin{pmatrix} + 1 & 2 & 3 \\ + p(1) & p(2) & p(3) +\end{pmatrix} += \begin{pmatrix} + 1 & 2 & 3 \\ + 2 & 3 & 1 +\end{pmatrix} +$$ + +For simplicity, the first row is kept as a list whose items match their indices. + +This notation makes it easier to identify the inverse of a group element. +All we have to do is sort the columns of the permutation by the second row, + then swap the two rows. + +$$ +\begin{pmatrix} + 1 & 2 & 3 \\ + 2 & 3 & 1 +\end{pmatrix}^{-1} += \begin{pmatrix} + 3 & 1 & 2 \\ + 1 & 2 & 3 +\end{pmatrix}^{-1} += \begin{pmatrix} + 1 & 2 & 3 \\ + 3 & 1 & 2 +\end{pmatrix} +$$ + +Note that the second row is now the same as the result from the naive notation. + +Since the first row will be the same in all cases, we can omit it, which results in + *one-line notation*. +The *n*th item in the list now describes the position in which *n* can be found after + the permutation. + +$$ +\begin{pmatrix} + 1 & 2 & 3 \\ + 2 & 3 & 1 +\end{pmatrix} +\equiv [\![2, 3, 1]\!] +$$ + +Double brackets are used to distinguish this as a permutation and not an ordinary list. + +These notations make it straightforward to encode symmetric group elements on a computer. +After all, we only have to read the items of a list by the indices in another. +Here's a compact definition in Haskell: + +```{haskell} +-- convenient wrapper type for below +newtype Permutation = P { unP :: [Int] } + +apply :: Permutation -> [a] -> [a] +apply = flip (\xs -> + map ( -- for each item of the permutation, map it to... + (xs !!) -- the nth item of the first list + . (+(-1)) -- (indexed starting with 1) + ) . unP) -- (after undoing the type wrapping) +-- written in a non-point free form +apply' (P xs) ys = map ( \n -> ys !! (n-1) ) xs + +print $ P [2,3,1] `apply` [1,2,3] +``` + +Note that this means `P [2,3,1]` is actually equivalent to $[\![2, 3, 1]\!]^{-1}$, + since we don't get $[3, 1, 2]$. + +While these notations are fairly explicit and easy to describe to a computer, + it's easy to misinterpret an element as its inverse. +There is also some redundancy: $[\![2, 1]\!]$ and $[\![2, 1, 3]\!]$ both describe a group element + which swaps the first two items of a list. +On one hand, the $S_n$ each belongs to is explicit, but on the other, + every element of a smaller symmetric group also belongs to a larger one. +The verbosity of these notations also makes composing group elements difficult[^1]. + +[^1]: Composition is relatively easy to describe in two-line notation. + Recall that we reordered columns when finding an inverse. + We can do so to match rows of two elements, then compose a new element + by looking at the first and last rows. + For example, with group inverses: + $$ + \begin{pmatrix} + 1 & 2 & 3 \\ + 2 & 3 & 1 + \end{pmatrix} + \begin{pmatrix} + 1 & 2 & 3 \\ + 2 & 3 & 1 + \end{pmatrix}^{-1} + = \begin{matrix} + \begin{pmatrix} + 1 & 2 & 3 \\ + \cancel{2} & \cancel{3} & \cancel{1} + \end{pmatrix} + \\ + \begin{pmatrix} + \cancel{2} & \cancel{3} & \cancel{1} \\ + 1 & 2 & 3 + \end{pmatrix} + \end{matrix} + = \begin{pmatrix} + 1 & 2 & 3 \\ + 1 & 2 & 3 + \end{pmatrix} + $$ + + +### Cycle Notation + +*Cycle notation* addresses all of these issues, but gets rid of the transparency with respect to lists. +Let's try phrasing the element we've been describing differently. + +> assign the first item to the second index, the second to the third, and the third to the first + +We start at index 1 and follow it to index 2, and from there follow it to index 3. +Continuing from index 3, we return to index 1, and from then we'd loop forever. +This describes a *cycle*, denoted as $(1 ~ 2 ~ 3)$. + +Cycle notation is much more delicate than list notation, since the notation is nonunique: + +- Naturally, the elements of a cycle may be cycled to produce an equivalent one. + - $(1 ~ 2 ~ 3) = (3 ~ 1 ~ 2) = (2 ~ 3 ~ 1)$ +- Cycles which have no common elements (i.e., are disjoint) commute, + since they act on separate parts of the list. + - $(1 ~ 2 ~ 3)(4 ~ 5) = (4 ~ 5)(1 ~ 2 ~ 3)$ + + +#### Cycle Algebra + +The true benefit of cycles is that they are easy to manipulate algebraically. +For some reason, [Wikipedia](https://en.wikipedia.org/wiki/Permutation#Cycle_notation) + does not elaborate on the composition rules for cycles, + and the text which I read as an introduction to group theory simply listed it as an exercise. +While playing around with them and deriving these rules oneself *is* a good idea, + I will list the most important here: + +- Cycles can be inverted by reversing their order. + - $(1 ~ 2 ~ 3)^{-1} = (3 ~ 2 ~ 1) = (1 ~ 3 ~ 2)$ +- Cycles may be composed if the last element in the first is the first index on the right. + Inversely, cycles may also be decomposed by partitioning on an index and duplicating. + - $(1 ~ 2 ~ 3) = (1 ~ 2)(2 ~ 3)$ +- If an index in a cycle is repeated twice, it may be omitted from the cycle. + - $(1 ~ 2 ~ 3)(1 ~ 3) = (1 ~ 2 ~ 3)(3 ~ 1) = (1 ~ 2 ~ 3 ~ 1) = (1 ~ 1 ~ 2 ~ 3) = (2 ~ 3)$ + +Going back to $(1 ~ 2 ~ 3)$, if we apply this permutation to the list $[1, 2, 3]$: + +$$ +(1 ~ 2 ~ 3) \left( \vphantom{0^{0^0}} [1, 2, 3] \right) + = (1 ~ 2)(2 ~ 3) \left( \vphantom{0^{0^0}} [1, 2, 3] \right) + = (1 ~ 2) \left( \vphantom{0^{0^0}} [1, 3, 2] \right) + = [3, 1, 2] +$$ + +Which is exactly what we expected with our naive notation. + + +Generators, Permutation Groups, and Sorting +------------------------------------------- + +If we have a group *G*, then we can select a set of elements + $\langle g_1, g_2, g_3, ... \rangle$ as *generators*. +If we form all possible products -- not only the pairwise ones $g_1 g_2$, + but also $g_1 g_2 g_3$ and all powers of any $g_n$ -- then the products form a subgroup of *G*. +Naturally, such a set is called a *generating set*. + +Symmetric groups are of primary interest because of their subgroups, also known as permutation groups. +[Cayley's theorem](https://en.wikipedia.org/wiki/Cayley%27s_theorem), + a fundamental result of group theory, states that all finite groups + are isomorphic to one of these subgroups. +This means that we can encode effectively any group using elements of the symmetric group. + +For example, consider the generating set $\langle (1 ~ 2) \rangle$, which contains a single element + that swaps the first two items of a list. +Its square is the identity, meaning that its inverse is itself. +The identity and $(1 ~ 2)$ are the only two elements of the generated group, + which is isomorphic to $C_2$, the cyclic group of order 2. +Similarly, the 3-cycle in the generating set $\langle (1 ~ 2 ~ 3) \rangle$ generates $C_3$, + the cyclic group of order 3. + +However, the generating set $\langle (1 ~ 2), (1 ~ 2 ~ 3)\rangle$, which contains both of these permutations, generates the entirety of $S_3$. +In fact, [ + every symmetric group can be generated by two elements + ](https://groupprops.subwiki.org/wiki/Symmetric_group_on_a_finite_set_is_2-generated): + a permutation which cycles all elements once, and the permutation which swaps the first two elements. + +$$ +S_n = \langle (1 ~ 2), (1 ~ 2 ~ 3 ~ 4 ~ ... ~ n) \rangle +$$ + + +### Sorting and 2-Cycles + +The proof linked to above, that every symmetric group can be generated by two elements, + uses the (somewhat obvious) result that one can produce any permutation of a list by picking two items, + swapping them, and repeating until the list is in the desired order. + +This is reminiscent of how sorting algorithms are able to sort a list by only comparing and swapping items. +As mentioned earlier when finding inverses using in two-line notation, sorting is almost the inverse of permuting. +However, not all 2-cycles are necessary to generate the whole symmetric group. + + +### Bubble Sort + +Consider [bubble sort](https://en.wikipedia.org/wiki/Bubble_sort). +In this algorithm, we swap two items when the latter is less than the former, + looping over the list until it is sorted. +Until the list is sorted, the algorithm finds all such adjacent inversions. +In the worst case, it will swap every pair of adjacent items, some possibly multiple times. +This corresponds to the generating set + $\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~\ ~ n) \rangle$. + +![ + Bubble sort ordering a reverse-sorted list +](./bubble_sort.gif){.narrow} + + +### Selection Sort + +Another method, [selection sort](https://en.wikipedia.org/wiki/Selection_sort), + searches through the list for the smallest item and swaps it to the beginning. +If this is the final item of the list, this results in the permutation $(1 ~ n)$. +Supposing that process is continued with the second item and we also want it + to swap with the final item, then the next swap corresponds to $(2 ~ n)$. +Continuing until the last item, this gives the generating set + $\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~\ ~ n) \rangle$. + +![ + Selection sort ordering a particular list, using only swaps with the final item +](./selection_sort.gif){.narrow} + +This behavior for selection sort is uncommon, and this animation omits the selection of a swap candidate. +The animation below shows a more destructive selection sort, in which the + candidate least item is placed at the end of the list (position 5). +Once the algorithm hits the end of the list, the candidate is swapped to the least unsorted position, + and the algorithm continues on the rest of the list. + +![ + "Destructive" selection sort. + The actual list being sorted consists of the initial four items, with the final item as temporary storage. +](./destructive_selection_sort.gif){.narrow} + + +Swap Diagrams +------------- + +Given a set of 2-cycles, it would be nice to know at a glance if the entire group is generated. +In cycle notation, a 2-cycle is an unordered pair of natural numbers which swap items of an *n*-list. +Similarly, the edges of an undirected graph on *n* vertices (labelled from 1 to *n*) + may be interpreted as an unordered pair of the vertices it connects. + +If we treat the two objects as the same, then we can convert between graphs and sets of 2-cycles. +Going from the latter to the former, we start on an empty graph on n vertices + (labelled from 1 to *n*). +Then, we connect two vertices with an edge when the set includes the permutation swapping + the indices labelled by the vertices. + +Returning to the generating sets we identified with sorting algorithms, + we identify with each a graph family. + +- Bubble sort: $\langle (1 ~ 2), (2 ~ 3), (3 ~ 4), (4 ~ 5), …, (n-1 ~~ n) \rangle$ + - The path graphs ($P_n$), which are precisely as they sound: + an unbranching path formed by *n* vertices. +- "Selection" sort: $\langle (1 ~ n), (2 ~ n), (3 ~ n), (4 ~ n), …, (n-1 ~~ n) \rangle$ + - The star graphs ($\bigstar_n$, as $S_n$ means the symmetric group), + one vertex connected to all others. +- Every 2-cycle in $S_n$ + - The complete graphs ($K_n$), which connect every vertex to every other vertex. + +![ ](./example_graphs.png) + +This interpretation of these objects doesn't have proper name, but I think the name "swap diagram" fits. +They allow us to answer at least one question about the generating set from a graph theory perspective. + + +### Connected Graphs + +A graph is connected if a path exists between all pairs of vertices. +The simplest possible path is simply a single edge, which we already know to be an available 2-cycle. + +The next simplest case is a path consisting of two edges. +Some cycle algebra shows that we can produce a third cycle which corresponds + to an edge connecting the two distant vertices. + +:::: {layout-ncol="2"} +![ ](./induced_edge.png) + +::: {} +$$ +\begin{align*} + &(m ~ n) (n ~ o) (m ~ n) \\ + &= (m ~ n ~ o) (m ~ n) \\ + &= (n ~ o ~ m) (m ~ n) \\ + &= (n ~ o ~ m ~ n) \\ + &= (o ~ m) = (m ~ o) +\end{align*} +$$ +Note that this is just the conjugation of $(n ~ o)$ by $(m ~ n)$ +::: +:::: + +In other words, if we have have two adjacent edges, the new edge corresponds to + a product of elements from the generating set. +Graph theory has a name for this operation: when we produce *all* new edges by linking vertices + that were separated by a distance of 2, the result is called the *square of that graph*. +In fact, higher [graph powers](https://en.wikipedia.org/wiki/Graph_power) will reflect connections + induced by more conjugations of adjacent edges. + +![ + As you might be able to guess, this implies that $(1 ~ 4) = (3 ~ 4)(2 ~ 3)(1 ~ 2)(2 ~ 3)(3 ~ 4)$. + Also, $\bigstar_n^2 \cong K_n$. +](./P4_powers.png) + +If our graph is connected, then repeating this operation will tend toward a complete graph. +Complete graphs contain every possible edge, and so correspond to all possible 2-cycles, + which trivially generate the symmetric group. +Conversely, if a graph has *n* vertices, then for it to be connected, it must have at least $n - 1$ edges. +Thus, a generating set of 2-cycles must have at least $n - 1$ items to generate the symmetric group. + +Picking a different vertex labelling will correspond to a different generating set. +For example, in the image of $P_4$ above, if the edge connecting vertices 1 and 2 + is replaced with an edge connecting 1 and 4, then the resulting graph + is still $P_4$, even though it describes a different generating set. +We can ignore these extra cases entirely -- either way, the graph power argument shows + that a connected graph corresponds to a set generating the whole symmetric group. + + +### Disconnected Graphs + +A disconnected graph is the + [disjoint union](https://en.wikipedia.org/wiki/Disjoint_union_of_graphs) of connected graphs. +Under graph powers, we know that each connected graph tends toward a complete graph, + meaning a disconnected graph as a whole tends toward a disjoint union of complete graphs, + or [cluster graph](https://en.wikipedia.org/wiki/Cluster_graph). +But what groups do cluster graphs correspond to? + +The simplest case to consider is what happens when the graph is $P_2 \oplus P_2$. +If there is an edge connecting vertices 1 and 2 and an edge connecting vertices 3 and 4, + it corresponds to the generating set $\langle (1 ~ 2), (3 ~ 4) \rangle$. +This is a pair of disjoint cycles, and the group they generate is + +$$ +\{e, (1 ~ 2), (3 ~ 4), (1 ~ 2)(3 ~ 4) \} + \cong S_2 \times S_2 + \cong C_2 \times C_2 +$$ + +One way to look at this is by considering paths on each component: + we can either cross an edge on the first component (corresponding to $(1 ~ 2)$), + the second component (corresponding to $(3 ~ 4)$), + or both at the same time. +This independence means that one group's structure is duplicated over the other's, + or more succinctly, gives the direct product. +In general, if we denote *γ* as the map which "runs" the swap diagram and produces the group, then + +$$ +\gamma( A \oplus B ) = S_{|A|} \times S_{|B|}, + ~ A, B \text{ connected} +$$ + +where $|A|$ is the number of vertices in *A*. + +*γ* has the interesting property of mapping a sum-like object onto a product-like object. +If we express a disconnected graph *U* as the disjoint union of its connected components $V_i$, then + +$$ +\begin{gather*} + U = \bigsqcup_i V_i + \\ + \gamma( U ) = \gamma \left( \bigsqcup_i V_i \right) = \prod_i S_{|V_i|} +\end{gather*} +$$ + +This describes *γ* for every simple graph. +It also shows that we're rather limited in the kinds of groups which can be expressed by a swap diagram. + + +Closing +------- + +This concludes the dry introduction to some investigations of mine into symmetric groups. +While I could have omitted the sections about permutation notation and generators, + I wanted to be thorough and tie it to concepts which were useful to my understanding. +The notion of a graph encoding a generating set in particular will be fairly important going forward. + +Originally, this post was half of a single, sprawling, meandering article. +I hope I've improved the organization by keeping the digression about sorting algorithms + to this initial article. +The [next post](../2) will cover some interesting structures which fill Euclidean space + and incredibly large graphs. + +Sorting and graph diagrams made with GeoGebra. diff --git a/posts/permutations/1/induced_edge.png b/posts/permutations/1/induced_edge.png new file mode 100644 index 0000000000000000000000000000000000000000..b9124d9f7165cf3f903ef3d931a8d41b20ed68e1 GIT binary patch literal 13487 zcmb_jcQ}^e+kWjrviFuU%1T!DmLf7ULxi$7Av;@zGLr0_O}6Zj6|#BlJtCX$di#C< ze1Ck$@%`}~M~-;jXWY+yU)On^=Xu@l+s7&j*KnzE5eURJB}G|v1Ojare%P;I!x4WY z8VUICs-2>)BLYFfg!(~4>Z=sOK`JLX9j7O@=1#899G)XwU0t~>ZLA$lpV>X`)9*wiuEnco%pg@%5d zCnPYPN2rB{kdmqPKM7@4Ao5le!wh`-05uOS@|$UK*Yr_gIKC#4=t%4+#lTwEyKmAt@^>dl<~5A}gEz7~h_mnK{rTgoKEQ 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!= 0 else "") + + str(root) + + ")^{" + + str(multiplicity) + # pad multiplicity + + ( + ("\\phantom{" + ("0" * (pad_to - len(str(multiplicity))) + "}")) + if pad_to is not None and len(str(multiplicity)) < pad_to + else "" + ) + + "}" + ) + + +@dataclass +class SymmetricSpectrum: + """ + Dataclass for symmetric spectra -- those for which, if a spectrum contains a root, + it also contains its negative. + + If a root polynomial is not symmetric, then its symmetric counterpart will *not* be specified. + """ + + # Either the integral root or a polynomial with the root, and the multiplicity + roots: list[tuple[sympy.Expr | int, int]] + not_shown_count: int = 0 + + def to_markdown(self) -> str: + """ + Display a spectrum as a markdown string + """ + integer_roots = [ + (root, multiplicity) + for root, multiplicity in self.roots + if isinstance(root, int) + ] + # symmetrify polynomials + polynomial_roots = [ + (root, multiplicity) + for base_root, multiplicity in self.roots + if isinstance(base_root, sympy.Expr) + for root in set( + [base_root, (-1) ** (sympy.degree(base_root)) * base_root.subs(x, -x)] + ) + ] + + shown_roots = len(polynomial_roots) > 0 or len(integer_roots) > 0 + not_shown_string = ( + ( + f"*Not shown: {self.not_shown_count} other roots*" + if shown_roots + else f"*Not shown: all {self.not_shown_count} roots*" + ) + if self.not_shown_count > 0 + else "" + ) + + # strictly integral spectra + if len(polynomial_roots) == 0 and len(integer_roots) != 0: + longest_integer = max( + len(str(multiplicity)) for _, multiplicity in integer_roots + ) + return ( + ( + "$$\\begin{gather*}" + + " \\\\ ".join( + display_integral_root(*rm, pad_to=longest_integer) + for rm in integer_roots + ) + + "\\end{gather*}$$" + ) + if shown_roots + else "" + ) + not_shown_string + + # conditionally split integral roots onto multiple lines + integral_roots_lines = ( + " ".join(display_integral_root(*rm) for rm in integer_roots) + if len(integer_roots) < 5 + else ( + " ".join( + display_integral_root(*rm) + for rm in integer_roots[: len(integer_roots) // 2] + ) + + "\\\\" + + " ".join( + display_integral_root(*rm) + for rm in integer_roots[len(integer_roots) // 2 :] + ) + ) + ) + + return ( + ( + "$$\\begin{gather*}" + + integral_roots_lines + + " \\\\ " + + " \\\\ ".join( + sympy.latex(root**multiplicity) + for root, multiplicity in polynomial_roots + ) + + "\\end{gather*}$$" + ) + if shown_roots + else "" + ) + not_shown_string + + +def count_roots_spectrum(spectrum: SymmetricSpectrum) -> int: + return ( + sum( + [ + ( + (multiplicity * 2 if root != 0 else multiplicity) + if isinstance(root, int) + else (int(sympy.degree(root, x)) * multiplicity) + ) + for root, multiplicity in spectrum.roots + ] + ) + + spectrum.not_shown_count + ) + + +@dataclass +class GraphData: + spectrum: SymmetricSpectrum + vertex_count: int | None = None + edge_count: int | None = None + distance_classes: list[int] | None = None diff --git a/posts/permutations/2/graph_data/complete.py b/posts/permutations/2/graph_data/complete.py new file mode 100644 index 0000000..e37ed6b --- /dev/null +++ b/posts/permutations/2/graph_data/complete.py @@ -0,0 +1,80 @@ +from .base import GraphData, SymmetricSpectrum + +# TODO: other non-spectral data +data = { + 3: GraphData( + vertex_count=6, + edge_count=9, + distance_classes=[1, 3, 2], + spectrum=SymmetricSpectrum([(0, 4), (3, 1)]), + ), + 4: GraphData( + vertex_count=24, + edge_count=72, + distance_classes=[1, 6, 11, 6], + spectrum=SymmetricSpectrum([(0, 4), (2, 9), (6, 1)]), + ), + 5: GraphData( + vertex_count=120, + edge_count=600, + distance_classes=[1, 10, 35, 50, 24], + spectrum=SymmetricSpectrum( + [ + (0, 36), + (2, 25), + (5, 16), + (10, 1), + ] + ), + ), + 6: GraphData( + vertex_count=720, + edge_count=5400, + distance_classes=[1, 15, 85, 225, 274, 120], + spectrum=SymmetricSpectrum( + [ + (0, 256), + (3, 125), + (5, 81), + (9, 25), + (15, 1), + ] + ), + ), + 7: GraphData( + vertex_count=5040, + edge_count=52920, + distance_classes=[1, 21, 175, 735, 1624, 1764, 720], + spectrum=SymmetricSpectrum( + [ + (0, 400), + (1, 441), + (3, 1225), + (6, 196), + (7, 225), + (9, 196), + (14, 36), + (21, 1), + ] + ), + ), + 8: GraphData( + vertex_count=40320, + spectrum=SymmetricSpectrum( + [ + (0, 9864), + (2, 3136), + (4, 6125), + (7, 4096), + (8, 196), + (10, 784), + (12, 441), + (14, 400), + (20, 49), + (28, 1), + ] + ), + ), +} + +# TODO (maybe): verify these quantities with outputs from Haskell diff --git a/posts/permutations/2/graph_data/path.py b/posts/permutations/2/graph_data/path.py new file mode 100644 index 0000000..3f951b1 --- /dev/null +++ b/posts/permutations/2/graph_data/path.py @@ -0,0 +1,100 @@ +from .base import SymmetricSpectrum, GraphData, x + +data = { + 3: GraphData( + vertex_count=6, + edge_count=6, + distance_classes=[1, 2, 2, 1], + spectrum=SymmetricSpectrum([(1, 2), (2, 1)]), + ), + 4: GraphData( + vertex_count=24, + edge_count=36, + distance_classes=[1, 3, 5, 6, 5, 3, 1], + spectrum=SymmetricSpectrum( + [ + (1, 3), + (3, 1), + (x**2 - 3, 2), + (x**2 - 2 * x - 1, 3), + (x**2 + 2 * x - 1, 3), + ] + ), + ), + 5: GraphData( + vertex_count=120, + edge_count=240, + distance_classes=[1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1], + spectrum=SymmetricSpectrum( + [ + (0, 12), + (1, 6), + (4, 1), + (x**2 - 5, 6), + (x**2 + 5 * x + 5, 4), + (x**2 + 3 * x + 1, 4), + (x**2 + 2 * x - 1, 5), + (x**3 + 2 * x**2 - 5 * x - 4, 5), + ] + ), + ), + 6: GraphData( + vertex_count=720, + edge_count=1800, + distance_classes=[1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1], + spectrum=SymmetricSpectrum( + [ + (0, 20), + (1, 25), + (2, 15), + (3, 5), + (4, 5), + (5, 1), + (x**2 - 3, 20), + ], + not_shown_count=558, + ), + ), + 7: GraphData( + vertex_count=5040, + edge_count=15120, + distance_classes=[ + 1, + 6, + 20, + 49, + 98, + 169, + 259, + 359, + 455, + 531, + 573, + 573, + 531, + 455, + 359, + 259, + 169, + 98, + 49, + 20, + 6, + 1, + ], + spectrum=SymmetricSpectrum( + [ + (0, 35), + (1, 20), + (2, 45), + (6, 1), + ], + not_shown_count=4873, + ), + ), + 8: GraphData( + vertex_count=40320, spectrum=SymmetricSpectrum([], not_shown_count=40320) + ), +} + +# TODO (maybe): verify these quantities with outputs from Haskell diff --git a/posts/permutations/2/graph_data/star.py b/posts/permutations/2/graph_data/star.py new file mode 100644 index 0000000..cd5276f --- /dev/null +++ b/posts/permutations/2/graph_data/star.py @@ -0,0 +1,85 @@ +from .base import SymmetricSpectrum, GraphData + +data = { + 3: GraphData( + vertex_count=6, + edge_count=6, + distance_classes=[1, 2, 2, 1], + spectrum=SymmetricSpectrum([(1, 2), (2, 1)]), + ), + 4: GraphData( + vertex_count=24, + edge_count=36, + distance_classes=[1, 3, 6, 9, 5], + spectrum=SymmetricSpectrum( + [ + (0, 4), + (1, 3), + (2, 6), + (3, 1), + ] + ), + ), + 5: GraphData( + vertex_count=120, + edge_count=240, + distance_classes=[1, 4, 12, 30, 44, 26, 3], + spectrum=SymmetricSpectrum( + [ + (0, 30), + (1, 4), + (2, 28), + (3, 12), + (4, 1), + ] + ), + ), + 6: GraphData( + vertex_count=720, + edge_count=1800, + distance_classes=[1, 5, 20, 70, 170, 250, 169, 35], + spectrum=SymmetricSpectrum( + [ + (0, 168), + (1, 30), + (2, 120), + (3, 105), + (4, 20), + (5, 1), + ] + ), + ), + 7: GraphData( + vertex_count=5040, + edge_count=15120, + distance_classes=[1, 6, 30, 135, 460, 1110, 1689, 1254, 340, 15], + spectrum=SymmetricSpectrum( + [ + (0, 840), + (1, 468), + (2, 495), + (3, 830), + (4, 276), + (5, 30), + (6, 1), + ] + ), + ), + 8: GraphData( + vertex_count=40320, + spectrum=SymmetricSpectrum( + [ + (0, 3960), + (1, 5691), + (2, 2198), + (3, 6321), + (4, 3332), + (5, 595), + (6, 42), + (7, 1), + ] + ), + ), +} + +# TODO (maybe): verify these quantities with outputs from Haskell diff --git a/posts/permutations/2/heap/heap_k_5.png b/posts/permutations/2/heap/heap_k_5.png new file mode 100644 index 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+--- + + + +```{python} +#| echo: false + +from IPython.display import Markdown +from tabulate import tabulate + +from graph_data import complete, path, star +``` + + +This post assumes you have read (or at least skimmed over parts of) the [first post](../1), + which talks about graphs and the symmetric group. +This post will contain some more "empirical" results, since I'm not an expert on graph theory. +However, one hardly needs to be an expert to learn or to make computations, observations, and predictions. + +We left off talking about producing a group from a graph, so we begin now by considering how to do the reverse. + + +Cayley Graphs +------------- + +For a given generating set, we can assign every element in the group it generates to a vertex in a graph. +Starting with each of the generators, we draw edges from one vertex to another when + the product of the initial vertex and a generator (in that order) is the product vertex. +This process continues until there are no more arrows to draw. +The resulting figure is known as a [Cayley graph](https://mathworld.wolfram.com/CayleyGraph.html)[^1]. + +[^1]: The construction implies a labelling of edges by its generating set and a labelling + of vertices by the generated elements. + It is also common to describe an unlabelled graph as "Cayley" if it could + be generated by this procedure. + +Cayley graphs depend on the generating set used, so they can take a wide variety of shapes. +Here are a few examples of Cayley graphs made from elements of $S_4$: + +![ + Left: $\{(1 ~ 3 ~ 2 ~ 4), (3 ~ 4), (1 ~ 4 ~ 2 ~ 3)\}$, cube graph
+ Middle: $\{(1 ~ 2 ~ 3), (2 ~ 3 ~ 4)\}$, cuboctahedral graph
+ Right: $\{(2 ~ 3), (3 ~ 4), (2 ~ 3 ~ 4)\}$, octahedral graph
+ Generating sets obtained from the previous MathWorld article. +](./s4_cayley_graphs.png) + +Owing to the way in which they are defined, Cayley graphs have a few useful properties as graphs. +At every vertex, we have as many outward edges as we do generators in the generating set, + so the outward (and in fact, inward) degree of each vertex is the same. +In other words, it is a regular graph. +More than that, it is [vertex-transitive](https://mathworld.wolfram.com/Vertex-TransitiveGraph.html), + since labelling a single vertex's outward edges will label that of the entire graph. + +In general, the Cayley graph is a directed graph. +However, if for every member of the generating set, we also include its inverse, + every directed edge will be matched by an edge in the opposite direction, + and the Cayley graph may be considered undirected. + + +Graphs to Graphs +---------------- + +All 2-cycles are their own inverse, so generating sets which include only them + produce undirected Cayley graphs. +Since this kind of generating set can itself be thought of as a graph, + we may consider an operation on graphs that maps a swap diagram to its Cayley graph. + +![ + An example of this operation, denoted as $\exp$. + The proper Cayley graphs for $(1 ~ 2)$ and $(3 ~ 4)$ are not shown; + they are isomorphic to the corresponding swap graphs, but have different vertex labels. +](./exp_p2_oplus_p2.png) + +It seems to be the case that $\exp( A \oplus B ) = \exp( A ) \times \exp( B )$, + where $\oplus$ signifies the disjoint uinion and $\times$ signifies the + [Cartesian (box) product of graphs](https://en.wikipedia.org/wiki/Cartesian_product_of_graphs)[^2]. +Unlike *γ* from the previous post, both the input and output of this operation are graphs. +Because of this and the sum/product relationship, I've taken to calling this operation the + "graph exponential"[^3]. + +[^2]: Graphs have many product structures, such as the tensor product and strong product. + The Cartesian product is (categorically) more natural when paired with disjoint unions. + +[^3]: Originally, I called this operation the "graph factorial", since it involves permutations + and the number of vertices in the resulting graph grows factorially. + +This operation is my own invention, so I am unsure whether or not + it constitutes anything useful. +In fact, the possible graphs grow so rapidly that computing anything about the exponential + of order 8 graphs starts to overwhelm a single computer. +It is, however, interesting, as I will hopefully be able to convince. + +A random graph will not generally correspond to an interesting generating set, + and therefore, will also generally have an uninteresting exponential graph. +Hence, I will continue with the examples used previously: paths, stars, and complete graphs. +They are among the simplest graphs one can consider, and as we will see shortly, + have exponentials which appear to have natural correspondences to other graph families. + + +Some Small Exponential Graphs +----------------------------- + +Because of [the difficulty in determining graph isomorphism]( + https://en.wikipedia.org/wiki/Graph_isomorphism_problem + ), it is challenging for a computer to find a graph in an encyclopedia. +Computers think of graphs as a list of vertices and their outward edges, + but this implementation faces inherent labelling issues. +These persist even if the graph is described as a list of (un)ordered pairs, + an adjacency matrix, or an incidence matrix, + the latter two of which have very large memory footprints[^4]. + +[^4]: I was able to locate a project named the + [Encyclopedia of Finite Graphs](https://github.com/thoppe/Encyclopedia-of-Finite-Graphs), + but it is only able to build a database simple connected graphs which + can be queried by invariants (and is outdated since it uses Python 2). + +However, as visual objects, humans can compare graphs fairly easily + -- the name means "drawing" after all. +Exponentials of order 3 and order 4 graphs are neither so small as to be uninteresting + nor so big as to be unparsable by humans. + + +### Order 3 + +![ ](./exp_order_3.png){.narrow} + +At this stage, we only really have two graphs to consider, since $P_3 = \bigstar_3$. +Immediately, one can see that $\exp( P_3 ) = \exp( \bigstar_3 ) = C_6$, + the 6-cycle graph (or hexagonal graph). +It is also apparent that $\exp( K_3 )$ is the utility graph, $K_{3,3}$. + +![ + Graph exponential of a disconnected graph.
+ Note that the red element commutes with the green and blue ones. + This produces two types of squares in the exponential graph. + Meanwhile, the green and blue elements have an order 3 product, and produce the hexagon. +](./exp_p3_oplus_p2.png) + +Here, we can again demonstrate the sum rule of the graph exponential with $\exp( P_3 \oplus P_2 )$. +Simplifying, since we know $\exp( P_3 ) = C_6$, the result is $C_6 \times P_2 = \text{Prism}_6$, + the hexagonal prism graph. + + +### Order 4 (and beyond) + +::: {layout="[[1,1],[1]]"} +![$\exp( P_4 )$](./exp_p4_networkx.png) + +![$\exp( \bigstar_4 )$](./exp_star4_networkx.png) + +![](./exp_order_4.png) +::: + +With some effort, $\exp( P_4 )$ can be imagined as a projection of a 3D object, + the [truncated octahedron](https://en.wikipedia.org/wiki/Truncated_octahedron). +Because of its correspondence to a 3D solid, this graph is planar. +Both the hexagon and this solid belong to a class of polytopes called + [*permutohedra*](https://en.wikipedia.org/wiki/Permutohedron), which are figures + that are also formed by permutations of the coordinate (1, 2, 3, ..., *n*) in Euclidean space[^5]. +Technically, there is a distinction between the Cayley graphs and permutohedra + since their labellings differ. +Both have edges generated by swaps, but in the latter case, the connected vertices are expected to be + separated by a certain distance. +More information about the distinction can be found at this article on [Wikimedia]( + https://commons.wikimedia.org/wiki/Category:Permutohedron_of_order_4_%28raytraced%29#Permutohedron_vs._Cayley_graph + )[^6]. + +[^5]: In fact, these figures are able to completely tessellate the $n-1$ dimensional subspace of + $\mathbb{R}^n$ where the coordinates sum to the $n-1$th triangular number. + Note also that the previous graph in the sequence of $\exp(P_n)$, the hexagonal graph, + is visible in the truncated octahedron. + This corresponds to the projection $(x,y,z,w) \mapsto (x,y,z)$ over + the coordinates of the permutohedra. + +[^6]: Actually, if one considers a *right* Cayley graph, where each generator is right-multiplied + to the permutation at a node rather than left-multiplied, then a true correspondence is obtained, + at least for order 4. + +Meanwhile, $\exp( \bigstar_4 )$ is more difficult to identify, at least without rearranging its vertices. +It turns out to be isomorphic to the [Nauru graph](https://mathworld.wolfram.com/NauruGraph.html), + a graph with many strange properties. +Notably, whereas the graph isomorphic to the permutohedron is obviously a spherical polyhedron, + the Nauru graph can be topologically embedded on a torus. +The Nauru graph also belongs to the family of + [permutation star graphs](https://mathworld.wolfram.com/PermutationStarGraph.html) + $PS_n$ (*n* = 4), which also includes the hexagonal graph (*n* = 3). +The MathWorld article confirms some kind of correspondence, stating graphs of this form + are generated by pairwise swaps. + +My attempts at finding a graph isomorphic to $\exp( K_4 )$ have thus far ended in failure. +It is certainly *not* isomorphic to $K_{4,4}$, since this graph has 8 vertices, + as opposed to 24 in $\exp( K_4 )$. + + +Graph Invariants +---------------- + +While I have managed to identify the families to which some of these graphs belong, + I am rather fond of computing (and conjecturing) sequences from objects. +Not only is it much easier to consult something like [the OEIS](https://oeis.org/) for these quantities, + but after finding a matching sequence, there are ample articles to consult for more information. +By linking to their respective entries, I hope you'll consider reading more there. + +Even though I have obtained these values empirically, I am certain that the sequences for + $\exp( P_n )$ and $\exp( \bigstar_n )$ match the corresponding OEIS entries. +I also have great confidence in the sequences I found for $\exp( K_n )$. + + +### Edge Counts + +Despite knowing how many vertices there are (*n*!, the order of the symmetric group), + we don't necessarily know how many edges there are. + +```{python} +#| echo: false +#| classes: plain + +Markdown(tabulate( + [ + [i, pathG.edge_count, starG.edge_count, completeG.edge_count] + for i in range(3, 8) + for pathG, starG, completeG in ( + [path.data[i], star.data[i], complete.data[i]], + ) + ] + [ + [ + "Sequence", + "Second column of Lah numbers
[OEIS A001286](http://oeis.org/A001286)", + "Same as previous", + "[OEIS 001809](http://oeis.org/A001809)", + ], + [ + "Rule", + r"$L(n,2) = n!{(n-1)(n-2) \over 4}$", + "", + r"$n!{n(n-1) \over 4}$" + ] + ], + headers=[ + "*n*", + r"$\#E(\exp( P_n ))$", + r"$\#E(\exp( \bigstar_n ))$", + r"$\#E(\exp( K_n ))$" + ] +)) +``` + + +### Radius and Distance Classes + +The radius of a graph is the smallest possible distance which separates two maximally-separated vertices. +Due to vertex transitivity, the greatest distance between two vertices is the same for every vertex. + +```{python} +#| echo: false +#| classes: plain + +Markdown(tabulate( + [ + [i, len(pathG), len(starG), len(completeG)] + for i in range(3, 8) + for pathG, starG, completeG in ( + [path.data[i].distance_classes, star.data[i].distance_classes, complete.data[i].distance_classes], + ) + ] + [ + [ + "Sequence", + "Triangular numbers
[OEIS A000217](http://oeis.org/A000217)", + "Integers not congruent to 2 (mod 3)
[OEIS A032766](http://oeis.org/A032766)", + "*n* - 1" + ], + [ + "Rule", + r"$\Delta_{n-1} = {n(n-1) \over 2}$", + r"$\lfloor {n-1 \over 2} \rfloor + n -\ 1$" + ] + ], + headers=[ + "*n*", + r"$r(\exp( P_n ))$", + r"$r(\exp( \bigstar_n ))$", + r"$r(\exp( K_n ))$" + ] +)) +``` + +These quantities are somewhat reductive. +If a vertex is distinguished, the remaining vertices may be partitioned into classes by their + distance from it. +Including the vertex itself (which is distance 0 away), there will be $r + 1$ such classes, + where *r* is the radius. +These classes are the same for every vertex due to transitivity. + +In the case of these graphs, they are a partition of *n*!. + +```{python} +#| echo: false +#| classes: plain + +half_break = lambda x: str(x[:len(x) // 2])[:-1] + "
" + str(x[len(x) // 2:])[1:] + +Markdown(tabulate( + [ + [i, pathG if i <= 5 else half_break(pathG), starG, completeG] + for i in range(3, 8) + for pathG, starG, completeG in ( + [path.data[i].distance_classes, star.data[i].distance_classes, complete.data[i].distance_classes], + ) + ] + [ + [ + "Sequence", + "Mahonian numbers
[OEIS A008302](http://oeis.org/A008302)", + "Whitney numbers of the second kind (star poset)
[OEIS A007799](http://oeis.org/A007799)", + "Stirling numbers of the first kind
[OEIS A132393](http://oeis.org/A132393)" + ] + ], + headers=[ + "*n*", + r"$\text{dists}(\exp( P_n ))$", + r"$\text{dists}(\exp( \bigstar_n ))$", + r"$\text{dists}(\exp( K_n ))$" + ] +)) +``` + +I am certain that the appearance of the Stirling numbers here is legitimate, + since these numbers count the number of permutations of *n* objects with *k* disjoint cycles. +Obviously, the identity element is distance 1 from all 2-cycles since they are all in the generating set; + likewise, all 3-cycles are distance 2 from the identity (but distance 1 from the 2-cycles), + and so on until the entire graph has been mapped. +The shapes induced by these classes were used to create the diagrams + of $\exp( K_3 )$ and $\exp( K_4 )$ above. + + +#### Spectrum + +The eigenvalues of the adjacency matrix of a graph can be interesting + and sometimes help in identifying a graph. +Unfortunately, eigenvalues are not necessarily integers, and therefore not easily + found in the OEIS (though they are always real for graphs). + +```{python} +#| echo: false +#| classes: plain + +Markdown(tabulate( + [ + [i, pathG.to_markdown(), starG.to_markdown(), completeG.to_markdown()] + for i in range(3, 9) + for pathG, starG, completeG in ( + [path.data[i].spectrum, star.data[i].spectrum, complete.data[i].spectrum], + ) + ], + headers=[ + "*n*", + r"$\text{Spec}(\exp( P_n ))$", + r"$\text{Spec}(\exp( \bigstar_n )$", + r"$\text{Spec}(\exp( K_n ))$" + ] +)) +``` + +From what I have been able to identify, the spectrum of an exponential graph is symmetric about 0, + by which I mean that the characteristic polynomial is even. +This has been the case for all graphs I have tried testing, even outside these graph families. + +Since all eigenvalues of $\exp( \bigstar_n )$ calculated are integers, + it appears they are integral graphs, a fact of which I am reasonably sure + because of the aforementioned correspondence to permutation star graphs. +Additionally, the eigenvalues are very conveniently the integers up to $n-1$ and down to $-n+1$. +Unfortunately, despite the ease of reading the eigenvalues themselves, + there isn't an OEIS entry for the multiplicities. +I was able to identify the multiplicity of the 0 eigenvalue with + [OEIS A217213](http://oeis.org/A217213), which counts orderings on Dyck paths. +If this is truly the sequence being generated, it means there is a 1:1 correspondence between + these orderings and a basis of the nullspace of the adjacency matrix. + +It seems to be the case that $\exp( K_n )$ are also integral graphs. +Perplexingly, the multiplicities for each of the eigenvalues appear to mostly be perfect powers. +This is the case until n = 8, which ruins the pattern because neither of + $9864 = 2^3 \cdot 3^2 \cdot 137$ or $6125 = 5^3 \cdot 7^2$ are perfect powers. +I find both this[^7] and the fact that such a large prime appears among the factorization of the former + rather creepy since all other primes which appear here are small -- 2, 3, 5, and 7. + +[^7]: Some physicists are fond of 137 for its closeness to the reciprocal + of the fine structure constant (a bit of mostly-harmless numerology). + + +### Notes about Spectral Computation + +For *n* = 3 through 6, exactly computing the spectrum + (or more accurately, the characteristic polynomial) + is possible, although it takes upwards of 10 minutes for *n* = 6 on my machine using `sympy`. +The spectra of *n* = 7, 8 are marked with an asterisk because they were computed numerically, + which still took nearly 8 hours in the case of the latter. +In fact, these graphs grow so quickly that it becomes nearly impossible to compute + the spectrum without an explicit formula. + +For *n* = 8, even storing the adjacency matrix in memory is a problem. +Assuming the use of single-precision floating point, this behemoth of a matrix is + $(8!)^2 \cdot 4 \text{ bytes} = 6.5\text{GB}$. +This doesn't even factor in additional space requirements for eigenvalue algorithms, + and is the reason I certainly won't be attempting to compute the spectrum for *n* = 9. + + +Gallery of Adjacency Matrices +----------------------------- + +The patterns in adjacency matrices depend on an enumeration of $S_n$ so that the vertices + can be labelled from 1 to *n*!. +While this is a fascinating topic unto itself, this post is already long enough, + and I feel comfortable with just sharing the pictures. + + +### [Plain Changes](https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm) + +::: {layout-ncol="3"} +![$P_5$](plain-changes/plain_path_5.png) + +![$\bigstar_5$](plain-changes/plain_star_5.png) + +![$K_5$](plain-changes/plain_k_5.png) + +![$P_6$](plain-changes/plain_path_6.png) + +![$\bigstar_6$](plain-changes/plain_star_6.png) + +![$K_6$](plain-changes/plain_k_6.png) +::: + + +### [Heap's Algorithm](https://en.wikipedia.org/wiki/Heap%27s_algorithm) + +:::: {} +::: {layout-ncol="3"} +![$P_5$](heap/heap_path_5.png) + +![$\bigstar_5$](heap/heap_star_5.png) + +![$K_5$](heap/heap_k_5.png) + +![$P_6$](heap/heap_path_6.png) + +![$\bigstar_6$](heap/heap_star_6.png) + +![$K_6$](heap/heap_k_6.png) +::: + +Note: GHC's `Data.List.permutations` is slightly different from Heap's algorithm as displayed on Wikipedia +:::: + + +Closing +------- + +As previously stated, I am only mostly sure of the validity of the exponential law for graphs. +It *seems* too good to be true, but testing it directly on some graphs by comparing the spectra + of the exponential of the sum against the product of the exponentials shows that they are at least cospectral. +Try it yourself, preferably with a better tool than [the ones I made in Haskell](https://github.com/queue-miscreant/SymmetricGraph). + +From the articles I was able to find, permutation star graphs have applications to parallel computing, + which is somewhat ironic considering how little care I had for the topic when writing this article. +If I needed ruthless efficiency, I probably could have used a library with GPU algorithms + (or taken a stab at writing a shader myself). +However, I *was* able to use this as a learning experience regarding mutable objects in Haskell. +With only immutable objects (and enough garbage to create an island in the Pacific), + I was running out of memory even with 16GB of RAM and 16GB of swap. +Introducing mutability not only brought improvements in space, but also a great deal of speedup, + enough to make rendering adjacency matrix images of order 8 graphs just barely doable within + a reasonable time span. + +Said images are cursed. +Remember, as raw bitmaps, these files are on the order of *gigabytes* big. +On a much weaker computer than I used to render the images, merely opening my file explorer + to the folder containing *the folder containing* the images + caused its all-too-eager thumbnailer to run. +This started consuming all of my system resources, crashed all of my browser's tabs, + distorted audio, and finally locked up the computer. +Despite this, PNG is a wonderful format that is able to compress them down to just 4MB, + which demonstrates just how sparse these matrices are. + +Despite everything I was able to find about permutation star graphs and permutohedra, + I was surprised that there is no information about the Cayley graphs generated by *all* + 2-cycles (or at least information which is easy to find). +This is especially disappointing considering the phantom pattern which gets destroyed by 137, + and I would love to know more about why this happens in the first place. + +Graph diagrams made with GeoGebra and NetworkX (GraphViz). + + +### Additional Links + +- [Whitney Numbers of the Second Kind for the Star Poset ( + Paper from ScienceDirect + )](https://www.sciencedirect.com/science/article/pii/S0195669813801278) + - This article includes a section about representing a list of generators as a graph, + making me wonder if someone has tried defining this operation before +- [Whitney Numbers (Josh Cooper's Mathpages)](https://people.math.sc.edu/cooper/graph.html) +- [ + The Many Faces of the Nauru Graph (Blogpost by David Eppstein) + ](https://11011110.github.io/blog/2007/12/12/many-faces-of.html) diff --git a/posts/permutations/2/plain-changes/plain_k_5.png b/posts/permutations/2/plain-changes/plain_k_5.png new file mode 100644 index 0000000000000000000000000000000000000000..67924f7d619a5af4b0119fe090d030e6f550984e GIT binary patch literal 980 zcmV;_11tQAP)sV}}rc+F(coS&s)`iSsMzAVYj{)O`ap^N!3DmL9IGMof4liO~ z{CdV%M<*d8#Ku!ISQm|q%SZ4RZF4hZ`rBi@HVlR{L_Kd{q58CFd54+5x513nxI9B* z^-D^{8W+Z9bFOV0Yh3Nv7an8TD@3^(#<=PMa*4qaHk_Kmy4OgHafFKfax4~h+7Ng- z=kmjRlye>5+bGQAAi>5X+Fc+s5cRejjh0VfSE!eB?L}S2j>TuI?#M9J0E`_-U&v)F z9yW{dVul%;ivNqkL+r5<>r4x}4{L^Do3_T)hK+Q_hQ)a-i5*Yhra9f_B9*&9b_3V2 zd*eFFxlW9$6>E*l&mj)#Tp8W^AtQ}lJAF5a&)tNzV)S%;0Xqx%2RWCgy(@Jt3>(1t zjW$`v#xBHyvFVktdI)wQck5ivus0%m*Xt5irvmR*nz2~l7wcTnDX!JIb{2Z8&P8D_ z6r7kJ-OX4nlqm4^*171Z?R73*c&&A=9JZ~_<{lfl 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zq6=p5q~UsUm_C_VfG$RR)xG4sj&t9dwz+!*y=h!!2=>*5i(*A33=8eaqs#VH?#<6k zO!|9H(s(m@3qJN`{(b13ns3wMB1@Qu6sTxu^(lf0Pe@7#TjRS$|J&$S>VqCpnPg|K z_vw)UH_v$GyK{U`;g&C^jZIVAW&bd&rrf_zN%*o->3odF``0JCQv(+J&Q9D=mwMVo zg}H#;Mjn6jj8 zpRay3_QK?dWB=C^yenfD%Y9pwt+R7-wRuQci_DQsKfqmnwYC9vtp9w;YU||+te@~t z+9@Zz@2ZYWh_)mxMFLf<;)Nn*weG|ee>!d{%Fa_SH_x#1(2I&u%}voHDopKGde8sX zQS|BqK60+v5_lQVf8F8X63Sm8|5J$n!~tAxNL(1fCL`#MQ>Z@Gf5%>jDh6EmiQTKa zYQ5bCMCO(+#mk>&VKOw782K3e(0|{{feMYq_m89d2QogqEn8$U9b31D@^_Ibeg}j0 zf#w6rgVhY7>Wa>Xt}-_#lp2`bhTx3}NwNz5*$Ent&k`=wC`081b=LpKK2|U)XBAZG zslKF*yr*>Pwc6wt=8yw7hqif2ri0V;ef+p@h?U4>AFYp%yUf6wPq%zG$nJoD`8eeG zl=bMe4gQ7W?hTgzPTq_4XRngP|9@XGbd8HF0snnQBvDchs!9LXe`?eGlNbE=-6sMV zJpbEd_~)8`^Oyf^?o6tGF2n!+@(u+5|Nb6QcDny}ME~zpsH^k;|G$*Hs{ij^Z`}&j Y(pT#ktIt8r6nq`fFi +.figure-img.narrow { + max-width: 768px; +} + +.figure-img.narrower { + max-width: 640px; +} + +.figure-img.narrowest { + max-width: 512px; +} + + + +This post assumes you have read the [first post](../1), which talks about swap diagrams + and the symmetric group. +The section about Cayley graphs from the [second post](../2) will also be useful. + +To summarize from the first post, a swap diagram is a graph where each vertex represents + a list index and each edge represents the permutation which swaps the two entries. +I singled out three graph families for their symmetry: + +- Paths, which link adjacent swaps +- Stars, in which all swaps contain a distinguished index +- Complete graphs, which contain all possible swaps + +Swap diagrams are ultimately limited by only being able to describe collections of 2-cycles. +If we wanted to include higher-order elements, we'd require a weaker + mathematical structure called a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). +These are even more difficult to draw than simple graphs, much less understand. + +Another way we are limited is in the kinds of groups which can be encoded. +One can rather quickly notice that swap diagrams will only generate symmetric groups + and direct products thereof. +Also, while edges are members of a generating set, the vertices contain information + which is ultimately irrelevant to the group. + + +Upgrading Diagrams +------------------ + +What if we forgot the roles of the vertices, allowed the edges to take their place? +In other words, we want to make a graph where the vertices represent generators, + rather than the edges. + +Ideally, we'd still like to preserve some information about coincident edges. +Graph-theoretically, we can elevate the role of the edges to that of the vertices + by making a [line graph](https://en.wikipedia.org/wiki/Line_graph). +This operation produces a graph such that + +- Every edge in the original graph becomes a vertex in the new graph +- Coincident edges in the new graph are connected by an edge in the new graph + - In other words, we convert adjacent edges to adjacent vertices. + +![ + While line graphs of paths simply shrink, + $\bigstar_5$ shrinks to $K_4$ before expanding to $K_{2,2,2}$ +](./line_graphs.png) + +The line graph of a swap diagram results in a graph where the vertices are the 2-cycles. +In a swap diagram, when two distinct 2-cycles share an index (vertex), their product has order 3. +Thus, in the line graph, the edges mean that the product of the two vertices has order 3. +Non-adjacent vertices still have a product with order 2, since they represent disjoint 2-cycles. + +To push the group/graph correspondence the even further, we can impose a final restriction. +Namely, a subgraph should represent a sub*group*. +Specifically, we want an *induced* subgraph + (one where if two connected vertices are included, then so is the edge joining them) + since this maintains the relationship between the generators. +Unfortunately, this restriction disqualifies many of the line graphs of swap diagrams. +At least our beloved paths are left untouched, since $L(P_n) = P_{n-1}$. + + +### Coxeter Diagrams + +As it turns out, these restrictions are significant and produce some very deep mathematical objects, + known as *Coxeter diagrams* (named for the same Coxeter as in [Goldberg-Coxeter](/posts/pentagons/1)). +In this domain, the aforementioned rules about vertices and edges apply: + each vertex corresponds to an order 2 element and each edge signifies + that the product of two elements has order 3. +Furthermore, *disconnected* vertices correspond to elements which commute with one another, + just like disjoint 2-cycles. +However, each vertex need not correspond to a single 2-cycle, + and can instead be *any* element of order 2, i.e., a product of disjoint 2-cycles. + + + +

+
+ + Finite coxeter.svg + + +
+ The most important Coxeter diagrams. + Note that the $A_n$ diagrams are just the familiar path graphs. +
+ Source: Rgugliel, via Wikimedia Commons +
+
+
+ +Coxeter diagrams are frequently extended to enable describing higher-order products + by labelling the edge[^1]. +For example, the product between $(1 ~ 2)(3 ~ 4)$ and $(2 ~ 3)$ has order 4. +A diagram for these elements would therefore include two vertices connected + with an edge labelled "4", corresponding to $B_2$. + +[^1]: Unlabelled edges still have order 3. + + +### How Big and What Shape? + +We already know how big the group a swap diagram generates. +Each connected component with *n* vertices generates the group $S_n$. +They combine via the direct product to give a group of order + +$$ +\left | \prod_i S_{n_i} \right | = \prod_i | S_{n_i} | = \prod_i n_i! +$$ + +Removing an edge from a swap diagram has the potential to make the graph disconnected. +In this case, the graph is split into two parts, which map to $S_m$ and $S_n$. +This tells us that $S_m \times S_n$ embeds in $S_{m + n}$. + +The subgroup restriction makes Coxeter diagrams a lot more sensitive to changes. +We at least know that if we remove a vertex, the resulting diagram is a subgroup of the whole diagram. +What we need is some way to measure the effect this has on the rest of the group. + + +Coxeter-Todd Algorithm +---------------------- + +This is exactly what the [*Coxeter-Todd algorithm*]( + https://en.wikipedia.org/wiki/Todd%E2%80%93Coxeter_algorithm + ) does. +It more-or-less generalizes the procedure for creating Cayley graphs. +Using the vertices of the Coxeter diagram as generators[^2], we convert products between + them to paths in a graph. + +[^2]: For clarity, I'll try to refer to these vertices specifically as "generators". + +Let's apply the algorithm to the diagram $A_3$. +First, we need to label our generators as *a*, *b*, and *c*. +Then, we select a select a generator to remove and create a graph according to the following steps: + + +### 1. Initialization + +In a Cayley graph, we start out with a single vertex representing the identity. +Here, the first vertex instead corresponds to the subgroup generated by all other generators. + +For each generator other than the one being removed, we attach loops to the vertex. +This is because the subgroup is closed -- when we multiply the set of all elements + of the subgroup by any of its elements, it doesn't change the set. +In other words, $hH = H, h \in H$. + +For the remaining generator, we draw an edge connecting to a new vertex, + which represents a new set of elements. +This is a coset of the subgroup -- that is, each vertex we make in this new graph + corresponds to a disjoint partition of elements of the whole group. + +![ + Left: the Coxeter diagram $A_3$
+ Right: the first step of generating a graph from removing generator *a*. + The (group generated by the) *bc* subdiagram is represented by the vertex with the *b* and *c* loops. +](./steps/coxeter_todd_1.png) + +Note that all loops and edges we draw will be undirected, since all generators have order 2. +Performing the same operation twice just takes us back to where we started. + + +### 2. Loop Transference + +At the new vertex, we transfer over every loop whose label is not connected to the label of the edge. +This is due to the rule stating that non-adjacent generators commute. +In other words, their product has order 2. +For our graph, this can be seen for $acac = a^2 c^2 = \text{id} = (ac)^2$, and means after following + labels *acac*, we end up at the same vertex we started at. + +![ ](./steps/coxeter_todd_step_2.png) + + +### 3. Extension, Branching, and Rejoining + +At the new vertex, we still need an edge labelled *b*, but it cannot be a loop. +Adjacent generators have a product whose order is the label (*n*) of the edge connecting them + (defaulting to 3 for unlabelled ones). +This means we need to form a path alternating between the two generators, + such that following it would loop back to the vertex we started at. +That is, + +$$ +(ab)^n = \stackrel{n \text{ times}}{\overbrace{(ab)(ab) ... (ab)(ab)}} = \text{id} +$$ + +![ + Attaching a *b* edge to a new vertex with an *a* loop so that $ababab = \text{id}$ +](./steps/coxeter_todd_step_3.png) + +If the Coxeter diagram branches, then so does the generated graph. +The labels of each pair of branching edges have a product with a known order, + so following a path alternating between two of them must return to the same vertex. +Since it alternates between the two labels *n* times, this ends up producing even-sided polygons + in our generated graph. +Commuting vertices form either a square or a pair of loops connected by an edge. +The latter is actually a special case of the former, which can be seen by matching + a pair of opposite edges. + +![ + Top: two nonadjacent vertices form a square
+ Bottom: two adjacent vertices form a hexagon +](./steps/coxeter_todd_cycles.png){.narrower} + + +### 4. Finalization + +If we ensure at each new vertex that we can follow a valid path for every pair of generators, + then this process will either terminate or continue indefinitely with a repeating pattern. +A quick sanity check on the resulting graph is to add the number of loops and edges at a vertex. +The sum should be the number of generators. + +![ + Final graph to the middle right. The *ab* and *bc* paths are shown above and below. +](./steps/coxeter_todd_step_4.png){.narrower} + +As stated earlier, each vertex in the graph corresponds to a coset of the group + under the subgroup being considered. +The number of cosets is the index of the subgroup, the quantity we were after the whole time. +In this case, the index of the three-vertex diagram ($A_3$) on the two-vertex diagram + ($A_2$, the same diagram without *a*) is 4. + + +### Results + +We can remove vertices one-by-one until we eventually reduce the graph to a single vertex, + which we know has order 2. +Multiplying all of the indices at each stage together gives us the order of the group we are after. + +![ + Coxeter diagrams (left) and the graphs generated by removing the a vertex (right).
+ Permutations shown above vertices which give a permutation group embedding. +](./coxeter_an.png){.narrow} + +It is readily shown that the group generated by $A_3$ has order $4 \cdot 3 \cdot 2 = 24$, + which coincides with the order of $S_4$. + +We can assign a permutation to each of our generators, so long as the edge condition + as it relates to their product is obeyed. +In the diagrams above, $(1 ~ 2)$ and $(2 ~ 3)$ have a product of order 3, + but $(1 ~ 2)$ and $(3 ~ 4)$ have a product of order 2, + since the cycles (and in the diagram, the vertices) are disjoint. +This argument generalizes nicely to higher $A_n$. + + +Other Diagrams for $A_3$ +------------------------ + +A path is hardly the most interesting diagram. +However, there are yet more graphs which can be produced from $A_3$. + + +### Starting in the Middle + +Selecting either endpoint of the diagram (generator *a* or *c*) will generate the same graph as shown above. +This is easy to see if you read the generated graph right-to-left rather than left-to-right. +However, the center point *b* has two neighbors. +If we try generating a graph by removing this generator, our graph branches. + +![ ](./coxeter_a3_generator_b.png){.narrow} + +The index of 6 given by this graph is correct, since the remaining diagram + $A_1 A_1$ has an order of $2 \cdot 2 = 4$. +This group is isomorphic to the [Klein four-group](https://en.wikipedia.org/wiki/Klein_four-group), + so this diagram shows that it occurs within $S_4$. + + +### Removing Multiple Vertices + +We can remove more than one generator from the Coxeter diagram at once, as long as we follow all the rules. +Instead, the initial vertex in the new graph branches immediately and has fewer loops. +For example, removing two generators from $A_3$ produces a figure with squares and hexagons: + +![ + Graph generated by removing any two vertices from $A_3$.
+ For the circled generators *a* and *b*, the diagram starts from the circled vertex + on the right (or its identical neighbor).
+ If *b* and *c* were removed instead, the diagram would start from + the similar vertices with *a* loops at the top of the graph.
+ If *a* and *c* were removed, the diagram would start from either the leftmost or rightmost vertex, + which both have a *b* loop. +](./coxeter_a3_pair.png){.narrow} + +A figure composed of squares and hexagons should be familiar from the previous post. +The truncated octahedron also contains only squares and hexagons and is generated by $S_4$. +In fact, the figure above is the truncated octahedron folded in half. +It is folded in "half" specifically because the single remaining vertex in the Coxeter diagram has order 2. +Consequently, the number of vertices in the graph must be half that + of the whole group (or complete permutahedron). + +If we remove both generators in $A_2$, we end up with a hexagon. + +![ + Graph generated by removing both generators from $A_2$. + All vertices are identical, so all may be considered "starting" vertices. +](./coxeter_a2_both.png){.narrow} + +Also from the previous post, we know this to be the Cayley graph for $S_3$ generated by a pair of swaps. +In this way, the Cayley graph can be thought of as a sort of "limiting" graph of the Coxeter diagram, + where each coset contains only a single element. + + +The Traditional Explanation +--------------------------- + +I must now admit that that I have not given anywhere near the whole explanation of Coxeter diagrams, + and have only succeeded in leading us down the proverbial algebraic rabbit hole. + +Coxeter diagrams also come equipped with a standard geometric interpretation. +The "order 2 elements" we discussed previously are replaced with mirrors, + and the edge labels are the angle between mirrors, + interpreted as half of the central angle of a regular *n*-gon (triangle: 60°, square: 45°, ...). + + +
+
+ + Dihedral symmetry domains 3 + + +
+ Fundamental domains of Coxeter diagram ![](coxeter_a2.png) +
+ Source: Tomuren, via Wikimedia Commons +
+
+
+ +The image above is composed of the mirrors specified by $A_2$. +One of the generators is the red reflection and the other is the the green reflection. +It is clear that alternating between red and green three times will take one back + to where they started, just like the algebra tells us. + +All of the plane ends up assigned to one of the domains. +Note also that points along mirrors which are equidistant from the center + form a hexagon made up of six equilateral triangles, one in each domain. + + +### Struggles in 3D + +$A_3$ has an additional vertex and edge. +This means that an additional mirror needs to be placed in along every preexisting reflection. +We end up reflecting into another dimension, and our domains shoot out into 3D space. +It is easier to imagine the additional domains by rotating two of the axes around each other, + so that the pairs will join together into equilateral triangular "cones". + +![ + Mirror axes in 3D for $A_3$. + The red, green, and blue lines all lie within the same plane, and correspond to $A_2$. + The cyan line is perpendicular to the red one, and is formed by rotating + either green about blue or blue about green by 60°. +](./roots/roots_a3.png){.narrowest} + +In 2D, equidistant points form a hexagon; in 3D, the points instead form a cuboctahedron. +Connecting the triangular faces of the cuboctahedron to the origin gives + a family of eight tetrahedra meeting at a point. +At this point the reflections become clearer, since certain reflections + do the same thing as in the plane above, while others will "double" a tetrahedron. + +::: {layout-ncol="2" layout-valign="center"} +![Cuboctahedron](./roots/cuboctahedron.png) + +![ + Six of the eight tetrahedra formed by connecting equidistant points ("roots") to the origin. + Note how whole tetrahedra are reflected in a mirror axis. +](./roots/roots_a3_with_domains.png) +::: + +Personally, I think the geometric way of thinking becomes is difficult even in 3D. +Every axis describes a separate rotation, which is simple enough. +But we're trying to create infinite cones of symmetry domains in 3D space, + and we can only try to interact with this using a 2D screen. +We're running out of room before even getting to 4D space, + where the visualization problems are even worse. +Also, while the angle between two mirrors is 60°, it's easy to slip into thinking + about the dihedral angle between two faces (domains) of the cones, which is *not* 60°. + + +Closing +------- + +When I initially wrote the article about swap diagrams (a name I hadn't given them at the time) + I had no idea about how closely it related to something else which I was having trouble understanding. +Now knowing better, I split the article and pushed the basics a little harder. +I find this closeness between swap diagrams and Coxeter diagrams interesting. +At the cost of simplicity, the latter enables the ability to encode higher-order cycles and give rise + to exceptional mathematical objects. + +It was difficult for me to find information documenting how to understand the algebraic perspective. +It also bears mentioning that I know very little about Lie theory, + another field of math where Coxeter diagrams turn up. +When I finally found [a group theory video](https://www.youtube.com/watch?v=BHezLvEH1DU) + by Professor Richard Borcherds, I was dismayed that there seems to be even less material + online which depicts the coset graphs. +I have written [an appendix](../appendix) cataloguing some graphs that appear when applying + the Coxeter-Todd algorithm, since I believe they are interesting in their own right. + +Images from Wikimedia belong to their respective owners. +All other diagrams created with GeoGebra. diff --git a/posts/permutations/3/line_graphs.png b/posts/permutations/3/line_graphs.png new file mode 100644 index 0000000000000000000000000000000000000000..674d29c1545e84ed27f8ea8367003e62b193a79b GIT binary patch literal 28340 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subgroup of only even permutations in W(B_4) +goodB4EvenPermsSubgroup = filter ((==1) . pParity) $ generatingSet $ goodB4Embedding + +-- Count the orders in the group xs +-- Should probably be done better in CayleyOps +countOrders xs = sortOn fst $ map ((,) <*> flip numOrder xs) (nub $ map order xs) + +commutatorSubgroup xs = nub $ [commutator x y | x <- xs, y <- xs] + +printGroupData x = do + putStr "\tOrder: " + putStrLn $ show $ length x + putStr "\tElement order count: " + putStrLn $ show $ countOrders x + putStr "\tCommutator subgroup order: " + putStrLn $ show $ length $ commutatorSubgroup x + + +main = do + putStrLn "Bad embedding:" + printGroupData $ generatingSet badb4Embedding + putStrLn "Even permutation subgroup of the good embedding:" + printGroupData goodB4EvenPermsSubgroup + putStrLn "Edge-generated subgroup of the good embedding:" + printGroupData $ generatingSet goodB4Edges diff --git a/posts/permutations/appendix/de/coxeter_d4_generator_a.png 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+.figure-img.narrower, img.narrower { + max-width: 512px; +} + +.figure-img.narrow, img.narrow { + max-width: 768px; +} + + + +This post is an appendix to [another post](../3) discussing the basics of Coxeter diagrams. +It focuses on transforming path-like swap diagrams into proper $A_n$ Coxeter diagrams, + which correspond to symmetric groups. +This post focuses on the graphs made by the cosets made by removing a single generator + (i.e., a vertex of the Coxeter diagram). + +For finite diagrams whose order is not prohibitively big, I will also provide an embedding + as a permutation group by labelling each generator in the Coxeter diagram. +Since each generator is the product of disjoint swaps, I will also show their swap diagrams, + as well as interactions via the edges. + + +Platonic Symmetry +----------------- + +The symmetric group $S_n$ also happens to describe the symmetries of an $(n-1)$-dimensional simplex. +The 3-simplex is simply a tetrahedron and has symmetry group $T_h$, which is isomorphic to $S_4$. +We know that $S_4$ can be encoded by the diagram $A_3$. + +The string {3, 3} can be read across the edges of $A_3$, denoting the order of certain symmetries. +This happens to coincide with another datum of the tetrahedron: + its [Schläfli symbol](https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol). +It describes triangles (the first 3) which meet in triples (the second 3) at a vertex. +It may also be interpreted as the symmetry of the 2-dimensional components (faces) and the vertex-centered symmetry. +[The Wikipedia article](https://en.wikipedia.org/wiki/Tetrahedron) + on the tetrahedron presents both of these objects in its information column. + +![ + From the Wikipedia article on the tetrahedron. + The Conway notation, Schläfli symbol, Coxeter diagram, and symmetry groups are shown. +](./tetrahedron_wikipedia.png) + +The image above shows more sophisticated diagrams alongside $A_3$, + which I will not attempt describing (mostly because I don't completely understand them myself). +Other Platonic solids and their higher-dimensional analogues have different Schläfli symbols, + and correspond to different Coxeter diagrams. + + +### $B_3$: Octahedral Group + +Adding an order-4 product into the mix makes things a lot more interesting. +The cube (Schläfli symbol {4, 3}) and octahedron ({3, 4}) share a symmetry group, $O_h$, + which corresponds to the $B_3$ diagram[^1]. + +[^1]: For groups without a common name, I'll instead use *W*(*diagram*) + (for [Weyl](https://en.wikipedia.org/wiki/Weyl_group)) to represent the generated group. + For example, in this case, $W(B_3) \cong O_h$. + +::: {layout-ncol="3" layout-valign="center"} +![](./b3/coxeter_b3_generator_a.png){.lightbox group="b3"} + +![](./b3/coxeter_b3_generator_b.png){.lightbox group="b3"} + +![](./b3/coxeter_b3_generator_c.png){.lightbox group="b3"} +::: + +The center graph is the first to have a hexagon created by removing a single generator. +Meanwhile, the third graph is entirely path-like, similar to the ones created + by removing an endpoint from the $A_n$ diagrams. +In the same vein, the graph for $B_3$ resembles the graph for $A_3$ made + by removing the center generator, albeit with two extra vertices. + +Going across left to right, the order suggested by each index is: + +- $8 \cdot |A_2| = 8 \cdot 6 = 48$ +- $12 \cdot |A_1 A_1| = 12 \cdot 4 = 48$ +- $6 \cdot |B_2| = 6 \cdot 8 = 48$ + - $B_2$ describes the symmetry of a square (i.e., $\text{Dih}_4$, the dihedral group of order 8) + +Each diagram suggests the same order, which is good. +A simple embedding which obeys the edge condition assigns $(1 ~ 2)(3 ~ 4)$ to *a*, + $(2 ~ 3)$ to *b*, and $(1 ~ 2)$ to *c*. +Then $ab = (1 ~ 2 ~ 4 ~ 3)$ has order 4, $bc = (1 ~ 3 ~ 2)$ has order 3, and *ac* obviously has order 2. + +There's a problem though. +These are all elements of $S_4$, and in fact, generate it + (in fact, $S_4 \cong O$, the rotational symmetries of a cube). +The group we want is $O_h \cong S_4 \times \mathbb{Z}_2$. +If we want to embed a group of order 48 in a symmetric group, we need one for which 48 divides its order. +48 divides $|S_6| = 720$, and indeed, a quick fix is just to multiply each generator + by another disjoint 2-cycle like $(5 ~ 6)$. + +These two embeddings generate different (proper) Cayley graphs. +The one for *O* has 24 vertices and is nonplanar. +On the other hand, the one for $O_h$ is planar, and is the skeleton of the + [truncated cuboctahedron](https://en.wikipedia.org/wiki/Truncated_cuboctahedron), + a figure composed of octagons, hexagons, and squares. +These shapes are exactly what those implied by the orders of the products in the Coxeter diagram. +Note also that the [cuboctahedron](https://en.wikipedia.org/wiki/Cuboctahedron) + is the figure produced halfway between dualizing + the cube and octahedron by shrinking faces (their *rectification*). + +In either case, the products of adjacent generators (the permutations on the edges) are the same. +When made into a Cayley graph, these products generate the + [rhombicuboctahedron](https://en.wikipedia.org/wiki/Rhombicuboctahedron), + which is another shape that is in some sense midway between the cube and octahedron. +Since all of these generators are in $S_4$, it only has half the number of vertices + as the truncated cuboctahedron. + + +::: {layout="[[1,1,1],[1]]"} +![ + False embedding +](./b3/false_octahedral_symmetry_graph.png) + +![ + Good embedding +](./b3/truncated_cuboctahedral_graph.png) + +![ + Edge-generated +](./b3/rhombicuboctahedral_graph.png) + +![ + Left: false embedding, right: good embedding +](./b3/octahedral_embedding.png){.narrow} +::: + + +### $H_3$: Icosahedral Group + +Continuing with groups based on 3D shapes, the dodecahedron (Schläfli symbol {5, 3}) + and icosahedron ({3, 5}) also share a symmetry group. +It is known as $I_h$ and corresponds to Coxeter diagram $H_3$. + +::: {layout-ncol="3" layout-valign="center"} +![](./h3/coxeter_h3_generator_a.png){.lightbox group="h3"} + +![](./h3/coxeter_h3_generator_b.png){.lightbox group="h3"} + +![](./h3/coxeter_h3_generator_c.png){.lightbox group="h3"} +::: + +Two of these graphs are similar to the cube/octahedron graphs. +The middle contains a decagon, corresponding to the order 5 product between *a* and *b*. + +We have the orders: + +- $20 \cdot |A_2| = 20 \cdot 6 = 120$ + - Graph resembles an extension of the graph of $B_3 / A_1 A_1$, as hexagons joining blocks of squares +- $30 \cdot |A_1 A_1| = 30 \cdot 4 = 120$ +- $12 \cdot |H_2| = 12 \cdot 10 = 120$ + - Graph resembles an extension of the graph of $B_3 / A_2$, as a single square joining paths + +The order 120 is the same as the order of $S_5$, which corresponds to diagram $A_4$. +However, these are not the same group, since $I_h \cong \text{Alt}_5 \times \mathbb{Z}_2 \ncong S_5$. +A naive (though slightly less obvious) embedding, found similarly to $B_3$'s, + incorrectly assigns the following: + +- $a = (1 ~ 2)(3 ~ 4)$ +- $b = (2 ~ 3)(4 ~ 5)$ +- $c = (2 ~ 3)(1 ~ 4)$ + +This is certainly wrong, since all these permutations are within $S_5$. +Actually, they are all even permutations and in fact generate $I \cong \text{Alt}_5$, + with order 60. +Yet again, multiplying a disjoint 2-cycle (in this case, $(6 ~ 7)$) to each generator + boosts the order to 120 and gives a proper embedding of $I_h$. + +Similarly to $B_3$, the first, incorrect embedding gives a nonplanar Cayley graph. +The second one gives a planar graph, the skeleton of the + [truncated icosidodecahedron](https://en.wikipedia.org/wiki/Truncated_icosidodecahedron). +It consists of decagons, hexagons, and squares, just like those which appear in the graphs above. +The icosidodecahedron is also the rectification of the dodecahedron and icosahedron. +In this case, the edges generate the + [rhombicosidodecahedronal graph](https://en.wikipedia.org/wiki/Rhombicosidodecahedron). + + +::: {layout="[[1,1,1],[1]]" layout-valign="center"} +![ + False embedding +](./h3/false_icosahedral_symmetry_graph.png) + +![ + Good embedding +](./h3/truncated_icosidodecahedral_graph.png) + +![ + Edge-generated +]( ./h3/rhomicosidodecahedral_graph.png) + +![ + Left: false embedding, right: good embedding +](./h3/icosahedral_embedding.png){.narrow} +::: + +It is remarkable that the truncations of the rectifications[^2] have skeleta that are the same + as the Cayley graphs generated by their respective Platonic solids' Coxeter diagrams. +In a way, these figures describe their own symmetry. +Also, both of these figures belong to a class of polyhedra known as + [zonohedra](https://en.wikipedia.org/wiki/Zonohedron). + +[^2]: This composition is also called "omnitruncation". + + +### $B_4$: Hyperoctahedral Group + +Up a dimension from the cube and octahedron lie their 4D counterparts: + the tesseract (Schläfli symbol {4, 3, 3}, interpreted as three cubes ({4, 3}) around an edge) + and 16-cell ({3, 3, 4}, four tetrahedra ({3, 3}) around an edge). +They correspond to Coxeter diagram $B_4$. + +::: {layout-ncol="2" layout-valign="center"} +![](./b4/coxeter_b4_generator_a.png){.lightbox group="b4"} + +![](./b4/coxeter_b4_generator_b.png){.lightbox group="b4"} + +![](./b4/coxeter_b4_generator_c.png){.lightbox group="b4"} + +![](./b4/coxeter_b4_generator_d.png){.lightbox group="b4"} +::: + +Three of these graphs (those starting with *a*, *c*, and *d*) are *also* similar + to those encountered one dimension lower. +The remaining graph is best understood in three dimensions, befitting the 4D symmetries it encodes. +It appears to have similar regions to the + [omnitruncated tesseract](https://en.wikipedia.org/wiki/Runcinated_tesseracts#Omnitruncated_tesseract), + featuring both the truncated octahedron and hexagonal prism cells. + +We have the orders: + +- $16 \cdot |A_3| = 16 \cdot 24 = 384$ + - Graph resembles an extension of $B_3 / A_2$, as squares connecting paths +- $32 \cdot |A_1 A_2| = 32 \cdot 2 \cdot 6 = 384$ +- $24 \cdot |B_2 A_1| = 24 \cdot 8 \cdot 2 = 384$ + - Graph resembles an extension of $B_3 / A_1 A_1$, as hexagons joining blocks of squares +- $8 \cdot |B_3| = 8 \cdot 48 = 384$ + - Graph resembles an extension of $A_3 / A_2$, a simple path + +The order of the group, 384, suggests that it needs to be embedded in at least $S_8$ + since $384 ~ \vert ~ 8! ~ ( = 40320)$. +Indeed, such an embedding exists (found by computer search rather than by hand): + +- $a = (1 ~ 3)$ +- $b = (1 ~ 2)(3 ~ 4)(5 ~ 6)(7 ~ 8)$ +- $c = (1 ~ 3)(2 ~ 6)(4 ~ 5)(7 ~ 8)$ +- $d = (1 ~ 3)(2 ~ 4)(5 ~ 7)(6 ~ 8)$ + +Notably, this embedding takes advantage of an order 4 product between an order 2 and an order 4 element. + +A similar computer search yielded an insufficient embedding in $S_8$, with order 192: + +- $a = (1 ~ 2)(3 ~ 4)(5 ~ 6)(7 ~ 8)$ +- $b = (1 ~ 3)(2 ~ 5)(4 ~ 7)(6 ~ 8)$ +- $c = (1 ~ 2)(3 ~ 4)(5 ~ 7)(6 ~ 8)$ +- $d = (1 ~ 2)(3 ~ 5)(4 ~ 6)(7 ~ 8)$ + +This false embedding *cannot* be "fixed" by multiplying some generators[^3] by $(9 ~ 10)$ + (implicitly embedding in $S_{10}$ instead). +Quickly "running" the generators shows that the order of the group is unchanged by this maneuver. +Much of the structure permutations ensures that nonadjacent vertices still have order-2 products. + +[^3]: The only candidate choices are *a* and nothing else, every generator but *a*, or all generators. + All other choices violate edge/product constraints from the diagram. + +![](./b4/hyperoctahedral_embedding.png){.narrow} + +I won't try to identify either of these generating sets' Cayley graphs since it is impractical + to try comparing graphs of this size (and they likely correspond to a 4D object's skeleton). +In fact, *H* appears to not be isomorphic to a subgroup of $W(B_4)$. +The latter has at least 2 subgroups of order 192: one generated by the edges in the above embedding, + and one containing only the even permutations. +These are distinct from one another, since the number of elements of a particular order is different. +The latter subgroup is closer to *H*, matching the number of elements of each order, + but the even permutations have commutator subgroup of order 96 while *H* has a commutator subgroup of order 48. + + +Other Finite Diagrams +--------------------- + +Higher-dimensional Platonic solids are hardly the limits of what these diagrams can encode. +The following three diagrams also give rise to finite graphs. + +### $D_4$ + +$D_4$ is the first Coxeter diagram with a branch. +Like $B_4$ before it, it is corresponds to the symmetries of a 4D object. +We only really have two choices in which generator to remove, which generate the following graphs: + +::: {layout-ncol="2" layout-valign="center"} +![](./de/coxeter_d4_generator_a.png){.lightbox group="d4"} + +![](./de/coxeter_d4_generator_b.png){.lightbox group="d4"} +::: + +While we could also remove *c* or *d*, this would just produce a graph identical + to the one on the left, just with different labelling. +The right diagram is rather interesting, as it can be described geometrically + as two cubes attached to three hexagons sharing an edge. + +Both of these cases give the order + +- $8 \cdot |A_3| = 8 \cdot 24 = 192$ +- $24 \cdot |A_1 A_1 A_1| = 24 \cdot 8 = 192$ + +The minimum degree of symmetric group is $S_8$, since $192 ~ \vert ~ 40320$. +Fortunately, a computer search yields a correct embedding immediately: + +- $a = (1 ~ 8)(2 ~ 3)(4 ~ 7)(5 ~ 6)$ +- $b = (1 ~ 2)(3 ~ 4)(5 ~ 6)(7 ~ 8)$ +- $c = (1 ~ 5)(2 ~ 7)(3 ~ 4)(6 ~ 8)$ +- $d = (1 ~ 8)(2 ~ 4)(3 ~ 7)(5 ~ 6)$ + +![ ](./de/d4_embedding.png){.narrower} + +In fact, the group generated by this diagram is isomorphic to the even permutation subgroup of $W(B_4)$. +This can be verified by selecting order 2 elements from the latter which obey the laws in $D_4$. +On the other hand, *H* and the edge-generated subgroup do not satisfy this diagram. + + + + +### $D_5$ + +*D*-type diagrams continue by elongating one of the paths. +The next diagram, $D_5$, has really only four distinct graphs, of which I will show only two: + +::: {layout-ncol="2" layout-valign="center"} +![](./de/coxeter_d5_generator_a.png){.lightbox group="d5"} + +![](./de/coxeter_d5_generator_de.png){.lightbox group="d5"} +::: + +First, note how the graph to the right is generated by removing the generator *e*, + but the left and right sides of the diagram are asymmetrical. +This is because *d* and *e* are equivalent with respect to the branch, + and shows us that removing either generator *d* or *e* results in the same graph. + +The order suggested by each is + +- $10 \cdot |D_4| = 10 \cdot 192 = 1920$ +- $16 \cdot |A_4| = 16 \cdot 120 = 1920$ + +If we were to remove generator *b*, we'd end up with the diagram $A_1 A_3$, + which generates a group of order 48. +The graph would have 1920 / 48 = 40 vertices, which would be fairly difficult to render. +Removing generator *c* would be even worse, since the resulting diagram, $A_2 A_1 A_1$, + has order 24, and the graph would require 80 vertices -- twice as many. + +Finding an embedding at this point is difficult. +The order 1920 also divides $40320 = |S_8|$, but computer search has failed + to find an embedding in up to $S_{11}$. + + +### $E_6$ + +If one of the shorter paths of $D_5$ is extended, then we end up with the diagram $E_6$. +I will only show one of its graphs. + +![ ](./de/coxeter_e6_generator_ae.png){.narrow} + +Similarly to one of the graphs for $D_5$, this graph goes in with *a* and comes out with *e*, + which are again symmetric with respect to the branch. +I particularly like how on this graph, most of the squares are structured + so that they can bridge to the *ae* commuting square (middle, bottom). + +The order of this group is $27 \cdot 1920 = 51840$, which borders on the + incomprehensible (if that threshold hasn't been crossed already). +The graphs not shown would have the following number of vertices + (which is precisely why I won't draw them out): + +- Removing *f*: $[E_6 : A_5] = 51840 / 720 = 72$ vertices +- Removing *b* or *d*: $[E_6 : A_1 A_4] = 51840 / 240 = 216$ vertices +- Removing *c*: $[E_6 : A_2 A_2 A_1] = 51840 / 72 = 720$ vertices + +Going by order alone, $E_6$ should embed in $S_9$ ($51840 ~ \vert ~ 9!$), + but since $S_{11}$ was too small for its subgroup $D_5$, this too optimistic. +I don't know what the minimum degree is required to embed $E_6$, + but finding it directly it is beyond my computational power. + +The *E* diagrams continue with $E_7$ and $E_8$. +Each of the three corresponds to the symmetries of semi-regular higher-dimensional objects, + whose significance and structure are difficult to comprehend. +Their size alone makes continuing onward by hand a fool's errand, + and I won't be attempting to draw them out right just now. + + +Infinite (Affine) Diagrams +-------------------------- + +Not every Coxeter diagram produces a finite, closed graph. +Instead, they may proliferate vertices forever. +They are termed either "affine" or "hyperbolic", depending respectively on whether the diagrams + join up with themselves or seem to require more and more room as the algorithm advances. +This is also related to the collection of fundamental domains and roots that the diagram describes. + +Since hyperbolic graphs are difficult to draw, I'll be restricting myself to the affine diagrams. +Unlike hyperbolic diagrams, which are unnamed, affine ones are typically named by altering finite diagrams. + +### $\widetilde A_2$ + +The Coxeter diagram associated to the triangle graph $K_3$ is called $\widetilde A_2$. +This graph is also the line graph of $\bigstar_3$ so it might make sense to assign + to the vertices the generators $(1 ~ 2)$, $(2 ~ 3)$, and $(1 ~ 3)$, + which we know to generate $S_3$. +However, $S_3$ already has a diagram, $A_2$, which is clearly a subdiagram of + $\widetilde A_2$, so the new group must be larger. + +![ ](./affine/coxeter_a2tilde_generator_a.png){.narrower} + +Attempting to make a graph by following the generators results in an infinite hexagonal tiling. +You might recall that $A_2$ generates a hexagon, so it is intriguing that this generates + a tiling based on this shape. +There are three distinct types of hexagon -- *ab*, *ac*, and *bc* + -- since each pair of elements has a product of order 3. +Removing a pair of vertices from the diagram would get rid of the initial edge (*a* in this case), + and the "limiting case" where all three vertices are removed is just the true hexagonal tiling. + + +### $\widetilde G_2$ + +The hexagonal tiling has Schläfli symbol {6, 3}, and is dual to the triangular tiling. +As a Coxeter diagram, this symbol matches the Coxeter diagram $\widetilde G_2$. + +![ ](./affine/coxeter_g3tilde_generator_b.png){.narrower} + +The graph generated by removing a generator is another infinite tiling, in this case the + [truncated trihexagonal tiling](https://en.wikipedia.org/wiki/Truncated_trihexagonal_tiling). +Once again, this tiling is the truncation of the rectification of the symmetry it typically describes. +Each pair of prodcuts corresponds to a distinct polygon in the graph: *ab* to dodecagons, *ac* to squares, + and *bc* to hexagons. + + +### $\widetilde C_2$ + +The only remaining regular 2D tiling is the square tiling (Schläfli symbol {4, 4}), + whose Coxeter diagram is named $\widetilde C_2$. + +![ ](./affine/coxeter_c2tilde_generator_b.png){.narrower} + +The tiling generated by this diagram is known as the + [truncated square tiling](https://en.wikipedia.org/wiki/Truncated_square_tiling). +Mirroring the other cases, it is also the truncated *rectified* square tiling, + since rectifying the square tiling merely produces another square tiling (rotated 45°). +In this tiling, the squares always correspond to the product *ac*, + but there are octagons for both order-4 products, *ab* or *bc*. + + +### $\widetilde A_3$ + +The above diagrams are the only rank-2 affine diagrams. +The simplest diagram of rank 3 is $\widetilde A_3$, which appears similar to 4-Cycle graph. + +::: {layout-ncol="2" layout-valign="center"} +![](./affine/coxeter_a3tilde_generator_a.png){.lightbox group="a3_tilde"} + +![](./affine/coxeter_a3tilde.png){.lightbox group="a3_tilde"} +::: + +Similar to how $\widetilde A_2$'s graph is the tiling of $A_2$'s, + its Cayley graph is the honeycomb of $A_3$'s. +The left image shows the initial branches, while the right image shows + the four distinct cells which form the honeycomb. +From left to right, the figures contain only edges labelled from + *abd*, *abc*, *bcd*, and *acd*, respectively. +The squares always represent commuting pairs on opposite ends of the diagram: *ac* and *bd*. + +I am not certain in general whether $\widetilde A_n$ generates the honeycomb formed by $A_n$, + but this tessellation should always exist, since the solids formed + by the generators of $A_n$ are always permutohedra. + + +Closing +------- + +The diagrams I have studied here are only the smaller ones which I can either visualize or compute. +I might have gotten carried away with studying the groups themselves in the 4D case, + since there appears to be so much contention. +Despite this, I examined only half of the available Platonic solids in 4D, missing out on + the 24-cell (Schläfli symbol {3, 4, 3}, $F_4$), + 120-cell (Schläfli symbol {5, 3, 3}, $H_4$), + and 600-cell (Schläfli symbol {3, 3, 5}, $H_4$). +If 4D symmetries are hard to understand, then things can only get worse in higher dimensions. + +One of the things I'm curious about is the minimal degree of symmetric group + needed to embed the finite diagrams (hereafter $\mu$). +While $S_6$ is minimal for the octahedral group, it doesn't appear to be big enough + for the icosahedral one. +There is a trivial embedding for groups which are direct products given by + +$$ +\begin{align*} + G &\hookrightarrow S_m \sub S_{m+n} \\ + H &\hookrightarrow S_n \sub S_{m+n} \\ + G \times H &\hookrightarrow S_{m+n} \\ + \mu(G \times H) &\le \mu(G) + \mu(H) = m + n +\end{align*} +$$ + +but the groups we're interested in don't necessarily have this property. + +Coset and embedding diagrams made with GeoGebra. +Cayley graph images made with NetworkX (GraphViz). + + +### Additional Links + +- [Point groups in three dimensions](https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions) (Wikipedia) +- [Point groups in four dimensions](https://en.wikipedia.org/wiki/Point_groups_in_four_dimensions) (Wikipedia) +- [ + Finding the minimal n such that a given finite group G is a subgroup of Sn + ](https://math.stackexchange.com/questions/1597347/finding-the-minimal-n-such-that-a-given-finite-group-g-is-a-subgroup-of-s-n) + (Mathematics Stack Exchange) +- [Omnitruncated](https://community.wolfram.com/groups/-/m/t/774393), by Clayton Shonkwiler diff --git a/posts/permutations/appendix/tetrahedron_wikipedia.png b/posts/permutations/appendix/tetrahedron_wikipedia.png new file mode 100644 index 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--git a/posts/polycount/index.qmd b/posts/polycount/index.qmd index c9fdf1e..9eac765 100644 --- a/posts/polycount/index.qmd +++ b/posts/polycount/index.qmd @@ -2,7 +2,7 @@ title: "Polynomial Counting" listing: contents: . - sort: "date desc" + sort: "date" --- Articles about generalizations of positional number systems, explained using the concept of counting on polynomials. From 5fbf8970c2f73703f8c583556bdb2f30f89b64b0 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Mon, 21 Jul 2025 23:45:41 -0500 Subject: [PATCH 07/14] add old c files for pade approximants --- posts/stereo/1/stereo_complex.c | 143 ++++++++++++++++++++++++++++++++ posts/stereo/1/stereo_math.c | 128 ++++++++++++++++++++++++++++ 2 files changed, 271 insertions(+) create mode 100644 posts/stereo/1/stereo_complex.c create mode 100644 posts/stereo/1/stereo_math.c diff --git a/posts/stereo/1/stereo_complex.c b/posts/stereo/1/stereo_complex.c new file mode 100644 index 0000000..a3a42f9 --- /dev/null +++ b/posts/stereo/1/stereo_complex.c @@ -0,0 +1,143 @@ +#include +#include +#include +#include +#include + +#define STRRED "\x1b[31m" +#define STRGREEN "\x1b[32m" +#define STRNORM "\x1b[m" + +#define NUM_LOOPS 100000 + +double complex complex_turn(double turn) +{ + return cexp(I*M_PI*turn); +} + +double complex approx_turn(double turn) +{ + const static double a = 8*M_SQRT2/3 - 3; + const static double b = 4 - 8*M_SQRT2/3; + + double p = turn*(b * turn * turn + a); + double q = p*p; + double r = 1 + q; + double c = (1 - q) / r, s = 2*p / r; + + return (c*c - s*s) + I*(2*c*s); +} + +void print_errors(const double *inputs, + const double complex *ideals, + const double complex *approxs, + int n) +{ + double c_error, s_error; + double largest_c_error, largest_s_error; + size_t largest_c_index, largest_s_index; + + double total_c_error = 0, total_s_error = 0; + + size_t i; + double complex ideal, approx; + + for (i = 0; i < n; i++) { + ideal = ideals[i]; + approx = approxs[i]; + + //squared error in c components + c_error = creal(ideal) - creal(approx); + c_error *= c_error; + //squared error in s components + s_error = cimag(ideal) - cimag(approx); + s_error *= s_error; + + if (largest_c_error < c_error) { + largest_c_error = c_error; + largest_c_index = i; + } + + if (largest_s_error < s_error) { + largest_s_error = s_error; + largest_s_index = i; + } + + total_c_error += c_error; + total_s_error += s_error; + } + //these now contain the *average* squared error + total_c_error /= (double) n; + total_s_error /= (double) n; + + printf("Squared error in cosines: \n"\ + "\tAverage: %f (%f%% error)\n""\tLargest: %f (%f%% error)" \ + "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n" + , total_c_error, sqrt(total_c_error) * 100 + , largest_c_error, sqrt(largest_c_error) * 100 + , inputs[largest_c_index] + , creal(ideals[largest_c_index]), creal(approxs[largest_c_index])); + printf("Squared error in sines: \n"\ + "\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)" \ + "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n" + , total_s_error, sqrt(total_s_error) * 100 + , largest_s_error, sqrt(largest_s_error) * 100 + , inputs[largest_c_index] + , cimag(ideals[largest_s_index]), cimag(approxs[largest_s_index])); +} + +//time the length of the computation `f` in nanoseconds +long time_computation( double complex (*f)(double), + const double *inputs, + double complex *results, + int n) +{ + size_t i; + long tick; + struct timespec tp; + + clock_gettime(CLOCK_MONOTONIC, &tp); + tick = tp.tv_nsec; + for (i = 0; i < n; i++) { + results[i] = f(inputs[i]); + } + //this isn't quite proper, since the clock may have ticked over a second + clock_gettime(CLOCK_MONOTONIC, &tp); + + return tp.tv_nsec - tick; +} + +int main(int argn, char **args) +{ + long trig_time, rat_time; + + double rands[NUM_LOOPS]; + double complex trigs[NUM_LOOPS]; + double complex rats[NUM_LOOPS]; + + size_t i; + for (i = 0; i < NUM_LOOPS; i++) { + rands[i] = rand() / (double) RAND_MAX; + } + + trig_time = time_computation(&complex_turn, rands, trigs, NUM_LOOPS); + printf("Timing for %d math.h sin and cos:\t%ldns\n", NUM_LOOPS, trig_time); + + rat_time = time_computation(&approx_turn, rands, rats, NUM_LOOPS); + printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); + + long diff = rat_time - trig_time; + double frac_speed; + if (diff > 0) { + frac_speed = rat_time / (double) trig_time; + printf(STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", + diff, frac_speed); + } else { + frac_speed = trig_time / (double) rat_time; + printf(STRGREEN "Approximation" STRNORM " faster, speedup: %ldns (%2.2fx)\n", + -diff, frac_speed); + print_errors(rands, trigs, rats, NUM_LOOPS); + } + + return 0; +} diff --git a/posts/stereo/1/stereo_math.c b/posts/stereo/1/stereo_math.c new file mode 100644 index 0000000..d841204 --- /dev/null +++ b/posts/stereo/1/stereo_math.c @@ -0,0 +1,128 @@ +#include +#include +#include +#include + +#define STRRED "\x1b[31m" +#define STRGREEN "\x1b[32m" +#define STRNORM "\x1b[m" + +struct circle { + double c; + double s; +}; + +double a = 8*M_SQRT2/3 - 3; +double b = 4 - 8*M_SQRT2/3; + +void trig(double turn, struct circle *ret) +{ + double arg = M_PI * turn; + ret->c = cos(arg); + ret->s = sin(arg); +} + +void rational(double turn, struct circle *ret) +{ + double p = turn*(b * turn * turn + a); + double q = p*p; + double r = 1 + q; + double c = (1 - q) / r, s = 2*p / r; + + ret->c = c*c - s*s; + ret->s = 2*c*s; +} + +double errors(int n, const struct circle *circles1, const struct circle *circles2) +{ + double c_error, s_error; + double largest_c_error, largest_s_error; + + double total_c_error = 0, total_s_error = 0; + + int i; + struct circle circle1, circle2; + + for (i = 0; i < n; i++) { + circle1 = circles1[i]; + circle2 = circles2[i]; + + //squared error in c components + c_error = circle1.c - circle2.c; + c_error *= c_error; + //squared error in s components + s_error = circle1.s - circle2.s; + s_error *= s_error; + + if (largest_c_error < c_error) + largest_c_error = c_error; + + if (largest_s_error < s_error) + largest_s_error = s_error; + + total_c_error += c_error; + total_s_error += s_error; + } + //these now contain the *average* squared error + total_c_error /= (double) n; + total_s_error /= (double) n; + + printf("Squared error in cosines: \n\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)\n" + , total_c_error, sqrt(total_c_error) * 100 + , largest_c_error, sqrt(largest_c_error) * 100); + printf("Squared error in sines: \n\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)\n" + , total_s_error, sqrt(total_s_error) * 100 + , largest_s_error, sqrt(largest_s_error) * 100); + + return 0; +} + +int main(int argn, char **args) +{ + //struct circle ret; + int i; + + struct timespec tp; + long tick; + long trig_time, rat_time; + + double rands[10000]; + struct circle trigs[10000]; + struct circle rats[10000]; + + for (i = 0; i < 10000; i++) { + rands[i] = rand() / (double) RAND_MAX; + } + + clock_gettime(CLOCK_MONOTONIC, &tp); + tick = tp.tv_nsec; + for (i = 0; i < 10000; i++) { + trig(rands[i], trigs+i); + } + clock_gettime(CLOCK_MONOTONIC, &tp); + // this isn't quite proper, since the clock may have ticked over a second + trig_time = tp.tv_nsec - tick; + printf("Timing for 10000 math.h sin and cos:\t%ldns\n", trig_time); + + clock_gettime(CLOCK_MONOTONIC, &tp); + tick = tp.tv_nsec; + for (i = 0; i < 10000; i++) { + rational(rands[i], rats+i); + } + clock_gettime(CLOCK_MONOTONIC, &tp); + rat_time = tp.tv_nsec - tick; + printf("Timing for 10000 approximations:\t%ldns\n", rat_time); + + double fracSpeed; + long linSpeed = rat_time - trig_time; + if (linSpeed > 0) { + fracSpeed = rat_time / (double) trig_time; + printf(STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", + linSpeed, fracSpeed); + } else { + fracSpeed = trig_time / (double) rat_time; + printf(STRGREEN "Approximation" STRNORM " faster, speedup: %ldns (%2.2fx)\n", + -linSpeed, fracSpeed); + errors(10000, rats, trigs); + } +} From 32294879c9750c58d1e4c01c535b63578edc55a2 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 00:16:00 -0500 Subject: [PATCH 08/14] clang-format stereography c files --- posts/stereo/1/.clang-format | 9 ++ posts/stereo/1/stereo_complex.c | 227 +++++++++++++++++--------------- posts/stereo/1/stereo_math.c | 194 ++++++++++++++------------- 3 files changed, 232 insertions(+), 198 deletions(-) create mode 100644 posts/stereo/1/.clang-format diff --git a/posts/stereo/1/.clang-format b/posts/stereo/1/.clang-format new file mode 100644 index 0000000..120882a --- /dev/null +++ b/posts/stereo/1/.clang-format @@ -0,0 +1,9 @@ +BasedOnStyle: llvm +IndentWidth: 4 + +AlignAfterOpenBracket: BlockIndent +BinPackParameters: OnePerLine +BreakBeforeBraces: Custom +BraceWrapping: + AfterFunction: true +PointerAlignment: Left diff --git a/posts/stereo/1/stereo_complex.c b/posts/stereo/1/stereo_complex.c index a3a42f9..b898262 100644 --- a/posts/stereo/1/stereo_complex.c +++ b/posts/stereo/1/stereo_complex.c @@ -1,143 +1,156 @@ +#include +#include #include #include #include -#include -#include -#define STRRED "\x1b[31m" +#define STRRED "\x1b[31m" #define STRGREEN "\x1b[32m" -#define STRNORM "\x1b[m" +#define STRNORM "\x1b[m" + +#define SECONDS_PER_NANOSECOND 1000000000 #define NUM_LOOPS 100000 -double complex complex_turn(double turn) -{ - return cexp(I*M_PI*turn); -} +double complex complex_turn(double turn) { return cexp(I * M_PI * turn); } double complex approx_turn(double turn) { - const static double a = 8*M_SQRT2/3 - 3; - const static double b = 4 - 8*M_SQRT2/3; + const static double a = 8 * M_SQRT2 / 3 - 3; + const static double b = 4 - 8 * M_SQRT2 / 3; - double p = turn*(b * turn * turn + a); - double q = p*p; - double r = 1 + q; - double c = (1 - q) / r, s = 2*p / r; - - return (c*c - s*s) + I*(2*c*s); + double p = turn * (b * turn * turn + a); + double q = p * p; + double r = 1 + q; + double c = (1 - q) / r, s = 2 * p / r; + + return (c * c - s * s) + I * (2 * c * s); } -void print_errors(const double *inputs, - const double complex *ideals, - const double complex *approxs, - int n) +void print_errors( + const double* inputs, + const double complex* ideals, + const double complex* approxs, + int n +) { - double c_error, s_error; - double largest_c_error, largest_s_error; - size_t largest_c_index, largest_s_index; + double c_error, s_error; + double largest_c_error, largest_s_error; + size_t largest_c_index, largest_s_index; - double total_c_error = 0, total_s_error = 0; + double total_c_error = 0, total_s_error = 0; - size_t i; - double complex ideal, approx; + size_t i; + double complex ideal, approx; - for (i = 0; i < n; i++) { - ideal = ideals[i]; - approx = approxs[i]; + for (i = 0; i < n; i++) { + ideal = ideals[i]; + approx = approxs[i]; - //squared error in c components - c_error = creal(ideal) - creal(approx); - c_error *= c_error; - //squared error in s components - s_error = cimag(ideal) - cimag(approx); - s_error *= s_error; - - if (largest_c_error < c_error) { - largest_c_error = c_error; - largest_c_index = i; - } - - if (largest_s_error < s_error) { - largest_s_error = s_error; - largest_s_index = i; - } + // squared error in c components + c_error = creal(ideal) - creal(approx); + c_error *= c_error; + // squared error in s components + s_error = cimag(ideal) - cimag(approx); + s_error *= s_error; - total_c_error += c_error; - total_s_error += s_error; - } - //these now contain the *average* squared error - total_c_error /= (double) n; - total_s_error /= (double) n; + if (largest_c_error < c_error) { + largest_c_error = c_error; + largest_c_index = i; + } - printf("Squared error in cosines: \n"\ - "\tAverage: %f (%f%% error)\n""\tLargest: %f (%f%% error)" \ - "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n" - , total_c_error, sqrt(total_c_error) * 100 - , largest_c_error, sqrt(largest_c_error) * 100 - , inputs[largest_c_index] - , creal(ideals[largest_c_index]), creal(approxs[largest_c_index])); - printf("Squared error in sines: \n"\ - "\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)" \ - "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n" - , total_s_error, sqrt(total_s_error) * 100 - , largest_s_error, sqrt(largest_s_error) * 100 - , inputs[largest_c_index] - , cimag(ideals[largest_s_index]), cimag(approxs[largest_s_index])); + if (largest_s_error < s_error) { + largest_s_error = s_error; + largest_s_index = i; + } + + total_c_error += c_error; + total_s_error += s_error; + } + // these now contain the *average* squared error + total_c_error /= (double)n; + total_s_error /= (double)n; + + printf( + "Squared error in cosines: \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)" + "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n", + total_c_error, sqrt(total_c_error) * 100, largest_c_error, + sqrt(largest_c_error) * 100, inputs[largest_c_index], + creal(ideals[largest_c_index]), creal(approxs[largest_c_index]) + ); + printf( + "Squared error in sines: \n" + "\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)" + "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n", + total_s_error, sqrt(total_s_error) * 100, largest_s_error, + sqrt(largest_s_error) * 100, inputs[largest_c_index], + cimag(ideals[largest_s_index]), cimag(approxs[largest_s_index]) + ); } -//time the length of the computation `f` in nanoseconds -long time_computation( double complex (*f)(double), - const double *inputs, - double complex *results, - int n) +// time the length of the computation `f` in nanoseconds +long time_computation( + double complex (*f)(double), + const double* inputs, + double complex* results, + int n +) { - size_t i; - long tick; - struct timespec tp; + size_t i; + long tick_s; + long tick_ns; + struct timespec tp; - clock_gettime(CLOCK_MONOTONIC, &tp); - tick = tp.tv_nsec; - for (i = 0; i < n; i++) { - results[i] = f(inputs[i]); - } - //this isn't quite proper, since the clock may have ticked over a second - clock_gettime(CLOCK_MONOTONIC, &tp); + clock_gettime(CLOCK_MONOTONIC, &tp); + tick_ns = tp.tv_nsec; + tick_s = tp.tv_sec; + for (i = 0; i < n; i++) { + results[i] = f(inputs[i]); + } + clock_gettime(CLOCK_MONOTONIC, &tp); - return tp.tv_nsec - tick; + return SECONDS_PER_NANOSECOND * (tp.tv_sec - tick_s) + + (tp.tv_nsec - tick_ns); } -int main(int argn, char **args) +int main(int argn, char** args) { - long trig_time, rat_time; + long trig_time, rat_time; - double rands[NUM_LOOPS]; - double complex trigs[NUM_LOOPS]; - double complex rats[NUM_LOOPS]; + double rands[NUM_LOOPS]; + double complex trigs[NUM_LOOPS]; + double complex rats[NUM_LOOPS]; - size_t i; - for (i = 0; i < NUM_LOOPS; i++) { - rands[i] = rand() / (double) RAND_MAX; - } + size_t i; + for (i = 0; i < NUM_LOOPS; i++) { + rands[i] = rand() / (double)RAND_MAX; + } - trig_time = time_computation(&complex_turn, rands, trigs, NUM_LOOPS); - printf("Timing for %d math.h sin and cos:\t%ldns\n", NUM_LOOPS, trig_time); + trig_time = time_computation(&complex_turn, rands, trigs, NUM_LOOPS); + printf("Timing for %d math.h sin and cos:\t%ldns\n", NUM_LOOPS, trig_time); - rat_time = time_computation(&approx_turn, rands, rats, NUM_LOOPS); - printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); + rat_time = time_computation(&approx_turn, rands, rats, NUM_LOOPS); + printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); - long diff = rat_time - trig_time; - double frac_speed; - if (diff > 0) { - frac_speed = rat_time / (double) trig_time; - printf(STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", - diff, frac_speed); - } else { - frac_speed = trig_time / (double) rat_time; - printf(STRGREEN "Approximation" STRNORM " faster, speedup: %ldns (%2.2fx)\n", - -diff, frac_speed); - print_errors(rands, trigs, rats, NUM_LOOPS); - } + long diff = rat_time - trig_time; + double frac_speed; + if (diff > 0) { + frac_speed = rat_time / (double)trig_time; + printf( + STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", diff, + frac_speed + ); + } else { + frac_speed = trig_time / (double)rat_time; + printf( + STRGREEN "Approximation" STRNORM + " faster, speedup: %ldns (%2.2fx)\n", + -diff, frac_speed + ); + print_errors(rands, trigs, rats, NUM_LOOPS); + } - return 0; + return 0; } diff --git a/posts/stereo/1/stereo_math.c b/posts/stereo/1/stereo_math.c index d841204..2086c1f 100644 --- a/posts/stereo/1/stereo_math.c +++ b/posts/stereo/1/stereo_math.c @@ -1,128 +1,140 @@ -#include #include +#include #include #include -#define STRRED "\x1b[31m" +#define STRRED "\x1b[31m" #define STRGREEN "\x1b[32m" -#define STRNORM "\x1b[m" +#define STRNORM "\x1b[m" struct circle { - double c; - double s; + double c; + double s; }; -double a = 8*M_SQRT2/3 - 3; -double b = 4 - 8*M_SQRT2/3; +double a = 8 * M_SQRT2 / 3 - 3; +double b = 4 - 8 * M_SQRT2 / 3; -void trig(double turn, struct circle *ret) +void trig(double turn, struct circle* ret) { - double arg = M_PI * turn; - ret->c = cos(arg); - ret->s = sin(arg); + double arg = M_PI * turn; + ret->c = cos(arg); + ret->s = sin(arg); } -void rational(double turn, struct circle *ret) +void rational(double turn, struct circle* ret) { - double p = turn*(b * turn * turn + a); - double q = p*p; - double r = 1 + q; - double c = (1 - q) / r, s = 2*p / r; + double p = turn * (b * turn * turn + a); + double q = p * p; + double r = 1 + q; + double c = (1 - q) / r, s = 2 * p / r; - ret->c = c*c - s*s; - ret->s = 2*c*s; + ret->c = c * c - s * s; + ret->s = 2 * c * s; } -double errors(int n, const struct circle *circles1, const struct circle *circles2) +double +errors(int n, const struct circle* circles1, const struct circle* circles2) { - double c_error, s_error; - double largest_c_error, largest_s_error; + double c_error, s_error; + double largest_c_error, largest_s_error; - double total_c_error = 0, total_s_error = 0; + double total_c_error = 0, total_s_error = 0; - int i; - struct circle circle1, circle2; + int i; + struct circle circle1, circle2; - for (i = 0; i < n; i++) { - circle1 = circles1[i]; - circle2 = circles2[i]; + for (i = 0; i < n; i++) { + circle1 = circles1[i]; + circle2 = circles2[i]; - //squared error in c components - c_error = circle1.c - circle2.c; - c_error *= c_error; - //squared error in s components - s_error = circle1.s - circle2.s; - s_error *= s_error; + // squared error in c components + c_error = circle1.c - circle2.c; + c_error *= c_error; + // squared error in s components + s_error = circle1.s - circle2.s; + s_error *= s_error; - if (largest_c_error < c_error) - largest_c_error = c_error; + if (largest_c_error < c_error) + largest_c_error = c_error; - if (largest_s_error < s_error) - largest_s_error = s_error; + if (largest_s_error < s_error) + largest_s_error = s_error; - total_c_error += c_error; - total_s_error += s_error; - } - //these now contain the *average* squared error - total_c_error /= (double) n; - total_s_error /= (double) n; + total_c_error += c_error; + total_s_error += s_error; + } + // these now contain the *average* squared error + total_c_error /= (double)n; + total_s_error /= (double)n; - printf("Squared error in cosines: \n\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)\n" - , total_c_error, sqrt(total_c_error) * 100 - , largest_c_error, sqrt(largest_c_error) * 100); - printf("Squared error in sines: \n\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)\n" - , total_s_error, sqrt(total_s_error) * 100 - , largest_s_error, sqrt(largest_s_error) * 100); + printf( + "Squared error in cosines: \n\tAverage: %f (%f%% error)\n\tLargest: %f " + "(%f%% error)\n", + total_c_error, sqrt(total_c_error) * 100, largest_c_error, + sqrt(largest_c_error) * 100 + ); + printf( + "Squared error in sines: \n\tAverage: %f (%f%% error)\n\tLargest: %f " + "(%f%% error)\n", + total_s_error, sqrt(total_s_error) * 100, largest_s_error, + sqrt(largest_s_error) * 100 + ); - return 0; + return 0; } -int main(int argn, char **args) +int main(int argn, char** args) { - //struct circle ret; - int i; + // struct circle ret; + int i; - struct timespec tp; - long tick; - long trig_time, rat_time; + struct timespec tp; + long tick; + long trig_time, rat_time; - double rands[10000]; - struct circle trigs[10000]; - struct circle rats[10000]; + double rands[10000]; + struct circle trigs[10000]; + struct circle rats[10000]; - for (i = 0; i < 10000; i++) { - rands[i] = rand() / (double) RAND_MAX; - } + for (i = 0; i < 10000; i++) { + rands[i] = rand() / (double)RAND_MAX; + } - clock_gettime(CLOCK_MONOTONIC, &tp); - tick = tp.tv_nsec; - for (i = 0; i < 10000; i++) { - trig(rands[i], trigs+i); - } - clock_gettime(CLOCK_MONOTONIC, &tp); - // this isn't quite proper, since the clock may have ticked over a second - trig_time = tp.tv_nsec - tick; - printf("Timing for 10000 math.h sin and cos:\t%ldns\n", trig_time); + clock_gettime(CLOCK_MONOTONIC, &tp); + tick = tp.tv_nsec; + for (i = 0; i < 10000; i++) { + trig(rands[i], trigs + i); + } + clock_gettime(CLOCK_MONOTONIC, &tp); + // this isn't quite proper, since the clock may have ticked over a second + trig_time = tp.tv_nsec - tick; + printf("Timing for 10000 math.h sin and cos:\t%ldns\n", trig_time); - clock_gettime(CLOCK_MONOTONIC, &tp); - tick = tp.tv_nsec; - for (i = 0; i < 10000; i++) { - rational(rands[i], rats+i); - } - clock_gettime(CLOCK_MONOTONIC, &tp); - rat_time = tp.tv_nsec - tick; - printf("Timing for 10000 approximations:\t%ldns\n", rat_time); + clock_gettime(CLOCK_MONOTONIC, &tp); + tick = tp.tv_nsec; + for (i = 0; i < 10000; i++) { + rational(rands[i], rats + i); + } + clock_gettime(CLOCK_MONOTONIC, &tp); + rat_time = tp.tv_nsec - tick; + printf("Timing for 10000 approximations:\t%ldns\n", rat_time); - double fracSpeed; - long linSpeed = rat_time - trig_time; - if (linSpeed > 0) { - fracSpeed = rat_time / (double) trig_time; - printf(STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", - linSpeed, fracSpeed); - } else { - fracSpeed = trig_time / (double) rat_time; - printf(STRGREEN "Approximation" STRNORM " faster, speedup: %ldns (%2.2fx)\n", - -linSpeed, fracSpeed); - errors(10000, rats, trigs); - } + double fracSpeed; + long linSpeed = rat_time - trig_time; + if (linSpeed > 0) { + fracSpeed = rat_time / (double)trig_time; + printf( + STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", + linSpeed, fracSpeed + ); + } else { + fracSpeed = trig_time / (double)rat_time; + printf( + STRGREEN "Approximation" STRNORM + " faster, speedup: %ldns (%2.2fx)\n", + -linSpeed, fracSpeed + ); + errors(10000, rats, trigs); + } } From 86938469af654dd6efdd5b12a7199dd988c82ea6 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 01:19:57 -0500 Subject: [PATCH 09/14] refactor c files --- posts/stereo/1/.gitignore | 2 + posts/stereo/1/Makefile | 10 ++ posts/stereo/1/main.c | 168 ++++++++++++++++++++++++++++++++ posts/stereo/1/stereo_complex.c | 143 ++------------------------- posts/stereo/1/stereo_math.c | 127 ++---------------------- 5 files changed, 194 insertions(+), 256 deletions(-) create mode 100644 posts/stereo/1/.gitignore create mode 100644 posts/stereo/1/Makefile create mode 100644 posts/stereo/1/main.c diff --git a/posts/stereo/1/.gitignore b/posts/stereo/1/.gitignore new file mode 100644 index 0000000..07337da --- /dev/null +++ b/posts/stereo/1/.gitignore @@ -0,0 +1,2 @@ +compare_complex +compare_math diff --git a/posts/stereo/1/Makefile b/posts/stereo/1/Makefile new file mode 100644 index 0000000..5cfffc1 --- /dev/null +++ b/posts/stereo/1/Makefile @@ -0,0 +1,10 @@ +default: compare_complex compare_math + +clean: + rm -rf compare_complex compare_math + +compare_complex: + gcc main.c stereo_complex.c -lm -DUSE_COMPLEX -o compare_complex + +compare_math: + gcc main.c stereo_math.c -lm -o compare_math diff --git a/posts/stereo/1/main.c b/posts/stereo/1/main.c new file mode 100644 index 0000000..074d565 --- /dev/null +++ b/posts/stereo/1/main.c @@ -0,0 +1,168 @@ +#include +#include +#include +#include +#include + +#define STR_RED "\x1b[31m" +#define STR_GREEN "\x1b[32m" +#define STR_NORM "\x1b[m" + +#define SECONDS_PER_NANOSECOND 1000000000 +#define NUM_LOOPS 100000 +#define SQRT_NUM_LOOPS 100 + +struct circle { + double c; + double s; +}; + +void turn_update(double turn, void* result); +void approx_turn_update(double turn, void* result); + +#ifdef USE_COMPLEX +#define EXTRACT_REAL(a) creal(a) +#define EXTRACT_IMAG(a) cimag(a) +#define CIRCLE_TYPE double complex +#else +#define EXTRACT_REAL(a) (a.c) +#define EXTRACT_IMAG(a) (a.s) +#define CIRCLE_TYPE struct circle +#endif + +void print_errors( + const double* inputs, + const CIRCLE_TYPE* ideals, + const CIRCLE_TYPE* approxs, + int n +) +{ + double c_error, s_error; + double largest_c_error, largest_s_error; + size_t largest_c_index, largest_s_index; + + double total_c_error = 0, total_s_error = 0; + + size_t i; + CIRCLE_TYPE ideal, approx; + + for (i = 0; i < n; i++) { + ideal = ideals[i]; + approx = approxs[i]; + + // squared error in c components + c_error = EXTRACT_REAL(ideal) - EXTRACT_REAL(approx); + c_error *= c_error; + // squared error in s components + s_error = EXTRACT_IMAG(ideal) - EXTRACT_IMAG(approx); + s_error *= s_error; + + if (largest_c_error < c_error) { + largest_c_error = c_error; + largest_c_index = i; + } + + if (largest_s_error < s_error) { + largest_s_error = s_error; + largest_s_index = i; + } + + total_c_error += c_error; + total_s_error += s_error; + } + // these now contain the *average* squared error + total_c_error /= (double)n; + total_s_error /= (double)n; + + printf( + "Squared error in cosines: \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + total_c_error, sqrt(total_c_error) * SQRT_NUM_LOOPS, largest_c_error, + sqrt(largest_c_error) * SQRT_NUM_LOOPS, inputs[largest_c_index], + EXTRACT_REAL(ideals[largest_c_index]), + EXTRACT_REAL(approxs[largest_c_index]) + ); + printf( + "Squared error in sines: \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + total_s_error, sqrt(total_s_error) * SQRT_NUM_LOOPS, largest_s_error, + sqrt(largest_s_error) * SQRT_NUM_LOOPS, inputs[largest_c_index], + EXTRACT_IMAG(ideals[largest_s_index]), + EXTRACT_IMAG(approxs[largest_s_index]) + ); +} + +// time the length of the computation `f` in nanoseconds +long time_computation( + void (*f)(double, void*), const double* inputs, CIRCLE_TYPE* results, int n +) +{ + size_t i; + struct timespec tp1; + struct timespec tp2; + + clock_gettime(CLOCK_MONOTONIC, &tp1); + for (i = 0; i < n; i++) { + f(inputs[i], results + i); + } + clock_gettime(CLOCK_MONOTONIC, &tp2); + + return SECONDS_PER_NANOSECOND * (tp2.tv_sec - tp1.tv_sec) + + (tp2.tv_nsec - tp1.tv_sec); +} + +int main(int argn, char** args) +{ + long trig_time, rat_time; + + double rands[NUM_LOOPS]; + CIRCLE_TYPE trigs[NUM_LOOPS]; + CIRCLE_TYPE rats[NUM_LOOPS]; + + size_t i; + for (i = 0; i < NUM_LOOPS; i++) { + rands[i] = rand() / (double)RAND_MAX; + } + + trig_time = time_computation(&turn_update, rands, trigs, NUM_LOOPS); + printf( +#ifdef USE_COMPLEX + "Timing for %d complex.h cexp:\t%ldns\n", +#else + "Timing for %d math.h sin and cos:\t%ldns\n", +#endif + NUM_LOOPS, + trig_time + ); + + rat_time = time_computation(&approx_turn_update, rands, rats, NUM_LOOPS); + printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); + + long diff = rat_time - trig_time; + double frac_speed; + if (diff > 0) { + frac_speed = rat_time / (double)trig_time; + printf( + STR_RED "stdlib" STR_NORM " faster, speedup: %ldns (%2.2fx)\n", + diff, frac_speed + ); + } else { + frac_speed = trig_time / (double)rat_time; + printf( + STR_GREEN "Approximation" STR_NORM + " faster, speedup: %ldns (%2.2fx)\n", + -diff, frac_speed + ); + print_errors(rands, trigs, rats, NUM_LOOPS); + } + + return 0; +} diff --git a/posts/stereo/1/stereo_complex.c b/posts/stereo/1/stereo_complex.c index b898262..c63e4b3 100644 --- a/posts/stereo/1/stereo_complex.c +++ b/posts/stereo/1/stereo_complex.c @@ -1,20 +1,7 @@ #include #include -#include -#include -#include -#define STRRED "\x1b[31m" -#define STRGREEN "\x1b[32m" -#define STRNORM "\x1b[m" - -#define SECONDS_PER_NANOSECOND 1000000000 - -#define NUM_LOOPS 100000 - -double complex complex_turn(double turn) { return cexp(I * M_PI * turn); } - -double complex approx_turn(double turn) +double complex complex_approx_turn(double turn) { const static double a = 8 * M_SQRT2 / 3 - 3; const static double b = 4 - 8 * M_SQRT2 / 3; @@ -27,130 +14,14 @@ double complex approx_turn(double turn) return (c * c - s * s) + I * (2 * c * s); } -void print_errors( - const double* inputs, - const double complex* ideals, - const double complex* approxs, - int n -) +double complex complex_turn(double turn) { return cexp(I * M_PI * turn); } + +void turn_update(double turn, double complex* result) { - double c_error, s_error; - double largest_c_error, largest_s_error; - size_t largest_c_index, largest_s_index; - - double total_c_error = 0, total_s_error = 0; - - size_t i; - double complex ideal, approx; - - for (i = 0; i < n; i++) { - ideal = ideals[i]; - approx = approxs[i]; - - // squared error in c components - c_error = creal(ideal) - creal(approx); - c_error *= c_error; - // squared error in s components - s_error = cimag(ideal) - cimag(approx); - s_error *= s_error; - - if (largest_c_error < c_error) { - largest_c_error = c_error; - largest_c_index = i; - } - - if (largest_s_error < s_error) { - largest_s_error = s_error; - largest_s_index = i; - } - - total_c_error += c_error; - total_s_error += s_error; - } - // these now contain the *average* squared error - total_c_error /= (double)n; - total_s_error /= (double)n; - - printf( - "Squared error in cosines: \n" - "\tAverage: %f (%f%% error)\n" - "\tLargest: %f (%f%% error)" - "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n", - total_c_error, sqrt(total_c_error) * 100, largest_c_error, - sqrt(largest_c_error) * 100, inputs[largest_c_index], - creal(ideals[largest_c_index]), creal(approxs[largest_c_index]) - ); - printf( - "Squared error in sines: \n" - "\tAverage: %f (%f%% error)\n\tLargest: %f (%f%% error)" - "\n\t\tInput:\t\t%f\n\t\tValue:\t\t%f\n\t\tApproximation:\t%f\n", - total_s_error, sqrt(total_s_error) * 100, largest_s_error, - sqrt(largest_s_error) * 100, inputs[largest_c_index], - cimag(ideals[largest_s_index]), cimag(approxs[largest_s_index]) - ); + *result = complex_turn(turn); } -// time the length of the computation `f` in nanoseconds -long time_computation( - double complex (*f)(double), - const double* inputs, - double complex* results, - int n -) +void approx_turn_update(double turn, double complex* result) { - size_t i; - long tick_s; - long tick_ns; - struct timespec tp; - - clock_gettime(CLOCK_MONOTONIC, &tp); - tick_ns = tp.tv_nsec; - tick_s = tp.tv_sec; - for (i = 0; i < n; i++) { - results[i] = f(inputs[i]); - } - clock_gettime(CLOCK_MONOTONIC, &tp); - - return SECONDS_PER_NANOSECOND * (tp.tv_sec - tick_s) + - (tp.tv_nsec - tick_ns); -} - -int main(int argn, char** args) -{ - long trig_time, rat_time; - - double rands[NUM_LOOPS]; - double complex trigs[NUM_LOOPS]; - double complex rats[NUM_LOOPS]; - - size_t i; - for (i = 0; i < NUM_LOOPS; i++) { - rands[i] = rand() / (double)RAND_MAX; - } - - trig_time = time_computation(&complex_turn, rands, trigs, NUM_LOOPS); - printf("Timing for %d math.h sin and cos:\t%ldns\n", NUM_LOOPS, trig_time); - - rat_time = time_computation(&approx_turn, rands, rats, NUM_LOOPS); - printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); - - long diff = rat_time - trig_time; - double frac_speed; - if (diff > 0) { - frac_speed = rat_time / (double)trig_time; - printf( - STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", diff, - frac_speed - ); - } else { - frac_speed = trig_time / (double)rat_time; - printf( - STRGREEN "Approximation" STRNORM - " faster, speedup: %ldns (%2.2fx)\n", - -diff, frac_speed - ); - print_errors(rands, trigs, rats, NUM_LOOPS); - } - - return 0; + *result = complex_approx_turn(turn); } diff --git a/posts/stereo/1/stereo_math.c b/posts/stereo/1/stereo_math.c index 2086c1f..a7025f6 100644 --- a/posts/stereo/1/stereo_math.c +++ b/posts/stereo/1/stereo_math.c @@ -1,29 +1,15 @@ #include -#include -#include -#include - -#define STRRED "\x1b[31m" -#define STRGREEN "\x1b[32m" -#define STRNORM "\x1b[m" struct circle { double c; double s; }; -double a = 8 * M_SQRT2 / 3 - 3; -double b = 4 - 8 * M_SQRT2 / 3; - -void trig(double turn, struct circle* ret) +void approx_turn_update(double turn, struct circle* ret) { - double arg = M_PI * turn; - ret->c = cos(arg); - ret->s = sin(arg); -} + static const double a = 8 * M_SQRT2 / 3 - 3; + static const double b = 4 - 8 * M_SQRT2 / 3; -void rational(double turn, struct circle* ret) -{ double p = turn * (b * turn * turn + a); double q = p * p; double r = 1 + q; @@ -33,108 +19,9 @@ void rational(double turn, struct circle* ret) ret->s = 2 * c * s; } -double -errors(int n, const struct circle* circles1, const struct circle* circles2) +void turn_update(double turn, struct circle* ret) { - double c_error, s_error; - double largest_c_error, largest_s_error; - - double total_c_error = 0, total_s_error = 0; - - int i; - struct circle circle1, circle2; - - for (i = 0; i < n; i++) { - circle1 = circles1[i]; - circle2 = circles2[i]; - - // squared error in c components - c_error = circle1.c - circle2.c; - c_error *= c_error; - // squared error in s components - s_error = circle1.s - circle2.s; - s_error *= s_error; - - if (largest_c_error < c_error) - largest_c_error = c_error; - - if (largest_s_error < s_error) - largest_s_error = s_error; - - total_c_error += c_error; - total_s_error += s_error; - } - // these now contain the *average* squared error - total_c_error /= (double)n; - total_s_error /= (double)n; - - printf( - "Squared error in cosines: \n\tAverage: %f (%f%% error)\n\tLargest: %f " - "(%f%% error)\n", - total_c_error, sqrt(total_c_error) * 100, largest_c_error, - sqrt(largest_c_error) * 100 - ); - printf( - "Squared error in sines: \n\tAverage: %f (%f%% error)\n\tLargest: %f " - "(%f%% error)\n", - total_s_error, sqrt(total_s_error) * 100, largest_s_error, - sqrt(largest_s_error) * 100 - ); - - return 0; -} - -int main(int argn, char** args) -{ - // struct circle ret; - int i; - - struct timespec tp; - long tick; - long trig_time, rat_time; - - double rands[10000]; - struct circle trigs[10000]; - struct circle rats[10000]; - - for (i = 0; i < 10000; i++) { - rands[i] = rand() / (double)RAND_MAX; - } - - clock_gettime(CLOCK_MONOTONIC, &tp); - tick = tp.tv_nsec; - for (i = 0; i < 10000; i++) { - trig(rands[i], trigs + i); - } - clock_gettime(CLOCK_MONOTONIC, &tp); - // this isn't quite proper, since the clock may have ticked over a second - trig_time = tp.tv_nsec - tick; - printf("Timing for 10000 math.h sin and cos:\t%ldns\n", trig_time); - - clock_gettime(CLOCK_MONOTONIC, &tp); - tick = tp.tv_nsec; - for (i = 0; i < 10000; i++) { - rational(rands[i], rats + i); - } - clock_gettime(CLOCK_MONOTONIC, &tp); - rat_time = tp.tv_nsec - tick; - printf("Timing for 10000 approximations:\t%ldns\n", rat_time); - - double fracSpeed; - long linSpeed = rat_time - trig_time; - if (linSpeed > 0) { - fracSpeed = rat_time / (double)trig_time; - printf( - STRRED "math.h" STRNORM " faster, speedup: %ldns (%2.2fx)\n", - linSpeed, fracSpeed - ); - } else { - fracSpeed = trig_time / (double)rat_time; - printf( - STRGREEN "Approximation" STRNORM - " faster, speedup: %ldns (%2.2fx)\n", - -linSpeed, fracSpeed - ); - errors(10000, rats, trigs); - } + double arg = M_PI * turn; + ret->c = cos(arg); + ret->s = sin(arg); } From b2b2dd4e04a31de13fa9d920ccbd182fc1ee3727 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 02:37:59 -0500 Subject: [PATCH 10/14] fixes for stereographic approximation reporting --- posts/stereo/1/main.c | 248 +++++++++++++++++++++--------------------- 1 file changed, 127 insertions(+), 121 deletions(-) diff --git a/posts/stereo/1/main.c b/posts/stereo/1/main.c index 074d565..28d2768 100644 --- a/posts/stereo/1/main.c +++ b/posts/stereo/1/main.c @@ -3,166 +3,172 @@ #include #include #include +#include #define STR_RED "\x1b[31m" #define STR_GREEN "\x1b[32m" #define STR_NORM "\x1b[m" #define SECONDS_PER_NANOSECOND 1000000000 -#define NUM_LOOPS 100000 -#define SQRT_NUM_LOOPS 100 +#define NUM_STATS 100000 +#define SQRT_NUM_STATS 100 +#define NUM_LOOPS 10000000 struct circle { - double c; - double s; + double c; + double s; }; -void turn_update(double turn, void* result); -void approx_turn_update(double turn, void* result); - #ifdef USE_COMPLEX -#define EXTRACT_REAL(a) creal(a) -#define EXTRACT_IMAG(a) cimag(a) +#define EXTRACT_COSINE(a) creal(a) +#define EXTRACT_SINE(a) cimag(a) #define CIRCLE_TYPE double complex #else -#define EXTRACT_REAL(a) (a.c) -#define EXTRACT_IMAG(a) (a.s) +#define EXTRACT_COSINE(a) (a.c) +#define EXTRACT_SINE(a) (a.s) #define CIRCLE_TYPE struct circle #endif -void print_errors( - const double* inputs, - const CIRCLE_TYPE* ideals, - const CIRCLE_TYPE* approxs, - int n -) -{ - double c_error, s_error; - double largest_c_error, largest_s_error; - size_t largest_c_index, largest_s_index; +void turn_update(double turn, CIRCLE_TYPE *result); +void approx_turn_update(double turn, CIRCLE_TYPE *result); - double total_c_error = 0, total_s_error = 0; +void print_errors(const double *inputs, const CIRCLE_TYPE *ideals, + const CIRCLE_TYPE *approxs, int n) { + double c_error, s_error; + double largest_c_error = 0, largest_s_error = 0; + size_t largest_c_index, largest_s_index; - size_t i; - CIRCLE_TYPE ideal, approx; + double total_c_error = 0, total_s_error = 0; - for (i = 0; i < n; i++) { - ideal = ideals[i]; - approx = approxs[i]; + size_t i; + CIRCLE_TYPE ideal, approx; - // squared error in c components - c_error = EXTRACT_REAL(ideal) - EXTRACT_REAL(approx); - c_error *= c_error; - // squared error in s components - s_error = EXTRACT_IMAG(ideal) - EXTRACT_IMAG(approx); - s_error *= s_error; + for (i = 0; i < n; i++) { + ideal = ideals[i]; + approx = approxs[i]; - if (largest_c_error < c_error) { - largest_c_error = c_error; - largest_c_index = i; - } + // squared error in c components + c_error = EXTRACT_COSINE(ideal) - EXTRACT_COSINE(approx); + c_error *= c_error; + // squared error in s components + s_error = EXTRACT_SINE(ideal) - EXTRACT_SINE(approx); + s_error *= s_error; - if (largest_s_error < s_error) { - largest_s_error = s_error; - largest_s_index = i; - } - - total_c_error += c_error; - total_s_error += s_error; + if (largest_c_error < c_error) { + largest_c_error = c_error; + largest_c_index = i; } - // these now contain the *average* squared error - total_c_error /= (double)n; - total_s_error /= (double)n; - printf( - "Squared error in cosines: \n" - "\tAverage: %f (%f%% error)\n" - "\tLargest: %f (%f%% error)\n" - "\t\tInput:\t\t%f\n" - "\t\tValue:\t\t%f\n" - "\t\tApproximation:\t%f\n", - total_c_error, sqrt(total_c_error) * SQRT_NUM_LOOPS, largest_c_error, - sqrt(largest_c_error) * SQRT_NUM_LOOPS, inputs[largest_c_index], - EXTRACT_REAL(ideals[largest_c_index]), - EXTRACT_REAL(approxs[largest_c_index]) - ); - printf( - "Squared error in sines: \n" - "\tAverage: %f (%f%% error)\n" - "\tLargest: %f (%f%% error)\n" - "\t\tInput:\t\t%f\n" - "\t\tValue:\t\t%f\n" - "\t\tApproximation:\t%f\n", - total_s_error, sqrt(total_s_error) * SQRT_NUM_LOOPS, largest_s_error, - sqrt(largest_s_error) * SQRT_NUM_LOOPS, inputs[largest_c_index], - EXTRACT_IMAG(ideals[largest_s_index]), - EXTRACT_IMAG(approxs[largest_s_index]) - ); + if (largest_s_error < s_error) { + largest_s_error = s_error; + largest_s_index = i; + } + + total_c_error += c_error; + total_s_error += s_error; + } + // these now contain the *average* squared error + total_c_error /= (double)n; + total_s_error /= (double)n; + + printf("Squared error in cosines (%d runs): \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + NUM_STATS, total_c_error, sqrt(total_c_error) * SQRT_NUM_STATS, + largest_c_error, sqrt(largest_c_error) * SQRT_NUM_STATS, + inputs[largest_c_index], EXTRACT_COSINE(ideals[largest_c_index]), + EXTRACT_COSINE(approxs[largest_c_index])); + printf("Squared error in sines (%d runs): \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + NUM_STATS, total_s_error, sqrt(total_s_error) * SQRT_NUM_STATS, + largest_s_error, sqrt(largest_s_error) * SQRT_NUM_STATS, + inputs[largest_s_index], EXTRACT_SINE(ideals[largest_s_index]), + EXTRACT_SINE(approxs[largest_s_index])); } // time the length of the computation `f` in nanoseconds -long time_computation( - void (*f)(double, void*), const double* inputs, CIRCLE_TYPE* results, int n -) -{ - size_t i; - struct timespec tp1; - struct timespec tp2; +// using a macro rather than a function taking a function pointer for +// more accurate time +#define TIME_COMPUTATION(f, inputs, results, n_stats, n_loop, time) \ + do { \ + size_t i; \ + struct timespec tp1; \ + struct timespec tp2; \ + \ + for (i = 0; i < n_stats; i++) { \ + f(inputs[i], results + i); \ + } \ + CIRCLE_TYPE temp; \ + clock_gettime(CLOCK_MONOTONIC, &tp1); \ + for (i = 0; i < n_stats; i++) { \ + f(rand() / (double)RAND_MAX, &temp); \ + } \ + clock_gettime(CLOCK_MONOTONIC, &tp2); \ + \ + time = SECONDS_PER_NANOSECOND * (tp2.tv_sec - tp1.tv_sec) + \ + (tp2.tv_nsec - tp1.tv_nsec); \ + } while (0) - clock_gettime(CLOCK_MONOTONIC, &tp1); - for (i = 0; i < n; i++) { - f(inputs[i], results + i); - } - clock_gettime(CLOCK_MONOTONIC, &tp2); +int main(int argn, char **args) { + long trig_time, rat_time; - return SECONDS_PER_NANOSECOND * (tp2.tv_sec - tp1.tv_sec) + - (tp2.tv_nsec - tp1.tv_sec); -} + double rands[NUM_STATS]; + CIRCLE_TYPE trigs[NUM_STATS]; + CIRCLE_TYPE rats[NUM_STATS]; -int main(int argn, char** args) -{ - long trig_time, rat_time; + size_t i; + for (i = 0; i < NUM_STATS; i++) { + rands[i] = rand() / (double)RAND_MAX; + } - double rands[NUM_LOOPS]; - CIRCLE_TYPE trigs[NUM_LOOPS]; - CIRCLE_TYPE rats[NUM_LOOPS]; - - size_t i; - for (i = 0; i < NUM_LOOPS; i++) { - rands[i] = rand() / (double)RAND_MAX; - } - - trig_time = time_computation(&turn_update, rands, trigs, NUM_LOOPS); - printf( + TIME_COMPUTATION(turn_update, rands, trigs, NUM_STATS, NUM_LOOPS, trig_time); + printf( #ifdef USE_COMPLEX "Timing for %d complex.h cexp:\t%ldns\n", #else "Timing for %d math.h sin and cos:\t%ldns\n", #endif - NUM_LOOPS, - trig_time - ); + NUM_LOOPS, trig_time); - rat_time = time_computation(&approx_turn_update, rands, rats, NUM_LOOPS); - printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); + TIME_COMPUTATION(approx_turn_update, rands, rats, NUM_STATS, NUM_LOOPS, + rat_time); + printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); - long diff = rat_time - trig_time; - double frac_speed; - if (diff > 0) { - frac_speed = rat_time / (double)trig_time; - printf( - STR_RED "stdlib" STR_NORM " faster, speedup: %ldns (%2.2fx)\n", - diff, frac_speed - ); + // Report results + long diff = rat_time - trig_time; + double frac_speed; + if (diff > 0) { + frac_speed = rat_time / (double)trig_time; + + // Disable colors for non-terminal output + if (isatty(STDOUT_FILENO)) { + printf(STR_RED "stdlib" STR_NORM " faster, speedup: %ldns (%2.2fx)\n", + diff, frac_speed); } else { - frac_speed = trig_time / (double)rat_time; - printf( - STR_GREEN "Approximation" STR_NORM - " faster, speedup: %ldns (%2.2fx)\n", - -diff, frac_speed - ); - print_errors(rands, trigs, rats, NUM_LOOPS); + printf("stdlib faster, speedup: %ldns (%2.2fx)\n", diff, frac_speed); + } + } else { + frac_speed = trig_time / (double)rat_time; + + // Disable colors for non-terminal output + if (isatty(STDOUT_FILENO)) { + printf(STR_GREEN "Approximation" STR_NORM + " faster, speedup: %ldns (%2.2fx)\n", + -diff, frac_speed); + } else { + printf("Approximation faster, speedup: %ldns (%2.2fx)\n", -diff, + frac_speed); } - return 0; + print_errors(rands, trigs, rats, NUM_STATS); + } + + return 0; } From 01f7014fcfad68b3f3515301f6a6b92b14cb99c0 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 02:40:52 -0500 Subject: [PATCH 11/14] move files to subdir, add recipe for results --- posts/stereo/1/Makefile | 10 - .../1/{ => approx-results}/.clang-format | 0 .../stereo/1/{ => approx-results}/.gitignore | 0 posts/stereo/1/approx-results/Makefile | 14 ++ posts/stereo/1/approx-results/main.c | 196 ++++++++++++++++++ .../1/approx-results/results_complex.txt | 15 ++ .../stereo/1/approx-results/results_math.txt | 15 ++ .../1/{ => approx-results}/stereo_complex.c | 0 .../1/{ => approx-results}/stereo_math.c | 0 posts/stereo/1/main.c | 174 ---------------- 10 files changed, 240 insertions(+), 184 deletions(-) delete mode 100644 posts/stereo/1/Makefile rename posts/stereo/1/{ => approx-results}/.clang-format (100%) rename posts/stereo/1/{ => approx-results}/.gitignore (100%) create mode 100644 posts/stereo/1/approx-results/Makefile create mode 100644 posts/stereo/1/approx-results/main.c create mode 100644 posts/stereo/1/approx-results/results_complex.txt create mode 100644 posts/stereo/1/approx-results/results_math.txt rename posts/stereo/1/{ => approx-results}/stereo_complex.c (100%) rename posts/stereo/1/{ => approx-results}/stereo_math.c (100%) delete mode 100644 posts/stereo/1/main.c diff --git a/posts/stereo/1/Makefile b/posts/stereo/1/Makefile deleted file mode 100644 index 5cfffc1..0000000 --- a/posts/stereo/1/Makefile +++ /dev/null @@ -1,10 +0,0 @@ -default: compare_complex compare_math - -clean: - rm -rf compare_complex compare_math - -compare_complex: - gcc main.c stereo_complex.c -lm -DUSE_COMPLEX -o compare_complex - -compare_math: - gcc main.c stereo_math.c -lm -o compare_math diff --git a/posts/stereo/1/.clang-format b/posts/stereo/1/approx-results/.clang-format similarity index 100% rename from posts/stereo/1/.clang-format rename to posts/stereo/1/approx-results/.clang-format diff --git a/posts/stereo/1/.gitignore b/posts/stereo/1/approx-results/.gitignore similarity index 100% rename from posts/stereo/1/.gitignore rename to posts/stereo/1/approx-results/.gitignore diff --git a/posts/stereo/1/approx-results/Makefile b/posts/stereo/1/approx-results/Makefile new file mode 100644 index 0000000..757487b --- /dev/null +++ b/posts/stereo/1/approx-results/Makefile @@ -0,0 +1,14 @@ +default: report + +clean: + rm -rf compare_complex compare_math results_complex.txt results_math.txt + +report: compare_complex compare_math + ./compare_complex > results_complex.txt + ./compare_math > results_math.txt + +compare_complex: + gcc main.c stereo_complex.c -lm -DUSE_COMPLEX -o compare_complex + +compare_math: + gcc main.c stereo_math.c -lm -o compare_math diff --git a/posts/stereo/1/approx-results/main.c b/posts/stereo/1/approx-results/main.c new file mode 100644 index 0000000..db1e705 --- /dev/null +++ b/posts/stereo/1/approx-results/main.c @@ -0,0 +1,196 @@ +#include +#include +#include +#include +#include +#include + +#define STR_RED "\x1b[31m" +#define STR_GREEN "\x1b[32m" +#define STR_NORM "\x1b[m" + +#define SECONDS_PER_NANOSECOND 1000000000 +#define NUM_STATS 100000 +#define SQRT_NUM_STATS 100 +#define NUM_LOOPS 10000000 + +struct circle { + double c; + double s; +}; + +#ifdef USE_COMPLEX +#define EXTRACT_COSINE(a) creal(a) +#define EXTRACT_SINE(a) cimag(a) +#define CIRCLE_TYPE double complex +#else +#define EXTRACT_COSINE(a) (a.c) +#define EXTRACT_SINE(a) (a.s) +#define CIRCLE_TYPE struct circle +#endif + +void turn_update(double turn, CIRCLE_TYPE* result); +void approx_turn_update(double turn, CIRCLE_TYPE* result); + +void print_errors( + const double* inputs, + const CIRCLE_TYPE* ideals, + const CIRCLE_TYPE* approxs, + int n +) +{ + double c_error, s_error; + double largest_c_error = 0, largest_s_error = 0; + size_t largest_c_index, largest_s_index; + + double total_c_error = 0, total_s_error = 0; + + size_t i; + CIRCLE_TYPE ideal, approx; + + for (i = 0; i < n; i++) { + ideal = ideals[i]; + approx = approxs[i]; + + // squared error in c components + c_error = EXTRACT_COSINE(ideal) - EXTRACT_COSINE(approx); + c_error *= c_error; + // squared error in s components + s_error = EXTRACT_SINE(ideal) - EXTRACT_SINE(approx); + s_error *= s_error; + + if (largest_c_error < c_error) { + largest_c_error = c_error; + largest_c_index = i; + } + + if (largest_s_error < s_error) { + largest_s_error = s_error; + largest_s_index = i; + } + + total_c_error += c_error; + total_s_error += s_error; + } + // these now contain the *average* squared error + total_c_error /= (double)n; + total_s_error /= (double)n; + + printf( + "Squared error in cosines (%d runs): \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + NUM_STATS, total_c_error, sqrt(total_c_error) * SQRT_NUM_STATS, + largest_c_error, sqrt(largest_c_error) * SQRT_NUM_STATS, + inputs[largest_c_index], EXTRACT_COSINE(ideals[largest_c_index]), + EXTRACT_COSINE(approxs[largest_c_index]) + ); + printf( + "Squared error in sines (%d runs): \n" + "\tAverage: %f (%f%% error)\n" + "\tLargest: %f (%f%% error)\n" + "\t\tInput:\t\t%f\n" + "\t\tValue:\t\t%f\n" + "\t\tApproximation:\t%f\n", + NUM_STATS, total_s_error, sqrt(total_s_error) * SQRT_NUM_STATS, + largest_s_error, sqrt(largest_s_error) * SQRT_NUM_STATS, + inputs[largest_s_index], EXTRACT_SINE(ideals[largest_s_index]), + EXTRACT_SINE(approxs[largest_s_index]) + ); +} + +// time the length of the computation `f` in nanoseconds +// using a macro rather than a function taking a function pointer for +// more accurate time +#define TIME_COMPUTATION(f, inputs, results, n_stats, n_loop, time) \ + do { \ + size_t i; \ + struct timespec tp1; \ + struct timespec tp2; \ + \ + for (i = 0; i < n_stats; i++) { \ + f(inputs[i], results + i); \ + } \ + CIRCLE_TYPE temp; \ + clock_gettime(CLOCK_MONOTONIC, &tp1); \ + for (i = 0; i < n_stats; i++) { \ + f(rand() / (double)RAND_MAX, &temp); \ + } \ + clock_gettime(CLOCK_MONOTONIC, &tp2); \ + \ + time = SECONDS_PER_NANOSECOND * (tp2.tv_sec - tp1.tv_sec) + \ + (tp2.tv_nsec - tp1.tv_nsec); \ + } while (0) + +int main(int argn, char** args) +{ + long trig_time, rat_time; + + double rands[NUM_STATS]; + CIRCLE_TYPE trigs[NUM_STATS]; + CIRCLE_TYPE rats[NUM_STATS]; + + size_t i; + for (i = 0; i < NUM_STATS; i++) { + rands[i] = rand() / (double)RAND_MAX; + } + + TIME_COMPUTATION( + turn_update, rands, trigs, NUM_STATS, NUM_LOOPS, trig_time + ); + printf( +#ifdef USE_COMPLEX + "Timing for %d complex.h cexp:\t%ldns\n", +#else + "Timing for %d math.h sin and cos:\t%ldns\n", +#endif + NUM_LOOPS, trig_time + ); + + TIME_COMPUTATION( + approx_turn_update, rands, rats, NUM_STATS, NUM_LOOPS, rat_time + ); + printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); + + // Report results + long diff = rat_time - trig_time; + double frac_speed; + if (diff > 0) { + frac_speed = rat_time / (double)trig_time; + + // Disable colors for non-terminal output + if (isatty(STDOUT_FILENO)) { + printf( + STR_RED "stdlib" STR_NORM " faster, speedup: %ldns (%2.2fx)\n", + diff, frac_speed + ); + } else { + printf( + "stdlib faster, speedup: %ldns (%2.2fx)\n", diff, frac_speed + ); + } + } else { + frac_speed = trig_time / (double)rat_time; + + // Disable colors for non-terminal output + if (isatty(STDOUT_FILENO)) { + printf( + STR_GREEN "Approximation" STR_NORM + " faster, speedup: %ldns (%2.2fx)\n", + -diff, frac_speed + ); + } else { + printf( + "Approximation faster, speedup: %ldns (%2.2fx)\n", -diff, + frac_speed + ); + } + + print_errors(rands, trigs, rats, NUM_STATS); + } + + return 0; +} diff --git a/posts/stereo/1/approx-results/results_complex.txt b/posts/stereo/1/approx-results/results_complex.txt new file mode 100644 index 0000000..27b4634 --- /dev/null +++ b/posts/stereo/1/approx-results/results_complex.txt @@ -0,0 +1,15 @@ +Timing for 10000000 complex.h cexp: 2864133ns +Timing for 10000000 approximations: 1321049ns +Approximation faster, speedup: 1543084ns (2.17x) +Squared error in cosines (100000 runs): + Average: 0.000051 (0.713743% error) + Largest: 0.000174 (1.320551% error) + Input: 0.729202 + Value: -0.659428 + Approximation: -0.672634 +Squared error in sines (100000 runs): + Average: 0.000070 (0.835334% error) + Largest: 0.000288 (1.698413% error) + Input: 0.842206 + Value: 0.475669 + Approximation: 0.458685 diff --git a/posts/stereo/1/approx-results/results_math.txt b/posts/stereo/1/approx-results/results_math.txt new file mode 100644 index 0000000..11dbb60 --- /dev/null +++ b/posts/stereo/1/approx-results/results_math.txt @@ -0,0 +1,15 @@ +Timing for 10000000 math.h sin and cos: 2822266ns +Timing for 10000000 approximations: 1223832ns +Approximation faster, speedup: 1598434ns (2.31x) +Squared error in cosines (100000 runs): + Average: 0.000051 (0.713743% error) + Largest: 0.000174 (1.320551% error) + Input: 0.729202 + Value: -0.659428 + Approximation: -0.672634 +Squared error in sines (100000 runs): + Average: 0.000070 (0.835334% error) + Largest: 0.000288 (1.698413% error) + Input: 0.842206 + Value: 0.475669 + Approximation: 0.458685 diff --git a/posts/stereo/1/stereo_complex.c b/posts/stereo/1/approx-results/stereo_complex.c similarity index 100% rename from posts/stereo/1/stereo_complex.c rename to posts/stereo/1/approx-results/stereo_complex.c diff --git a/posts/stereo/1/stereo_math.c b/posts/stereo/1/approx-results/stereo_math.c similarity index 100% rename from posts/stereo/1/stereo_math.c rename to posts/stereo/1/approx-results/stereo_math.c diff --git a/posts/stereo/1/main.c b/posts/stereo/1/main.c deleted file mode 100644 index 28d2768..0000000 --- a/posts/stereo/1/main.c +++ /dev/null @@ -1,174 +0,0 @@ -#include -#include -#include -#include -#include -#include - -#define STR_RED "\x1b[31m" -#define STR_GREEN "\x1b[32m" -#define STR_NORM "\x1b[m" - -#define SECONDS_PER_NANOSECOND 1000000000 -#define NUM_STATS 100000 -#define SQRT_NUM_STATS 100 -#define NUM_LOOPS 10000000 - -struct circle { - double c; - double s; -}; - -#ifdef USE_COMPLEX -#define EXTRACT_COSINE(a) creal(a) -#define EXTRACT_SINE(a) cimag(a) -#define CIRCLE_TYPE double complex -#else -#define EXTRACT_COSINE(a) (a.c) -#define EXTRACT_SINE(a) (a.s) -#define CIRCLE_TYPE struct circle -#endif - -void turn_update(double turn, CIRCLE_TYPE *result); -void approx_turn_update(double turn, CIRCLE_TYPE *result); - -void print_errors(const double *inputs, const CIRCLE_TYPE *ideals, - const CIRCLE_TYPE *approxs, int n) { - double c_error, s_error; - double largest_c_error = 0, largest_s_error = 0; - size_t largest_c_index, largest_s_index; - - double total_c_error = 0, total_s_error = 0; - - size_t i; - CIRCLE_TYPE ideal, approx; - - for (i = 0; i < n; i++) { - ideal = ideals[i]; - approx = approxs[i]; - - // squared error in c components - c_error = EXTRACT_COSINE(ideal) - EXTRACT_COSINE(approx); - c_error *= c_error; - // squared error in s components - s_error = EXTRACT_SINE(ideal) - EXTRACT_SINE(approx); - s_error *= s_error; - - if (largest_c_error < c_error) { - largest_c_error = c_error; - largest_c_index = i; - } - - if (largest_s_error < s_error) { - largest_s_error = s_error; - largest_s_index = i; - } - - total_c_error += c_error; - total_s_error += s_error; - } - // these now contain the *average* squared error - total_c_error /= (double)n; - total_s_error /= (double)n; - - printf("Squared error in cosines (%d runs): \n" - "\tAverage: %f (%f%% error)\n" - "\tLargest: %f (%f%% error)\n" - "\t\tInput:\t\t%f\n" - "\t\tValue:\t\t%f\n" - "\t\tApproximation:\t%f\n", - NUM_STATS, total_c_error, sqrt(total_c_error) * SQRT_NUM_STATS, - largest_c_error, sqrt(largest_c_error) * SQRT_NUM_STATS, - inputs[largest_c_index], EXTRACT_COSINE(ideals[largest_c_index]), - EXTRACT_COSINE(approxs[largest_c_index])); - printf("Squared error in sines (%d runs): \n" - "\tAverage: %f (%f%% error)\n" - "\tLargest: %f (%f%% error)\n" - "\t\tInput:\t\t%f\n" - "\t\tValue:\t\t%f\n" - "\t\tApproximation:\t%f\n", - NUM_STATS, total_s_error, sqrt(total_s_error) * SQRT_NUM_STATS, - largest_s_error, sqrt(largest_s_error) * SQRT_NUM_STATS, - inputs[largest_s_index], EXTRACT_SINE(ideals[largest_s_index]), - EXTRACT_SINE(approxs[largest_s_index])); -} - -// time the length of the computation `f` in nanoseconds -// using a macro rather than a function taking a function pointer for -// more accurate time -#define TIME_COMPUTATION(f, inputs, results, n_stats, n_loop, time) \ - do { \ - size_t i; \ - struct timespec tp1; \ - struct timespec tp2; \ - \ - for (i = 0; i < n_stats; i++) { \ - f(inputs[i], results + i); \ - } \ - CIRCLE_TYPE temp; \ - clock_gettime(CLOCK_MONOTONIC, &tp1); \ - for (i = 0; i < n_stats; i++) { \ - f(rand() / (double)RAND_MAX, &temp); \ - } \ - clock_gettime(CLOCK_MONOTONIC, &tp2); \ - \ - time = SECONDS_PER_NANOSECOND * (tp2.tv_sec - tp1.tv_sec) + \ - (tp2.tv_nsec - tp1.tv_nsec); \ - } while (0) - -int main(int argn, char **args) { - long trig_time, rat_time; - - double rands[NUM_STATS]; - CIRCLE_TYPE trigs[NUM_STATS]; - CIRCLE_TYPE rats[NUM_STATS]; - - size_t i; - for (i = 0; i < NUM_STATS; i++) { - rands[i] = rand() / (double)RAND_MAX; - } - - TIME_COMPUTATION(turn_update, rands, trigs, NUM_STATS, NUM_LOOPS, trig_time); - printf( -#ifdef USE_COMPLEX - "Timing for %d complex.h cexp:\t%ldns\n", -#else - "Timing for %d math.h sin and cos:\t%ldns\n", -#endif - NUM_LOOPS, trig_time); - - TIME_COMPUTATION(approx_turn_update, rands, rats, NUM_STATS, NUM_LOOPS, - rat_time); - printf("Timing for %d approximations:\t%ldns\n", NUM_LOOPS, rat_time); - - // Report results - long diff = rat_time - trig_time; - double frac_speed; - if (diff > 0) { - frac_speed = rat_time / (double)trig_time; - - // Disable colors for non-terminal output - if (isatty(STDOUT_FILENO)) { - printf(STR_RED "stdlib" STR_NORM " faster, speedup: %ldns (%2.2fx)\n", - diff, frac_speed); - } else { - printf("stdlib faster, speedup: %ldns (%2.2fx)\n", diff, frac_speed); - } - } else { - frac_speed = trig_time / (double)rat_time; - - // Disable colors for non-terminal output - if (isatty(STDOUT_FILENO)) { - printf(STR_GREEN "Approximation" STR_NORM - " faster, speedup: %ldns (%2.2fx)\n", - -diff, frac_speed); - } else { - printf("Approximation faster, speedup: %ldns (%2.2fx)\n", -diff, - frac_speed); - } - - print_errors(rands, trigs, rats, NUM_STATS); - } - - return 0; -} From 8fb3ddad8f4ef015db82dc5e97c0cdab5decf713 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 03:26:00 -0500 Subject: [PATCH 12/14] add programmatic output for approximation in stereo.1 --- posts/stereo/1/index.qmd | 38 +++++++++++++++++--------------------- 1 file changed, 17 insertions(+), 21 deletions(-) diff --git a/posts/stereo/1/index.qmd b/posts/stereo/1/index.qmd index bdeaecc..7e00740 100644 --- a/posts/stereo/1/index.qmd +++ b/posts/stereo/1/index.qmd @@ -24,6 +24,7 @@ categories: ```{python} #| echo: false +from IPython.display import display, Code import sympy from sympy.abc import t ``` @@ -200,7 +201,7 @@ This action of squaring *z* means that as *t* ranges from $-\infty$ to $\infty$, Let's go on a brief tangent and compare these expressions to their transcendental trigonometric counterparts. Graphing the real and imaginary parts separately shows how much they resemble - $\cos(\pi t)$ and $\sin(\pi t)$ around 0[^1]. + $\cos(\pi t)$ and $\sin(\pi t)$ around zero[^1]. [^1]: An approximation of a function as the ratio of two polynomials is called a [Padé approximant](https://en.wikipedia.org/wiki/Pad%C3%A9_approximant). @@ -376,28 +377,21 @@ A best-case analysis of the above approximation, given some value *t* is... ...or 15 FLOPs. On a more optimistic note, a Monte Carlo test of the above approximation on my computer yields - promising results when compared with GCC's `libm` implementations of `sin` and `cos` - (also tested with `cexp`). + promising results when compared with GCC's `libm` implementations of `sin`, `cos`, and `cexp`. -```{} -Timing for 100000 math.h sin and cos: 5409619ns -Timing for 100000 approximations: 2408645ns -Approximation faster, speedup: 3000974ns (2.25x) -Squared error in cosines: - Average: 0.000051 (0.713743% error) - Largest: 0.000174 (1.320551% error) - Input: 0.729202 - Value: -0.659428 - Approximation: -0.672634 -Squared error in sines: - Average: 0.000070 (0.835334% error) - Largest: 0.000288 (1.698413% error) - Input: 0.729202 - Value: 0.475669 - Approximation: 0.458685 +```{python} +#| echo: false + +for file in [ + "approx-results/results_math.txt", + "approx-results/results_complex.txt", +]: + with open(file) as a: + display(Code(a.read())) ``` -For the source which I used to generate this output, see the repository linked below. +For the source which I used to generate this output, see the repository linked at the bottom + of this article. Even though results can be inaccurate, this exercise in the algebraic manipulation of complex numbers is fairly interesting since it requires no calculus to define, @@ -453,6 +447,7 @@ $$ These are exactly the same expressions which appear in the real and imaginary components of the ratio between $1 + ti$ and its conjugate. + Compared to the algebra using complex numbers, this is quite a bit more work. One might ask whether, given a proper arithmetic setting, this can be extended to three dimensions. In other words, we want to find the projection of a sphere using some new number system @@ -535,7 +530,7 @@ If you know a little group theory, you might know there are only two nonabelian dihedral group of degree 4 ](https://en.wikipedia.org/wiki/Examples_of_groups#dihedral_group_of_order_8). -In the latter group, *j* and *k* are both imaginary, but square to 1 (*i* still squares to -1)[^4] . +In the latter group, *j* and *k* are both imaginary, but square to 1 (*i* still squares to -1)[^4]. [^4]: I'm being a bit careless with the meanings of "1" and "-1" here. Properly, these are the group identity and another group element which commutes with all others. @@ -707,5 +702,6 @@ However, I feel like it fails to provide an algebraic motivation for quaternions or even stereography in the first place. Hopefully, my remarks on the algebraic approach can be used to augment the information in the video. + Diagrams created with GeoGebra and Matplotlib. Repository with approximations (as well as GeoGebra files) available [here](https://github.com/queue-miscreant/approx-trig). From 94dee54cdb7d0b3ccc1adaf364a14eca103d7d09 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 03:28:17 -0500 Subject: [PATCH 13/14] add videos in stereo.2 --- posts/stereo/2/anim.py | 2 +- posts/stereo/2/index.qmd | 31 ++++++++++++++++++++++++------- 2 files changed, 25 insertions(+), 8 deletions(-) diff --git a/posts/stereo/2/anim.py b/posts/stereo/2/anim.py index 07b77f7..357714f 100644 --- a/posts/stereo/2/anim.py +++ b/posts/stereo/2/anim.py @@ -34,7 +34,7 @@ def bindfig(fig): class SympyAnimationWrapper: """ - Context manager for binding a figurebinfoinfoinfoinfoinfoinfo + Context manager for binding a figure for animation. """ def __init__(self, filename: Path | str): self._filename = filename if isinstance(filename, Path) else Path(filename) diff --git a/posts/stereo/2/index.qmd b/posts/stereo/2/index.qmd index c73201a..475b609 100644 --- a/posts/stereo/2/index.qmd +++ b/posts/stereo/2/index.qmd @@ -44,7 +44,7 @@ def generate_os(amt): yield ret ret = (ret * o).expand().cancel().simplify() -os = list(generate_os(10)) +os = list(generate_os(11)) cs = [sympy.re(i).simplify() for i in os] ss = [sympy.im(i).simplify() for i in os] ``` @@ -365,9 +365,7 @@ will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$. ```{python} #| echo: false -#| fig-cap: | -#| p/1 polar roses as rational curves. -#| Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0). +#| output: false @dataclass class Rose: @@ -384,7 +382,7 @@ class Rose: ) -def animate_roses(filename: str, arguments: list[Rose], interval=200): +def animate_roses(filename: str, arguments: list[Rose], interval=500): with SympyAnimationWrapper(filename) as animate: @animate(list(range(len(arguments))), interval=interval) def ret(fr): @@ -408,6 +406,13 @@ animate_roses( ) ``` +::: {#fig-polar-roses-1} +{{< video "./polar_roses_1.mp4" >}} + +p/1 polar roses as rational curves. +Since *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0)." +::: + $q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder whether the other polar curves can be obtained by allowing it to vary as well. And you'd be right. @@ -420,7 +425,7 @@ will plot a $p/q$ polar rose as t ranges over $(-\infty, \infty)$. ```{python} #| echo: false -#| fig-cap: p/q polar roses as rational curves +#| output: false animate_roses( "polar_roses_2.mp4", @@ -431,6 +436,11 @@ animate_roses( ) ``` +::: {#fig-polar-roses-2} +{{< video "./polar_roses_2.mp4" >}} + +p/q polar roses as rational curves +::: Just as with the prior calculus examples, doubling all subscripts of *c* and *s* will only require *t* to range over $(-1, 1)$, which removes the ugly bite mark. @@ -700,7 +710,7 @@ Indeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate t ```{python} #| echo: false -#| fig-cap: Approximations to the Archimedean spiral +#| output: false with SympyAnimationWrapper("approximate_archimedes.mp4") as animate: @animate(list(range(10)), interval=500) @@ -734,6 +744,13 @@ with SympyAnimationWrapper("approximate_archimedes.mp4") as animate: ret.save() # type: ignore ``` +::: {#fig-approx-archimedes} +{{< video ./approximate_archimedes.mp4 >}} + +Approximations to the Archimedean spiral +::: + + Since R necessarily defines a rational curve, the curves will never be equal, just as any stretching of $c_n$ will never exactly become cosine. From 097431afdca147b0b7181b0eb7646cb801e0aa6b Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 22 Jul 2025 03:31:59 -0500 Subject: [PATCH 14/14] add stereo index --- .../stereo/2/index/execute-results/html.json | 12 ++++++++++++ .../2/index/figure-html/cell-5-output-1.png | Bin 0 -> 15018 bytes .../2/index/figure-html/cell-6-output-1.png | Bin 0 -> 11162 bytes .../2/index/figure-html/cell-7-output-1.png | Bin 0 -> 19508 bytes index.qmd | 1 + posts/stereo/2/index.qmd | 2 +- posts/stereo/_metadata.yml | 5 +++++ posts/stereo/index.qmd | 8 ++++++++ 8 files changed, 27 insertions(+), 1 deletion(-) create mode 100644 _freeze/posts/stereo/2/index/execute-results/html.json create mode 100644 _freeze/posts/stereo/2/index/figure-html/cell-5-output-1.png create mode 100644 _freeze/posts/stereo/2/index/figure-html/cell-6-output-1.png create mode 100644 _freeze/posts/stereo/2/index/figure-html/cell-7-output-1.png create mode 100644 posts/stereo/_metadata.yml create mode 100644 posts/stereo/index.qmd diff --git a/_freeze/posts/stereo/2/index/execute-results/html.json b/_freeze/posts/stereo/2/index/execute-results/html.json new file mode 100644 index 0000000..c657e0f --- /dev/null +++ b/_freeze/posts/stereo/2/index/execute-results/html.json @@ -0,0 +1,12 @@ +{ + "hash": "e723e799442b3e52f9543809a686e1f9", + "result": { + "engine": "jupyter", + "markdown": "---\ntitle: \"Further Notes on Algebraic Stereography\"\ndescription: |\n How do you rotate in 2D and 3D without standard trigonometry?\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2021-10-10\"\ndate-modified: \"2025-06-30\"\ncategories:\n - algebra\n - complex analysis\n - polar roses\n - generating functions\n---\n\n\n\n\n\nIn my previous post, I discussed the stereographic projection of a circle as it pertains\n to complex numbers, as well as its applications in 2D and 3D rotation.\nIn an effort to document more interesting facts about this mathematical object\n (of which scarce information is immediately available online),\n I will now elaborate on more of its properties.\n\n\nChebyshev Polynomials\n---------------------\n\n[Previously](/posts/chebyshev/1), I derived the\n [Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials)\n with the archetypal complex exponential.\nThese polynomials express the sines and cosines of a multiple of an angle from\n the sine and cosine of the base angle.\nWhere $T_n(t)$ are Chebyshev polynomials of the first kind and $U_n(t)$ are those of the second kind,\n\n$$\n\\begin{gather*}\n \\cos(n \\theta) = T_n(\\cos(\\theta))\n \\\\\n \\sin(n \\theta) = U_{n - 1}(\\cos(\\theta)) \\sin(\\theta)\n\\end{gather*}\n$$\n\nThe complex exponential derivation begins by squaring and developing a second-order recurrence.\n\n$$\n\\begin{align*}\n (e^{i\\theta})^2 &= (\\cos + i\\sin)^2\n \\\\\n &= \\cos^2 + 2i\\cos \\cdot \\sin - \\sin^2 + (0 = \\cos^2 + \\sin^2 - 1)\n \\\\\n &= 2\\cos^2 + 2i\\cos \\cdot \\sin - 1\n \\\\\n &= 2\\cos \\cdot (\\cos + i\\sin) - 1\n \\\\\n &= 2\\cos(\\theta)e^{i\\theta} - 1\n \\\\\n (e^{i\\theta})^{n+2} &= 2\\cos(\\theta)(e^{i\\theta})^{n+1} - (e^{i\\theta})^n\n\\end{align*}\n$$\n\nThis recurrence relation can then be used to obtain the Chebyshev polynomials, and hence,\n the expressions using sine and cosine above.\nPresented this way with such a simple derivation, it appears as though these relationships\n are inherently trigonometric.\n\nHowever, these polynomials actually have *nothing* to do with sine and cosine on their own.\nFor one, [they appear in graph theory](/posts/chebyshev/2), and for two,\n the importance of the complex exponential is overstated.\n$e^{i\\theta}$ really just specifies a point on the complex unit circle.\nThis property is used on the second line to coax the equation into a quadratic in $e^{i\\theta}$.\nThis is also the *only* property upon which the recurrence depends; all else is algebraic manipulation.\n\n\n### Back to the Stereograph\n\nKnowing this, let's start over with the stereographic projection of the circle:\n\n$$\no_1(t) = {1 + it \\over 1 - it}\n = {1 - t^2 \\over 1 + t^2} + i {2t \\over 1 + t^2}\n = \\text{c}_1 + i\\text{s}_1\n$$\n\nThe subscript \"1\" is because as *t* ranges over $(-\\infty, \\infty)$, the function loops once\n around the unit circle.\nTaking this to higher powers keeps points on the circle since all points on the circle\n have a norm of 1.\nIt also makes more loops around the circle, which we can denote by larger subscripts:\n\n$$\n\\begin{align*}\n o_n &= (o_1)^n\n = \\left( {1 + it \\over 1 - it} \\right)^n\n \\\\\n \\text{c}_n + i\\text{s}_n\n &= (\\text{c}_1 + i\\text{s}_1)^n\n\\end{align*}\n$$\n\nThis mirrors raising the complex exponential to a power\n (which loops over the range $(-\\pi, \\pi)$ instead).\nThe final line is analogous to de Moivre's formula, but in a form where everything is\n a ratio of polynomials in *t*.\nThis means that the Chebyshev polynomials can be obtained directly from these rational expressions:\n\n$$\n\\begin{align*}\n o_2 = (o_1)^2 &= (\\text{c}_1 + i\\text{s}_1)^2\n \\\\\n &= \\text{c}_1^2 + 2i\\text{c}_1\\text{s}_1 - \\text{s}_1^2\n + (0 = \\text{c}_1^2 + \\text{s}_1^2 - 1)\n \\\\\n &= 2\\text{c}_1^2 + 2i\\text{c}_1\\text{s}_1 - 1\n \\\\\n &= 2\\text{c}_1(\\text{c}_1 + i\\text{s}_1) - 1\n \\\\\n &= 2\\text{c}_1 o_1 - 1\n \\\\\n o_2 \\cdot (o_1)^n &= 2\\text{c}_1 o_1 \\cdot (o_1)^n - (o_1)^n\n \\\\\n o_{n+2} &= 2\\text{c}_1 o_{n+1} - o_n\n\\end{align*}\n$$\n\nThis matches the earlier recurrence relation with the complex exponential and therefore\n the recurrence relation of the Chebyshev polynomials.\nIt also means that the the rational functions obey the same relationship as sine and cosine:\n\n$$\n\\begin{matrix}\n \\begin{gather*}\n \\text{c}_n = T_n(\\text{c}_1)\n \\\\\n \\text{s}_n = U_{n-1}(\\text{c}_1) \\text{s}_1\n \\end{gather*}\n & \\text{where }\n \\text{c}_1 = {1 - t^2 \\over 1 + t^2}, &\n \\text{s}_1 = {2t \\over 1 + t^2}\n\\end{matrix}\n$$\n\nThus, the Chebyshev polynomials are tied to (coordinates on) circles,\n rather than explicitly to the trigonometric functions.\nIt is a bit strange that these polynomials are in terms of rational functions, but no stranger\n than them being in terms of *ir*rational functions like sine and cosine.\n\n\nCalculus\n--------\n\nSince these functions behave similarly to sine and cosine, one might wonder about\n the nature of these expressions in the context of calculus.\n\nFor comparison, the complex exponential (as it is a parallel construction) has a simple derivative[^1].\nSince the exponential function is its own derivative, the expression acquires\n an imaginary coefficient through the chain rule.\n\n[^1]: This is forgoing the fact that complex derivatives require more care than their real counterparts.\n It matters slightly less in this case since this function is complex-valued, but has a real parameter.\n\n$$\n\\begin{align*}\n e^{it} &= \\cos(t) + i\\sin(t)\n \\\\\n {d \\over dt} e^{it}\n &= {d \\over dt} \\cos(t) + {d \\over dt} i\\sin(t)\n \\\\\n i e^{it} &= -\\sin(t) + i\\cos(t)\n \\\\\n i[\\cos(t) + i\\sin(t)]\n &\\stackrel{\\checkmark}{=} -\\sin(t) + i\\cos(t)\n\\end{align*}\n$$\n\nMeanwhile, the complex stereograph has derivative\n\n$$\n\\begin{align*}\n {d \\over dt} o_1(t) &= {d \\over dt} {1 + it \\over 1 - it}\n = {i(1 - it) + i(1 + it) \\over (1 - it)^2}\n \\\\\n &= {2i \\over (1 - it)^2}\n = {2i(1 + it)^2 \\over (1 + t^2)^2}\n = {2i(1 - t^2 + 2it) \\over (1 + t^2)^2}\n \\\\\n &= {-4t \\over (1 + t^2)^2} + i {2(1 - t^2) \\over (1 + t^2)^2}\n \\\\\n &= {-2 \\over 1 + t^2}s_1 + i {2 \\over 1 + t^2}c_1\n \\\\\n &= -(1 + c_1)s_1 + i(1 + c_1)c_1\n \\\\\n &= i(1 + c_1)o_1\n\\end{align*}\n$$\n\nJust like the complex exponential, an imaginary coefficient falls out.\nHowever, the expression also accrues a $1 + c_1$ term, almost like an adjustment factor\n for its failure to be the complex exponential.\nSine and cosine obey a simpler relationship with respect to the derivative,\n and thus need no adjustment.\n\n\n### Complex Analysis\n\nSince $o_n$ is a curve which loops around the unit circle *n* times, that possibly suits it\n to showing a simple result from complex analysis.\nIntegrating along a contour which wraps around a sufficiently nice function's pole\n (i.e., where its magnitude grows without bound) yields a familiar value.\nThis is easiest to see with $f(z) = 1 / z$:\n\n$$\n\\oint_\\Gamma {1 \\over z} dz\n = \\int_a^b {\\gamma'(t) \\over \\gamma(t)} dt\n = 2\\pi i\n$$\n\nIn this example, Γ is a counterclockwise curve parametrized by γ which loops once around\n the pole at *z* = 0.\nMore loops will scale this by a factor according to the number of loops.\n\nNormally this equality is demonstrated with the complex exponential, but will $o_1$ work just as well?\nIf Γ is the unit circle, the integral is:\n\n$$\n\\oint_\\Gamma {1 \\over z} dz\n = \\int_{-\\infty}^\\infty {o_1'(t) \\over o_1(t)} dt\n = \\int_{-\\infty}^\\infty i(1 + c_1(t)) dt\n = 2i\\int_{-\\infty}^\\infty {1 \\over 1 + t^2} dt\n$$\n\nIf one has studied their integral identities, the indefinite version of the final integral\n will be obvious as $\\arctan(t)$, which has horizontal asymptotes of $\\pi / 2$ and $-\\pi / 2$.\nTherefore, the value of the integral is indeed $2\\pi i$.\n\nIf there are *n* loops, then naturally there are *n* of these $2\\pi i$s.\nSince powers of *o* are more loops around the circle, the chain and power rules show:\n\n$$\n\\begin{gather*}\n {d \\over dt} (o_1)^n = n(o_1)^{n-1} {d \\over dt} o_1\n \\\\[14pt]\n \\oint_\\Gamma {1 \\over z} dz\n = \\int_{-\\infty}^\\infty {n o_1(t)^{n-1} o_1'(t) \\over o_1(t)^n} dt\n = n \\int_{-\\infty}^\\infty {o_1'(t) \\over o_1(t)} dt\n = 2 \\pi i n\n\\end{gather*}\n$$\n\nIt is certainly possible to perform these contour integrals along straight lines;\n in fact, integrating along lines from 1 to *i* to -1 to -*i* deals with a\n similar integral involving arctangent.\nHowever, the best one can do to construct more loops with lines is to count each line\n multiple times, which isn't extraordinarily convincing.\n\nPerhaps the use of $\\infty$ in the integral bounds is also unconvincing.\nThe integral can be shifted back into the realm of plausibility by considering simpler bounds on $o_2$:\n\n$$\n\\begin{align*}\n \\oint_\\Gamma {1 \\over z} dz\n &= \\int_{-1}^1 {2 o_1(t) o_1'(t) \\over o_1(t)^2} dt\n \\\\\n &= 2 \\int_{-1}^1 {o_1'(t) \\over o_1(t)} dt\n \\\\\n &= 2(2i\\arctan(1) - 2i\\arctan(-1))\n \\\\\n &= 2\\pi i\n\\end{align*}\n$$\n\nThis has an additional benefit: using the series form of $1 / (1 + t^2)$ and integrating,\n one obtains the series form of the arctangent.\nThis series converges for $-1 \\le t \\le 1$, which happens to match the bounds of integration.\nThe convergence of this series is fairly important, since it is tied to formulas for π,\n in particular [Leibniz's formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80).\n\nWere one to integrate with the complex exponential, we would instead use the bounds $(0, 2\\pi)$,\n since at this point a full loop has been made.\nBut think to yourself -- how do you know the period of the complex exponential?\nHow do you know that 2π radians is equivalent to 0 radians?\nThe result using stereography relies on neither of these prior results and is directly pinned\n to a formula for π instead an apparent detour through the number *e*.\n\n\nPolar Curves\n------------\n\nPolar coordinates are useful for expressing for which the distance from the origin is\n a function of the angle with respect to the positive *x*-axis.\nThey can also be readily converted to parametric forms:\n\n$$\n\\begin{gather*}\n r(\\theta) &\\Longleftrightarrow&\n \\begin{matrix}\n x(\\theta) = r \\cos(\\theta) \\\\\n y(\\theta) = r \\sin(\\theta)\n \\end{matrix}\n\\end{gather*}\n$$\n\nPolar curves frequently loop in on themselves, and so it is necessary to choose appropriate bounds\n for θ (usually as multiples of π) when plotting.\nEvidently, this is due to the use of sine and cosine in the above parametrization.\nFortunately, $s_n$ and $c_n$ (as shown by the calculus above) have much simpler bounds.\nSo what happens when one substitutes the rational functions in place of the trig ones?\n\n\n### Polar Roses\n\n[Polar roses](https://en.wikipedia.org/wiki/Rose_(mathematics)) are beautiful shapes which have\n a simple form when expressed in polar coordinates.\n\n$$\nr(\\theta) = \\cos \\left( {p \\over q} \\cdot \\theta \\right)\n$$\n\nThe ratio $p/q$ in least terms uniquely determines the shape of the curve.\n\nIf you weren't reading this post, you might assume this curve is transcendental since it uses cosine,\n but you probably know better at this point.\nThe Chebyshev examples above demonstrate the resemblance between $c_n$ and $\\cos(n\\theta)$.\nThe subscript of $c$ is easiest to work with as an integer, so let $q = 1$.\n\n$$\nx(t) = c_p(t) c_1(t) \\qquad y(t) = c_p(t) s_1(t)\n$$\n\nwill plot a $p/1$ polar rose as t ranges over $(-\\infty, \\infty)$.\n\n\n\n::: {#fig-polar-roses-1}\n{{< video \"./polar_roses_1.mp4\" >}}\n\np/1 polar roses as rational curves.\nSince *t* never reaches infinity, a bite appears to be taken out of the graphs near (-1, 0).\"\n:::\n\n$q = 1$ happens to match the subscript *c* term of *x* and *s* term of *y*, so one might wonder\n whether the other polar curves can be obtained by allowing it to vary as well.\nAnd you'd be right.\n\n$$\nx(t) = c_p(t) c_q(t) \\qquad y(t) = c_p(t) s_q(t)\n$$\n\nwill plot a $p/q$ polar rose as t ranges over $(-\\infty, \\infty)$.\n\n\n\n::: {#fig-polar-roses-2}\n{{< video \"./polar_roses_2.mp4\" >}}\n\np/q polar roses as rational curves\n:::\n\nJust as with the prior calculus examples, doubling all subscripts of *c* and *s* will\n only require *t* to range over $(-1, 1)$, which removes the ugly bite mark.\nPerhaps it is also slightly less satisfying, since the fraction $p/q$ directly appears in the\n typical polar incarnation with cosine.\nOn the other hand, it exposes an important property of these curves: they are all rational.\n\nThis approach lends additional precision to a prospective pseudo-polar coordinate system.\nIn the next few examples, I will be using the following notation for compactness:\n\n$$\n \\begin{gather*}\n R_n(t) = f(t) &\\Longleftrightarrow&\n \\begin{matrix}\n x(t) = f(t) c_n(t) \\\\\n y(t) = f(t) s_n(t)\n \\end{matrix}\n\\end{gather*}\n$$\n\n\n### Conic Sections\n\nThe polar equation for a conic section (with a particular unit length occurring somewhere)\n in terms of its eccentricity $\\varepsilon$ is:\n\n$$\nr(\\theta) = {1 \\over 1 - \\varepsilon \\cos(\\theta)}\n$$\n\nCorrespondingly, the rational polar form can be expressed as\n\n$$\nR_1(t) = {1 \\over 1 - \\varepsilon c_1}\n$$\n\nSince polynomial arithmetic is easier to work with than trigonometric identities,\n it is a matter of pencil-and-paper algebra to recover the implicit form from a parametric one.\n\n\n#### Parabola ($|\\varepsilon| = 1$)\n\nThe conic section with the simplest implicit equation is the parabola.\nSince $c_n$ is a simple ratio of polynomials in *t*, it is much simpler to recover the implicit equation.\nFor $\\varepsilon = 1$,\n\n:::: {layout-ncol=\"2\"}\n\n::: {#39eea2a2 .cell execution_count=5}\n\n::: {.cell-output .cell-output-display}\n![$x = {c_1 \\over 1 - c_1} \\quad y = {s_1 \\over 1 - c_1}$](index_files/figure-html/cell-5-output-1.png){}\n:::\n:::\n\n\n::: {}\n$$\n\\begin{align*}\n 1 - c_1 &= 1 - {1 - t^2 \\over 1 + t^2}\n = {2 t^2 \\over 1 + t^2}\n \\\\\n y &= {s_1 \\over 1 - c_1}\n = {2t \\over 1 + t^2} {1 + t^2 \\over 2 t^2}\n = {1 \\over t}\n \\\\\n x &= {c_1 \\over 1 - c_1}\n = {1 - t^2 \\over 1 + t^2} \\cdot {1 + t^2 \\over 2 t^2}\n = {1 - t^2 \\over 2t^2}\n \\\\\n &= {1 \\over 2t^2} - {1 \\over 2}\n = {y^2 \\over 2} - {1 \\over 2}\n\\end{align*}\n$$\n:::\n::::\n\n*x* is a quadratic polynomial in *y*, so trivially the figure formed is a parabola.\nTechnically it is missing the point where $y = 0 ~ (t = \\infty)$, and this is not a circumstance\n where using a higher $c_n$ would help.\nIt is however, similar to the situation where we allow $o_1(\\infty) = -1$, and an argument\n can be made to waive away any concerns one might have.\n\n\n#### Ellipse ($|\\varepsilon| < 1$)\n\nEllipses are next.\nThe simplest fraction between zero and one is 1/2, so for $\\varepsilon = 1/2$,\n\n:::: {layout-ncol = \"2\"}\n\n::: {#f9484e49 .cell execution_count=6}\n\n::: {.cell-output .cell-output-display}\n![$x = {c_1 \\over 1 - c_1 / 2} \\quad y = {s_1 \\over 1 - c_1 / 2}$](index_files/figure-html/cell-6-output-1.png){}\n:::\n:::\n\n\n::: {}\n$$\n\\begin{align*}\n 1 - {1 \\over 2}c_1 &= 1 - {1 \\over 2} \\cdot {1 - t^2 \\over 1 + t^2}\n = {3 t^2 + 1 \\over 2 + 2t^2}\n \\\\\n y &= {s_1 \\over 1 - {1 \\over 2}c_1}\n = {4t \\over 3t^2 + 1}\n \\\\\n x &= {c_1 \\over 1 - {1 \\over 2}c_1}\n = {2 - 2t^2 \\over 3t^2 + 1}\n\\end{align*}\n$$\n:::\n::::\n\nThere isn't an obvious way to combine products of *x* and *y* into a single equation.\nThe general form of a conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, so\n we know that the implicit equation for the curve almost certainly involves $x^2$ and $y^2$.\n\n$$\nx^2 = {4 - 8t^2 + 4t^4 \\over (3t^2 + 1)^2} \\qquad\n y^2 = {16t^2 \\over (3t^2 + 1)^2}\n$$\n\nSquaring produces some $t^4$ terms which cannot exist outside of these terms and *xy*.\nA linear combination of $x^2$ and $y^2$ never includes any cubic terms in the numerator\n which would appear in *xy*, so $B = 0$.\nSince all remaining terms are linear in *x* and *y*, their denominator must appear as a factor\n in the numerator of $Ax^2 + Cy^2$, whatever *A* and *C* are.\n\nSince the coefficient of $t^4$ in $x^2$ is 4, *A* must be multiple of 3.\nThrough trial and error, $A = 3, C = 4$ gives:\n\n$$\n\\begin{align*}\n 3x^2 + 4y^2\n &= {12 - 24t^2 + 12t^4 + 64t^2 \\over (3t^2 + 1)^2}\n \\\\\n &= {12 - 40t^2 + 12t^4 \\over (3t^2 + 1)^2}\n \\\\\n &= {(4t^2 + 12) (3t^2 + 1) \\over (3t^2 + 1)^2}\n \\\\\n &= {4t^2 + 12 \\over 3t^2 + 1}\n\\end{align*}\n$$\n\nSince the numerator of *y* has a *t*, this is clearly some combination of *x* and a constant.\nBy the previous line of thought, the constant term must be a multiple of 4, and picking the smallest\n option finally results in the implicit form:\n\n$$\n\\begin{align*}\n 4 &= {4(3t^2 + 1) \\over 3t^2 + 1}\n = {12t^2 + 4 \\over 3t^2 + 1}\n \\\\\n {4t^2 + 12 \\over 3t^2 + 1} - 4\n &= {8 - 8t^2 \\over 3t^2 + 1} = 4x\n \\\\[14pt]\n 3x^2 + 4y^2 &= 4x + 4\n \\\\\n 3x^2 + 4y^2 - 4x - 4 &= 0\n\\end{align*}\n$$\n\nNotably, the coefficients of *x* and *y* are 3 and 4.\nSimultaneously, $o_1(\\varepsilon) = o_1(1/2) = {3 \\over 5} + i{4 \\over 5}$.\nThis binds together three concepts: the simplest case of the Pythagorean theorem,\n the 3-4-5 right triangle; the coefficients of the implicit form; and the role of eccentricity\n with respect to stereography.\n\n\n#### Hyperbola ($|\\varepsilon| > 1$)\n\nAs evidenced by the bound on the eccentricity above, hyperbolae are in some way the inverses of ellipses.\nSince $o_1(2)$ is a reflection of $o_1(1/2)$, you might think the implicit equation for\n $\\varepsilon = 2$ to be the same, but with a flipped sign or two.\nUnfortunately, you'd be wrong.\n\n:::: {layout-ncol=\"2\"}\n\n::: {#226c9935 .cell execution_count=7}\n\n::: {.cell-output .cell-output-display}\n![$x = {c_1 \\over 1 - 2c_1} \\quad y = {s_1 \\over 1 - 2c_1}$](index_files/figure-html/cell-7-output-1.png){}\n:::\n:::\n\n\n::: {}\n$$\n\\begin{gather*}\n \\begin{align*}\n x &= {c_1 \\over 1 - 2c_1} = {1 - t^2 \\over 3t^2 - 1}\n \\\\\n y &= {s_1 \\over 1 - 2c_1} = {2t \\over 3t^2 - 1}\n \\\\[14pt]\n 3x^2 - y^2 &= {3 - 6t^2 + 3t^4 - 4t^2 \\over 3t^2 - 1}\n \\\\\n &= {(t^2 - 3)(3t^2 - 1) \\over (3t^2 - 1)^2 }\n \\\\\n &= {t^2 - 3 \\over 3t^2 - 1 }\n = ... = -4x - 1\n \\end{align*}\n \\\\[14pt]\n 3x^2 - y^2 + 4x + 1 = 0\n\\end{gather*}\n$$\n:::\n::::\n\nAt the very least, the occurrences of 1 in the place of 4 have a simple explanation: 1 = 4 - 3.\n\n\n### Archimedean Spiral\n\nArguably the simplest (non-circular) polar curve is $r(\\theta) = \\theta$, the unit\n [Archimedean spiral](https://en.wikipedia.org/wiki/Archimedean_spiral).\nSince the curve is defined by a constant turning, this is a natural application of the properties\n of sine and cosine.\nThe closest equivalent in rational polar coordinates is $R_1(t) = t$.\nBut this can be converted to an implicit form:\n\n$$\n\\begin{gather*}\n x = tc_1 \\qquad y = ts_1\n \\\\[14pt]\n x^2 + y^2 = t^2(c_1^2 + s_1^2) = t^2\n \\\\\n y = {2t^2 \\over 1 + t^2} = {2(x^2 + y^2) \\over 1 + (x^2 + y^2)}\n \\\\[14pt]\n (1 + x^2 + y^2)y = 2(x^2 + y^2)\n\\end{gather*}\n$$\n\nThe curve produced by this equation is a\n [right strophoid](https://mathworld.wolfram.com/RightStrophoid.html)\n with a node at (0, 1) and asymptote $y = 2$.\nThis form suggests something interesting about this curve: it approximates the Archimedean spiral\n (specifically the one with polar equation $r(\\theta) = \\theta/2$).\nIndeed, the sequence of curves with parametrization $R_n(t) = 2nt$ approximate the (unit) spiral\n for larger *n*, as can be seen in the following video.\n\n\n\n::: {#fig-approx-archimedes}\n{{< video ./approximate_archimedes.mp4 >}}\n\nApproximations to the Archimedean spiral\n:::\n\n\nSince R necessarily defines a rational curve, the curves will never be equal,\n just as any stretching of $c_n$ will never exactly become cosine.\n\n\nClosing\n-------\n\nSine, cosine, and the exponential function, are useful in a calculus setting precisely\n because of their constant \"velocity\" around the circle.\nAlso, nearly every modern scientific calculator in the world features buttons\n for trigonometric functions, so there seems to be no reason *not* to use them.\n\nWe can however be misled by their apparent omnipresence.\nStereographic projection has been around for *millennia*, and not every formula needs to be rewritten\n in its language.\nFor example (and as previously mentioned), defining the Chebyshev polynomials really only requires\n understanding the multiplication of two complex numbers whose norm cannot grow,\n not trigonometry and dividing angles.\nMany other instances of sine and cosine merely rely on a number (or ratio) of loops around a circle.\nWhen velocity does not factor, it will obviously do to \"stay rational\".\n\nOne of my favorite things to plot as a kid were polar roses, so I was somewhat intrigued\n to see that they are, in fact, rational curves.\nOn the other hand, their rationality follows immediately from the rationality of the circle\n (which itself follows from the existence of Pythagorean triples).\nIf I were more experienced with manipulating Chebyshev polynomials or willing to set up a\n linear system in (way too) many terms, I might have considered attempting to find\n an implicit form for them as well.\n\nDiagrams created with Sympy and Matplotlib.\n\n", + "supporting": [ + "index_files" + ], + "filters": [], + "includes": {} + } +} \ No newline at end of file diff --git a/_freeze/posts/stereo/2/index/figure-html/cell-5-output-1.png b/_freeze/posts/stereo/2/index/figure-html/cell-5-output-1.png new file mode 100644 index 0000000000000000000000000000000000000000..35d0d33b5406f46f0373801307e5fb8d2abe3814 GIT binary patch literal 15018 zcmd^mWmHt%+xG!cFc1(?l$Mf4Qd&X8k?s-@kxuCb1wl|j2|;3z?(RmA5@`mcL2|$$ zgdv7{_d);9$LD=MJRjb*-g_u$4 zw{UiLa1!O=vHhEm`OMH zC)mXG&E9vObei%dH4|^2&j%6P`Qx3zyAm-;kZ_muN`iDm{BJWEX`wrAvk@hoi=Afn zSE{V(nQU$*s+jz)5*~H@c(DS-rR%~py28J7%wSn$^P|Hdjep_DlUB+j_J;2C)1v9a zp8dq6Zr)2WK%l1J=art|6Cya*d9w;p%X{ zx|Y`Pfp0})qtTf&XON9%ugRf@!Sb`vtL9$c9sbs~ww1No^&Q-3FrBXsI0__PXJ_Xe 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a/posts/stereo/2/index.qmd b/posts/stereo/2/index.qmd index 475b609..7631557 100644 --- a/posts/stereo/2/index.qmd +++ b/posts/stereo/2/index.qmd @@ -12,7 +12,7 @@ categories: - algebra - complex analysis - polar roses - - generating function + - generating functions ---

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