finish up stereo.3; minor revisions to stereo.2

posts/math/stereo/2/index.qmd

@@ -111,36 +111,37 @@ This is also the *only* property upon which the recurrence depends; all else is
 Knowing this, let's start over with the stereographic projection of the circle:
 
 $$
-o_1(t) = {1 + it \over 1 - it}
+o(t) = {1 + it \over 1 - it}
   = {1 - t^2 \over 1 + t^2} + i {2t \over 1 + t^2}
   = \text{c}_1 + i\text{s}_1
 $$
 
-The subscript "1" is because as *t* ranges over $(-\infty, \infty)$, the function loops once
+The subscript "1" for *c* and *s* is because as *t* ranges over $[-\infty, \infty)$,
-  around the unit circle.
+  the function loops once around the unit circle.
-Taking this to higher powers keeps points on the circle since all points on the circle
+Taking *o* 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:
 
 $$
 \begin{align*}
-  o_n &= (o_1)^n
+  o^n
-    = \left( {1 + it \over 1 - it} \right)^n
+  &= \left( {1 + it \over 1 - it} \right)^n
   \\
-  \text{c}_n + i\text{s}_n
   &= (\text{c}_1 + i\text{s}_1)^n
+  \\
+  &= \text{c}_n + i\text{s}_n
 \end{align*}
 $$
 
 This mirrors raising the complex exponential to a power
-  (which loops over the range $(-\pi, \pi)$ instead).
+  (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:
 
 $$
 \begin{align*}
-  o_2 = (o_1)^2 &= (\text{c}_1 + i\text{s}_1)^2
+  o^2 &= (\text{c}_1 + i\text{s}_1)^2
   \\
   &= \text{c}_1^2 + 2i\text{c}_1\text{s}_1 - \text{s}_1^2
     + (0 = \text{c}_1^2 + \text{s}_1^2 - 1)
@@ -149,11 +150,11 @@ $$
   \\
   &= 2\text{c}_1(\text{c}_1 + i\text{s}_1) - 1
   \\
-  &= 2\text{c}_1 o_1 - 1
+  &= 2\text{c}_1 o - 1
   \\
-  o_2 \cdot (o_1)^n &= 2\text{c}_1 o_1 \cdot (o_1)^n - (o_1)^n
+  o^2 \cdot o^n &= 2\text{c}_1 o \cdot o^n - o^n
   \\
-  o_{n+2} &= 2\text{c}_1 o_{n+1} - o_n
+  o^{n+2} &= 2\text{c}_1 o^{n+1} - o^n
 \end{align*}
 $$
 
@@ -211,7 +212,7 @@ Meanwhile, the complex stereograph has derivative
 
 $$
 \begin{align*}
-  {d \over dt} o_1(t) &= {d \over dt} {1 + it \over 1 - it}
+  {d \over dt} o(t) &= {d \over dt} {1 + it \over 1 - it}
     = {i(1 - it) + i(1 + it) \over (1 - it)^2}
   \\
   &= {2i \over (1 - it)^2}
@@ -224,20 +225,19 @@ $$
   \\
   &= -(1 + c_1)s_1 + i(1 + c_1)c_1
   \\
-  &= i(1 + c_1)o_1
+  &= i(1 + c_1)o
 \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
+However, the expression also accrues $1 + c_1$ as an adjustment factor.
-  for its failure to be the complex exponential.
+The complex exponential doesn't need this term because it goes around the circle at a constant rate,
-Sine and cosine obey a simpler relationship with respect to the derivative,
+  resulting in 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
+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.
@@ -249,16 +249,16 @@ $$
   = 2\pi i
 $$
 
-In this example, *Γ* is a counterclockwise curve parametrized by *γ* which loops once around
+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?
+Normally this equality is demonstrated with Γ as the unit circle and *γ* as the complex exponential?
-If *Γ* is the unit circle, the integral is:
+But will *o* work just as well in place of the latter?
 
 $$
 \oint_\Gamma {1 \over z} dz
-  = \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt
+  = \int_{-\infty}^\infty {o'(t) \over o(t)} dt
   = \int_{-\infty}^\infty i(1 + c_1(t)) dt
   = 2i\int_{-\infty}^\infty {1 \over 1 + t^2} dt
 $$
@@ -272,30 +272,30 @@ Since powers of *o* are more loops around the circle, the chain and power rules
 
 $$
 \begin{gather*}
-  {d \over dt} (o_1)^n = n(o_1)^{n-1} {d \over dt} o_1
+  {d \over dt} o^n = no^{n-1} {d \over dt} o
   \\[14pt]
   \oint_\Gamma {1 \over z} dz
-    = \int_{-\infty}^\infty {n o_1(t)^{n-1} o_1'(t) \over o_1(t)^n} dt
+    = \int_{-\infty}^\infty {n o(t)^{n-1} o'(t) \over o(t)^n} dt
-    = n \int_{-\infty}^\infty {o_1'(t) \over o_1(t)} dt
+    = n \int_{-\infty}^\infty {o'(t) \over o(t)} dt
     = 2 \pi i n
 \end{gather*}
 $$
 
 It is certainly possible to perform these contour integrals along straight lines
-  in the complex plane; in fact, making *Γ* a diamond-shaped contour from
+  in the complex plane; in fact, making Γ a diamond-shaped contour from
   1 to *i* to -1 to -*i* produces 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.
-The integral can be shifted back into the realm of plausibility by considering simpler bounds on $o_2$:
+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
+    &= \int_{-1}^1 {2 o(t) o'(t) \over o(t)^2} dt
   \\
-  &= 2 \int_{-1}^1 {o_1'(t) \over o_1(t)} dt
+  &= 2 \int_{-1}^1 {o'(t) \over o(t)} dt
   \\
   &= 2(2i\arctan(1) - 2i\arctan(-1))
   \\
@@ -309,7 +309,7 @@ This series converges for $-1 \le t \le 1$, which happens to match the bounds of
 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, we would instead use the bounds $(0, 2\pi)$,
+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?
@@ -361,7 +361,7 @@ $$
 x(t) = c_p(t) c_1(t) \qquad y(t) = c_p(t) s_1(t)
 $$
 
-will plot a $p/1$ polar rose as t ranges over $(-\infty, \infty)$.
+will plot a $p/1$ polar rose as t ranges over $[-\infty, \infty)$.
 
 ```{python}
 #| echo: false
@@ -413,15 +413,15 @@ 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
+$q = 1$ happens to match the subscript of a *c* term of *x* and the *s* term of *y*,
-  whether the other polar curves can be obtained by allowing it to vary as well.
+  so one might wonder whether the other polar curves can be obtained by allowing it to vary.
 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 $(-\infty, \infty)$.
+will plot a $p/q$ polar rose as t ranges over $[-\infty, \infty)$.
 
 ```{python}
 #| echo: false
@@ -443,7 +443,7 @@ 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.
+  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.
@@ -522,9 +522,9 @@ $$
 ::::
 
 *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
+Technically, it is missing the point where $y = 0 ~ (t = \infty)$.
-  where using a higher $c_n$ would help.
+Unfortunately, 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
+It is however, similar to the situation where we allow $o(\infty) = -1$, and an argument
   can be made to waive away any concerns one might have.
 
 
@@ -562,9 +562,9 @@ $$
 :::
 ::::
 
-There isn't an obvious way to combine products of *x* and *y* into a single equation.
+There isn't an obvious way write *x* and *y* above as an implicit equation for the curve.
-The general form of a conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, so
+The general form of a conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$,
-   we know that the implicit equation for the curve almost certainly involves $x^2$ and $y^2$.
+  so we know that such an equation curve almost certainly involves $x^2$ and $y^2$.
 
 $$
 x^2 = {4 - 8t^2 + 4t^4 \over (3t^2 + 1)^2} \qquad
@@ -612,7 +612,7 @@ $$
 $$
 
 Notably, the coefficients of *x* and *y* are 3 and 4.
-Simultaneously, $o_1(\varepsilon) = o_1(1/2) = {3 \over 5} + i{4 \over 5}$.
+Simultaneously, $o(\varepsilon) = o(1/2) = {3 \over 5} + i{4 \over 5}$.
 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.
@@ -621,7 +621,7 @@ This binds together three concepts: the simplest case of the Pythagorean theorem
 #### 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
+Since $o(2)$ is a reflection of $o(1/2)$, 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.
 
@@ -751,7 +751,7 @@ Approximations to the Archimedean spiral
 :::
 
 
-Since R necessarily defines a rational curve, the curves will never be equal,
+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.
 
 

posts/math/stereo/3/anim.py

@@ -0,0 +1 @@
+../2/anim.py

posts/math/stereo/3/index.qmd

@@ -1,12 +1,12 @@
 ---
 title: "Stereographic Hyperspheres"
 description: |
-  TODO
+  How do you explicitly describe *n*-dimensional spheres?
 format:
   html:
     html-math-method: katex
 jupyter: python3
-date: "2026-09-11"
+date: "2026-10-04"
 categories:
   - algebra
   - topology
@@ -14,22 +14,43 @@ categories:
 draft: true
 ---
 
+<style>
+.figure-img {
+  max-width: 512px;
+  object-fit: contain;
+  height: 100%;
+}
+</style>
 
-In [the first post of this series](../1/), we explored the application of the quaternions
+```{python}
-  to rotation in three dimensions.
+#| echo: false
-Rotation is one problem, but the quaternions and the characterization of the sphere given
-  correspond to another: how do we parameterize higher-dimensional spheres?
 
-It's relatively easy to describe spheres implicitly using coordinates.
+import numpy as np
-The natural definition is the locus of points which all have the same distance to the origin
+import matplotlib.pyplot as plt
-  in Euclidean space.
+import sympy
-In other words, a point $(x_0, x_1, x_2, ... x_n)$ is on a unit hypersphere
+import sympy.plotting as plot
-  in *n*+1-dimensional space if
+from sympy.abc import t
+
+from anim import SympyAnimationWrapper
+```
+
+In [the first post of this series](../1/), we explored a definition of quaternions
+  and their application to rotation in three dimensions.
+In doing so, we used stereography to define a point on the 2-sphere, i.e.,
+  one whose coordinates satisfy $x^2 + y^2 + z^2 = 1$.
+
+It's easy to extend this implicit equation to higher-dimensional spheres (or hyperspheres).
+A point $(x_0, x_1, x_2, ... x_n)$ is on a unit hypersphere in *n*+1-dimensional
+  Euclidean space if
 
 $$
 x_0^2 + x_1^2 + x_2^2 + ... + x_n^2 = 1
 $$
 
+This follows naturally from the definition of the sphere as the locus of points which
+  all have the same distance to the origin.
+Since this has an implicit equation, there's a natural question: how do we parameterize hyperspheres?
+
 
 Guidance from Lower Dimensions
 ------------------------------
@@ -54,6 +75,9 @@ o_2(s, t) = {1 + is + jt \over 1 - is - jt}
 $$
 
 These are valid constructions because division works for both complex numbers and quaternions.
+But the ability to divide
+  [isn't very common in higher dimensions](https://mathworld.wolfram.com/DivisionAlgebra.html),
+  so without justification, we can't extend it and hope the math works out.
 
 
 ### Topological Insights
@@ -100,13 +124,16 @@ In another sense, the real space is the extra dimension into which the sphere ex
 Topologically, this description of the resulting space is called the
   [one-point compactification](https://mathworld.wolfram.com/One-PointCompactification.html).
 In other words, the circle is the one-point compactification of the line,
-  and in general, an *n* sphere is the one-point compactification of Euclidean *n*-space.
+  and in general, an *n*-sphere is the one-point compactification of Euclidean *n*-space.
 
 $$
 \mathbb{E}^{n} \cup \{ \infty \} \cong S^n
 $$
 
-![]()
+![
+  One-point compactification of 1- (top) and 2- (bottom) dimensional Euclidean space.
+  Both spaces extend indefinitely in the indicated directions before joining up at $\infty$.
+](./one-point_compactification.png)
 
 
 ### Invariance of Dimension
@@ -118,31 +145,32 @@ In fact, when constructing the 2-sphere, all we cared about was that *i* and *j*
 
 [Geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra) gives some tools to generalize
   this argument to higher dimensions.
-In an *n*-dimensional algebra, we have unit vectors $e_0, e_1, ..., e_{n-1}$
+In an *n*-dimensional algebra, we have unit vectors ${\vec e_0}, {\vec e_1}, ..., {\vec e_{n-1}}$
   and the following properties:
 
 - Scalars and vectors can be added and multiplied together,
   and all possibilities comprise the algebra
 - The product of a unit vector with itself is a scalar, generally chosen among -1, 0, or 1
 - Scalars commute, but the product of two different unit vectors anticommutes
-  - e.g., $e_0 e_1 = - e_1 e_0$
+  - e.g., ${\vec e_0} {\vec e_1} = - {\vec e_1} {\vec e_0}$
   - Consequently, the square of the product is the negative of the product of the squares
-    - e.g., $e_0 e_1 e_0 e_1 = - e_0 e_1 e_1 e_0 = - e_0^2 e_1^2$
+    - e.g., ${\vec e_0} {\vec e_1} {\vec e_0} {\vec e_1} = - {\vec e_0} {\vec e_1} {\vec e_1} {\vec e_0} = - {\vec e_0^2} {\vec e_1^2}$
 - Division by anything other than scalars is undefined
 
-A consequence is that the square of a general vector ***v*** with components $x_k e_k$ is a scalar.
+A consequence is that the square of a general vector $\vec v$ with components
+  ${\vec e_k}x_k$ is a scalar.
 This can be seen by arranging the components of the product after distributing as a square:
 
 $$
 \begin{align*}
-  {\bm v} &= e_0 x_0 + e_1 x_1 + ... e_{n-1} x_{n-1} = \sum_k e_k x_k
+  {\vec v} &= {\vec e_0} x_0 + {\vec e_1} x_1 + ... {\vec e_{n-1}} x_{n-1} = \sum_k {\vec e_k} x_k
   \\
-  {\bm v}^2 &= (\sum_k^{n-1} e_k x_k) (\sum_l^{n-1} e_l x_l)
+  {\vec v^2} &= (\sum_k^{n-1} {\vec e_k} x_k) (\sum_l^{n-1} {\vec e_l} x_l)
-  = \sum_k^{n-1} \sum_l^{n-1} e_k e_l x_k x_l
+  = \sum_k^{n-1} \sum_l^{n-1} {\vec e_k} {\vec e_l} x_k x_l
   \\
-  &= \underset{\diagdown}{\sum_k^{n-1} e_k^2 x_k^2}
+  &= \underset{\diagdown}{\sum_k^{n-1} {\vec e_k^2} x_k^2}
-  + \underset{◥}{ \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l }
+  + \underset{◥}{ \sum_k^{n-1} \sum_{l > k} {\vec e_k} {\vec e_l} x_k x_l }
-  + \underset{◣}{ \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l }
+  + \underset{◣}{ \sum_k^{n-1} \sum_{l < k} {\vec e_k} {\vec e_l} x_k x_l }
 \end{align*}
 $$
 
@@ -150,18 +178,18 @@ Due to anticommutativity, we can cancel the upper and lower triangles, leaving o
 
 $$
 \begin{align*}
-  ◣ &= \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l
+  ◣ &= \sum_k^{n-1} \sum_{l < k} {\vec e_k} {\vec e_l} x_k x_l
-  = \sum_k^{n-1} \sum_{k < l} e_l e_k x_l x_k
+  = \sum_k^{n-1} \sum_{k < l} {\vec e_l} {\vec e_k} x_l x_k
   \\
-  &= - \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l
+  &= - \sum_k^{n-1} \sum_{l > k} {\vec e_k} {\vec e_l} x_k x_l
   = - ◥
   \\[10pt]
   &\implies \diagdown + ◥ + ◣ = \diagdown + ◥ - ◥ = \diagdown
 \end{align*}
 $$
 
-To align with the prior examples *i* and *j*, we'll assume that $e_k^2 = -1$ for all *k*.
+To align with the prior examples *i* and *j*, we'll assume that ${\vec e_k^2} = -1$ for all *k*.
-This means that ${\bm v}^2 = - ||{\bm v}||$, the sum of squares of the extent
+This means that ${\vec v}^2 = - ||{\vec v}||$, the sum of squares of the extent
   in each basis (or Euclidean norm).
 
 
@@ -170,36 +198,37 @@ This means that ${\bm v}^2 = - ||{\bm v}||$, the sum of squares of the extent
 Finally, we can consider an expression analogous to the one from which we derived
   the 1- and 2-spheres.
 
-Suppose that a vector and a scalar are added together, as $a + {\bm v}$.
+Suppose that a vector and a scalar are added together, as $a + {\vec u}$.
 If this point is on a sphere and the scalar component is considered the extent in a new dimension,
   then the norm of the entire quantity should be
 
 $$
-||a + {\bm v}|| = a^2 + ||{\bm v}|| = a^2 - {\bm v}^2 = 1
+||a + {\vec u}|| = a^2 + ||{\vec u}|| = a^2 - {\vec u}^2 = 1
 $$
 
-As a vector ***u*** ranges over *n*-dimensional space, the expression...
+Now let a vector $\vec v$ range over *n*-dimensional space.
+The expression...
 
 $$
-o_n({\bm u}) = {1 + {\bm u} \over 1 - {\bm u}}
+{\bm o_n}({\vec v}) = a + {\vec u} = {1 + {\vec v} \over 1 - {\vec v}}
 $$
 
 ...seems to be a ratio between two distinct quantities with the same norm,
-  since $1^2 - {\bm u}^2 = 1^2 - (-{\bm u})^2$.
+  since $1^2 - {\vec v}^2 = 1^2 - (-{\vec v})^2$.
-This expression is actually ill-defined since there is a vector in the denominator,
+However, it's ill-defined since there is a vector we can't divide by in the denominator.
-  but we can use a conjugation trick to clear it:
+Due to the properties of the algebra, we can use a conjugation trick to clear it:
 
 $$
 \begin{align*}
-  {1 + {\bm u} \over 1 - {\bm u}}
+  {1 + {\vec v} \over 1 - {\vec v}}
-  &= \left( {1 + {\bm u} \over 1 - {\bm u}} \right)
+  &= \left( {1 + {\vec v} \over 1 - {\vec v}} \right)
-  \left( {1 + {\bm u} \over 1 + {\bm u}} \right)
+  \left( {1 + {\vec v} \over 1 + {\vec v}} \right)
-  = {(1 + {\bm u})^2 \over (1 - {\bm u})(1 + {\bm u})}
+  = {(1 + {\vec v})^2 \over (1 - {\vec v})(1 + {\vec v})}
   \\
-  &= {1 + 2{\bm u} + {\bm u}^2 \over 1 - {\bm u}^2}
+  &= {1 + 2{\vec v} + {\vec v}^2 \over 1 - {\vec v}^2}
   \\
-  &= {1 - ||{\bm u}|| \over 1 + ||{\bm u}||} + {2{\bm u} \over 1 + ||{\bm u}||}
+  &= {1 - ||{\vec v}|| \over 1 + ||{\vec v}||} + {2{\vec v} \over 1 + ||{\vec v}||}
-  = a + {\bm v}
+  = a + {\vec u}
 \end{align*}
 $$
 
@@ -209,20 +238,20 @@ We can also show that the norm of this expression is 1, as desired:
 
 $$
 \begin{align*}
-  a^2 - {\bm v}^2 &= 1
+  a^2 - {\vec v}^2 &= 1
 \\
   \implies
-  \stackrel{\text{Numerator of } a}{(1 + {\bm u}^2)^2}
+  \stackrel{\text{Numerator of } a}{(1 + {\vec v}^2)^2}
-  - \stackrel{\text{Numerator of } \bm v}{(2{\bm u})^2}
+  - \stackrel{\text{Numerator of } \vec u}{(2{\vec v})^2}
-  &= \stackrel{\text{Common denominator}}{1 - {\bm u}^2}
+  &= \stackrel{\text{Common denominator}}{1 - {\vec v}^2}
 \end{align*}
 $$
 
-This is true no matter how many dimensions ***v*** has[^1], justifying our earlier abuse of notation.
+This is true no matter how many dimensions $\vec v$ has[^1], justifying our earlier abuse of notation.
 
 [^1]: Technically, this should only hold for finitely many dimensions.
   The $\infty$-sphere, composed of vectors with only finitely many nonzero components,
-  is probably also valid under this construction, but I haven't proven this.
+  is probably also valid under this construction, but it warrants a proper proof.
 
 
 Inducing an Alternative
@@ -243,100 +272,120 @@ This can (topologically) be turned into a 1-sphere $S^1$ (the circle) by an oper
   all points in the space to two new, auxiliary points.
 Subsequently, we can take the circle and repeat the operation to build the 2-sphere $S^2$.
 
+![
+  Example of suspension of the 0- and 1-spheres, forming the 1- and 2-spheres, respectively.
+  The space is duplicated along the blue lines except at the two blue endpoints.
+](suspension.png)
+
 In general,
 
 $$
 \text{Susp}(S^{n-1}) = S^n
 $$
 
-![]()
 
+### Algebraic Suspension
 
-### Algebraic Dual, Part 2
+Let's compare the topological definition with what we have algebraically.
-
+We first definied the circle, or 1-sphere as
-Let's look at the first interesting case.
-We first definied the circle, or 1-dimensional sphere as
 
 $$
 o(t) = {1 + it \over 1 - it}
 = {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2}
 $$
 
-If the real part remains fixed, then in most cases,
+Negating *t* keeps the real part the same, but negates the imaginary part.
-  the space looks two discrete points; to wit, a 0-dimensional sphere.
+So in most cases, where there *is* an imaginary part,
-The remaining two points 1 and -1 are in some sense "new" to the space.
+  the space looks two discrete points; to wit, a 0-sphere.
-
+The remaining two points 1 and -1 are the exception.
-![]()
 
 Similarly, when we intersect the 2-sphere with a plane along a line of latitude,
-  the space looks like a 1-dimensional sphere except at two points, also 1 and -1.
+  the space looks like a 1-sphere except at the two poles, also 1 and -1.
 
-![]()
+![
+  The sphere, as parametrized by ${\bm o_2}({\vec v})$, being intersected by a plane.
+  The plane is perpendicular to the line connecting -1 and 1 and intersects the sphere
+    in a circle (1-sphere).
+](parametrized_suspension.png)
 
-If we create a 2D vector with components in the real and imaginary parts of *o*,
+Halfway between the two poles, at the equator, the scalar component is 0 and the sphere is a pure vector.
-  we can immediately create an expression for the 2-sphere:
+For a general sphere, this happens when $||{\vec v}|| = 1$:
+
+$$
+{\bm o_n}({\vec v})
+= {1 - ||{\vec v}|| \over 1 + ||{\vec v}||}
++ {2{\vec v} \over 1 + ||{\vec v}||}
+= {1 - 1 \over 1 + 1} + {2 \over 1 + 1}{\vec v}
+= {\vec v}
+$$
+
+In general, this happens when $\vec v$ is a point on a unit sphere of one dimension lower.
+For the 2-sphere, this is a 1-sphere, which we can easily parametrize using *o*.
+Being a unit sphere, all points on it behave similarly to *i* in that their square is -1.
+Thus, we can construct an expression for the 2-sphere by replacing *i* with the vector in question.
 
 $$
 \begin{align*}
-  {\bm w}_1(t) &= {1 - t^2 \over 1 + t^2} e_0 + {2t \over 1 + t^2} e_1
+  {\vec v} = {\vec w}_1(s)
+  &= {1 - s^2 \over 1 + s^2} e_0 + {2s \over 1 + s^2} e_1
   \\[10pt]
-  \varsigma_2(s,t) &= {1 + {\bm w}_1(t)s \over 1 - {\bm w}_1(t)s}
+  {\bm \varsigma}_2(s,t) &= {1 + {\vec w}_1(s)t \over 1 - {\vec w}_1(s)t}
-  = {1 - s^2 \over 1 + s^2} + {2s \over 1 + s^2} {\bm w}_1(t)
+  = {1 - t^2 \over 1 + t^2} + {2t \over 1 + t^2} {\vec w}_1(s)
 \end{align*}
 $$
 
-Note that if *s* is exchanged with *-s*, then the scalar part remains the same,
+Just like with *o*, if *t* is replaced with *-t* in the above expression,
-  but the vector part, which corresponds to latitudinal circles, is negated.
+  then the scalar part remains the same, but the vector part
-In effect, this means that if *s* is allowed to range over negative numbers,
+  (which corresponds to latitudinal circles) is negated.
-  we will produce two duplicate circles.
+Since the circle is a connected space, we only need one of the two circles this generates,
+  and *s* must range over $[0, \infty]$.
+*s*, however, ranges over $[-\infty, \infty)$, since that's the domain of *o*.
 
 This process can be continued indefinitely -- at each stage,
-  $\varsigma_k$[^2] describes a *k*-dimensional unit sphere.
+  $\bm \varsigma_n$[^2] describes a *n*-dimensional unit sphere.
-It be converted to a pure vector ${\bm w}_k$ by multiplying the scalar component
+It can be converted to a pure vector ${\vec w}_n$ by multiplying the scalar component
-  with a new unit vector $e_k$.
+  with a new unit vector $e_n$.
-In this form, ${\bm w}_k^2 = -1$ for any *k*-dimensional algebra[^3].
+In this form, ${\vec w}_n^2 = -1$ for any *n*-dimensional algebra[^3].
 This provides an inductive construction parallel to the topological one.
 
-[^2]: For "σφαίρα", sphere. I'm using ς in hope that it'll be less prone to confusion with "o".
+[^2]: For "σφαίρα", sphere. I'm using ς rather than σ in hope that it's less prone to confusion with "o".
 [^3]: This should sound familiar from the first post -- it matches the "unit quaternions".
 
 $$
 \begin{align*}
-  {\bm w}_k(x_0, x_1, ..., x_{k-1})
+  {\vec w}_n(x_0, x_1, ..., x_{n-1})
-  &= \text{Scalar}(\varsigma_k(x_0, x_1, ..., x_{k-1}))e_k
+  &= V({\bm \varsigma}_n(x_0, x_1, ..., x_{n-1}))
   \\
-  &+ \text{Vector}(\varsigma_k(x_0, x_1, ..., x_{k-1}))
+  &= \text{Scalar}({\bm \varsigma}_n)e_n + \text{Vector}({\bm \varsigma}_n)
   \\
-  \varsigma_{k+1}(x_0, x_1, ..., x_{k-1}, x_k)
+  {\bm \varsigma}_{n+1}(x_0, x_1, ..., x_{n-1}, x_n)
-  &= {1 + {\bm w}_k(x_0, x_1, ..., x_{k-1})x_k \over 1 - {\bm w}_k(x_0, x_1, ..., k_{k-1})x_k}
+  &= {1 + {\vec w}_n(x_0, x_1, ..., x_{n-1})x_n \over 1 - {\vec w}_n(x_0, x_1, ..., n_{n-1})x_n}
+  \\
+  &= {1 - x_n^2 \over 1 + x_n^2} + {2x_n \over 1 + x_n^2}{\vec w_n}
 \end{align*}
 $$
 
-When the new parameter $x_k$ is 0 or $\infty$, the vector part collapses,
+When the new parameter $x_n$ is 0 or $\infty$, the vector part collapses,
   and we get either 1 or -1, the "new points" of the suspension.
 
 $$
 \begin{align*}
-  \varsigma_{k+1}(..., 0)
+  {\bm \varsigma}_{n+1}(..., 0)
-  &= {1 + {\bm w}_k(...)\cdot 0 \over 1 - {\bm w}_k(...) \cdot 0}
+  &= {1 + {\vec w}_n(...)\cdot 0 \over 1 - {\vec w}_n(...) \cdot 0}
   - {1 \over 1} = 1
  \\
-  \varsigma_{k+1}(..., \infty)
+  {\bm \varsigma}_{n+1}(..., \infty)
-  &\approx {1 + {\bm w}_k(...)\cdot \infty \over 1 - {\bm w}_k(...) \cdot \infty}
+  &\approx {1 + {\vec w}_n(...)\cdot \infty \over 1 - {\vec w}_n(...) \cdot \infty}
   \approx {\infty \over -\infty} \approx -1
 \end{align*}
 $$
 
-The phenomenon of duplicate latitudinal spheres is a recurring one in dimensions greater than 1.
+All spheres but the 0-sphere are connected spaces, so duplicate latitudinal spheres
-This can be seen from the 0-sphere being unique among spheres --
+  occur in all dimensions greater than 1.
-  as two discrete, disconnected points, negative numbers are needed for the expected duplication.
+Only in dimension 1 are negative numbers required for the expected duplication.
 More directly, this means that the parameter attached to the 1D case ($x_0$) ranges over
   positive and negative numbers, but all others range over only positve numbers.
 
-:::{TODO}
-We could also construct $\bm w$ from the results in the previous section.
-:::
-
 
 Multiple Wrappings
 ------------------
@@ -350,32 +399,34 @@ Starting with the scalar/vector form of the sphere, we can square the sphere and
 
 $$
 \begin{align*}
-  o_n &= a + {\bm v}
+  {\bm o}_n &= a + {\vec u}
   \\
-  o_n^2 &= (a + {\bm v})^2 = a^2 + {\bm v}^2 + 2a{\bm v}
+  {\bm o}_n^2 &= (a + {\vec u})^2
   \\
-  &= a^2 + {\bm v}^2 + \textcolor{red}{(a^2 - {\bm v}^2 - 1 = 0)} + 2a{\bm v}
+  &= a^2 + {\vec u}^2 + 2a{\vec u} +\textcolor{red}{(0 = a^2 - {\vec u}^2 - 1)}
   \\
-  &= 2a^2 - 1 + 2a{\bm v} = 2a(a + {\bm v}) - 1 = 2a o_n - 1
+  &= 2a^2 - 1 + 2a{\vec u} = 2a(a + {\vec u}) - 1
+  \\
+  &= 2a {\bm o_n} - 1
 \end{align*}
 $$
 
 This gives the familiar recurrence relation...
 
 $$
-o_n^{m+2} = 2a o_n^{m+1} - o_n^m
+{\bm o}_n^{m+2} = 2a {\bm o}_n^{m+1} - {\bm o}_n^m
 $$
 
-...and thus a sphere can be wrapped around itself any number of times, as given by:
+...and thus a sphere can be wrapped around itself any number of times, as given by
 
 $$
-o_n^m = T_m(a) + U_{m-1}(a){\bm u}
+{\bm o}_n^m = T_m(a) + U_{m-1}(a){\vec u}
 $$
 
-*T* and *U* are the standard Chebyshev polynomials.
+where *T* and *U* are the standard Chebyshev polynomials.
 
 
-### Negative Indices and Beyond
+### Negative Indices
 
 The topological equivalent to this statement is
 
@@ -385,23 +436,24 @@ $$
 
 More directly, a map from the *n*-sphere to itself can be characterized by an integer,
   the *degree*, and these compose as integers add.
-Since this is an integer, there's the notion maps in an opposite direction
+
+Since this is an integer, there's the notion of maps in an opposite direction
   which correspond to negative degrees.
-This seems to align with the behavior of the exponent in $o_n^m$.
+This seems to align with the behavior of the exponent *m* in ${\bm o}_n^m$.
 However, we've only defined *m* over positive integers;
-  after all, $o_n$ contains a vector, so we can't really divide by it.
+  after all, $\bm o_n$ contains a vector, so we can't really divide by it.
 
 Fortunately, it's pretty easy to make sense of this.
-Since we have a recurrence relation for the powers of the sphere, we
+Since we have a recurrence relation for the powers of the *n*-sphere, we
   can extend it backwards to define it over negative indices[^4].
 
 $$
 \begin{align*}
-  o_n^1 &= 2a o_n^{0} - o_n^{-1}
+{\bm o}_n^1 &= 2a {\bm o}_n^{0} - {\bm o}_n^{-1}
   \\
-  a + {\bm v} &= 2a - o_n^{-1}
+  a + {\vec u} &= 2a - {\bm o}_n^{-1}
   \\
-  o_n^{-1} &= a - {\bm v}
+  {\bm o}_n^{-1} &= a - {\vec u}
 \end{align*}
 $$
 
@@ -414,84 +466,144 @@ $$
 This actually aligns with what we'd expect according to adding powers, since:
 
 $$
-o_n^1 \cdot o_n^{-1} = (a + {\bm v})(a - {\bm v}) = a^2 - {\bm v}^2 = 1 = o_n^0
+({\bm o}_n^1)({\bm o}_n^{-1})
+= (a + {\vec u})(a - {\vec u}) = a^2 - {\vec u}^2
+= 1 = {\bm o}_n^0
 $$
 
 
 ### Degrees and Induction
 
-In the inductive case, since ${\bm w}^2 = -1$, we can pull a similar trick.
+The inductive case was established by noticing that a vector $\vec w$ lying on a unit sphere
-Replacing *i* with ***w***, the only thing that needs changing from the previous article is
+  behaves similarly to *i* in that that ${\vec w}^2 = -1$.
-  replacing "real" with "scalar" and "nonreal" with "vector".
+We can use the same trick for higher-order wrappings -- the only thing that needs changing
+  from the previous article is replacing "real" with "scalar" and "nonreal" with "vector".
 
 $$
-\varsigma_n^m
+{\bm \varsigma}_n^m
-= ( c + s \cdot { {\bm w}_{n-1}} )^m
+= ( c + s { {\vec w}_{n-1}} )^m
-= T_m(c) + s U_m(c) \cdot {\bm w}_{n-1}
+= T_m(c) + s U_m(c) {\vec w}_{n-1}
 $$
 
 Similarly,
 
 $$
 \begin{align*}
-\varsigma_n^{-1} &= ( c - s \cdot { {\bm w}_{n-1}} )
+{\bm \varsigma}_n^{-1} &= ( c - s{\vec w}_{n-1} )
 \\
-\varsigma_n^{1} \cdot \varsigma_n^{-1}
+({\bm \varsigma}_n^{1}) ({\bm \varsigma}_n^{-1})
-  &= ( c + s \cdot {{\bm w}_{n-1}} )( c - s \cdot {{\bm w}_{n-1}} )
+  &= ( c + s{\vec w}_{n-1} )( c - s{\vec w}_{n-1} )
   \\
-  &= c^2 - s^2 {\bm w}_{n-1}^2 = c^2 + s^2
+  &= c^2 - s^2 {\vec w}_{n-1}^2 = c^2 + s^2 = 1
   \\
-  &= 1 = \varsigma_n^0
+  &= {\bm \varsigma}_n^0
 \end{align*}
 $$
 
+Technically, $\bm \varsigma_n$ is already a higher-degree map when all parameters
+  (besides the one from the base case) are allowed to range over negative numbers.
+In this case, $\bm \varsigma_2^\pm$ is a degree-2 map, $\bm \varsigma_3^\pm$ is a degree-4 map,
+  and $\bm \varsigma_n^\pm$ is a degree-$2^{n-1}$ map.
 
-Degree 2 and the Equator
-------------------------
 
-The prior discussion about degree gives us the tools to address "points at infinity".
+De-infinitizing
-As a reminder, if we let a vector ***u*** range over the entirety of Euclidean space,
+---------------
-  the sphere is only closed by allowing such an extra point.
 
-We can get rid of this point for the circle by considering a degree 2 map rather than a degree 1 map.
+The degree also gives us the tools to address "points at infinity".
-The former wraps around once $-\infty$ to $\infty$, while the latter wraps around once from -1 to 1.
+If $\vec w_n$ is a point on the equatorial unit *n-1*-sphere, then $\bm o_n$
-Conveniently, -1 and 1 are both the points at which the real part of the degree 1 map becomes 0.
+  behaves as the identity.
-Points in this range lie on the semicircle bounded by these two points (which form a 0-sphere).
+But we also know that it squares to -1, and that squaring $\bm o_n$ produces a degree-2 map.
-
-![]()
-
-For higher-dimensional spheres we just need to replace "real part" with "scalar part".
-For a sphere $o_n({\bm u}_n)$, this is precisely when ${\bm u}_n$ has a norm of 1.
 
 $$
-o_n({\bm u}_n)
+{\bm o}_n({\vec w}_n)^2 = {\vec w}_n^2 = -1
-= {1 - ||{\bm u_n}|| \over 1 + ||{\bm u_n}||}
-+ {2{\bm u_n} \over 1 + ||{\bm u_n}||}
-= {1 - 1 \over 1 + 1} + {2 \over 1 + 1}{\bm u_n}
-= {\bm u}_n
 $$
 
-Let *n* = 2 so we can plot it.
+This means that the degree-2 map can be interpreted as collapsing the equator
-Then we're actually talking about place in which the unit sphere in 3D space looks like the unit circle.
+  to a single point, the pole -1.
-Specifically, this unit circle can be considered an equator bounding the hemisphere around the scalar 1.
+The hemi-*n*-sphere surrounding the antipode 1 gets closed, resulting in the whole *n*-sphere.
 
-![]()
+```{python}
+#| code-fold: true
+#| output: false
 
-In general ${\bm u}_n$ has a norm of 1 exactly when it's a point on the equatorial *n-1*-sphere.
+# circle map
-Conveniently, under the degree 2 map, this equatorial sphere collapses to a single point.
+s,t = sympy.symbols("s t", real=True)
+o = (1 + sympy.I*s) / (1 - sympy.I*s)
 
-$$
+# doubled map for finite range
-o_n({\bm u}_n)^2
+o2 = o**2
-= {\bm u}_n^2 = -1
+o2_real, o2_imag = o2.as_real_imag()
-$$
 
-This point was formerly the image of the point at infinity,
+# inductive 2-sphere
-  eliminating its necessity in the description of the *n*-sphere.
+sphere_x = o2_real.subs(s,t)
+sphere_y = o2_imag.subs(s,t)*o2_real
+sphere_z = o2_imag.subs(s,t)*o2_imag
+
+def animate_sphere(filename: str, n=30, interval=80):
+  lerp_steps = np.linspace(0, 1, n)
+  t_hemisphere = 2**0.5 - 1
+
+  with SympyAnimationWrapper(filename) as animate:
+    @animate(len(lerp_steps), interval=interval)
+    def ret(fr):
+        plt.clf()
+        lerp = lerp_steps[fr]
+
+        t_upper = t_hemisphere*(1 - lerp) + 1*lerp
+        p = plot.plot3d_parametric_surface(
+            sphere_x, sphere_y, sphere_z,
+            (s, -1, 1), (t, 0, t_upper),
+            xlim=(-1,1), ylim=(-1,1), zlim=(-1,1),
+            show=False,
+            backend="matplotlib",
+        )
+        p2 = plot.plot3d_parametric_line(
+            sphere_x.subs(t, t_upper), sphere_y.subs(t, t_upper), sphere_z.subs(t, t_upper),
+            (s, -1, 1),
+            show=False,
+            backend="matplotlib",
+        )
+        p.append(p2[0])
+        p.show()
+
+    ret.save()  # type: ignore
+
+animate_sphere("close_equatorial_sphere.mp4")
+```
+
+::: {#fig-hemisphere-closure}
+{{< video "./close_equatorial_sphere.mp4" >}}
+
+Effect of the degree-2 map on the hemisphere containing the scalar 1.
+:::
+
+In the one-point construction, this region can only be described using all components of the input vector,
+  since the scalar component depends on it.
+Thus, the domain is made finite just by squaring ***o***.
+
+However, in the inductive construction, the scalar component only depends on
+  the new free parameter, leaving the domain of lower-dimensional spheres unaffected,
+  and potentially still unbounded.
+
+The layered nature of the inductive construction means there are different "levels"
+  at which wraps can be placed.
+For example, for the 2-sphere, the smallest domain for which the entire sphere is parametrized
+  is shown in the table below:
+
+| Sphere                                 | Domain for first wrap around the sphere            |
+|----------------------------------------|----------------------------------------------------|
+| ${\bm o}_2^2({\vec e_0} s + {\vec e_1} t)$      | $s^2 + t^2 \le 1$                            |
+| ${\bm \varsigma}_2^2({\bm \varsigma}_1(s),t)$   | $s \in [-\infty, \infty] \quad t \in [0, 1)$ |
+| ${\bm \varsigma}_2({\bm \varsigma}_1^2(s),t)$   | $s \in [-1, 1] \quad t \in [0, \infty]$      |
+| ${\bm \varsigma}_2^2({\bm \varsigma}_1^2(s),t)$ | $s \in [-1, 1] \quad t \in [0, 1]$           |
+
+A finite domain is only achieved in the final case, corresponding to the combination of
+  two separate degree-2 maps (i.e., a degree-4 map).
 
 
-### Closing the Sphere
+### Closing the Disc
 
-Of course, this comes with another topological analogue.
+Of course, the behavior of the equator comes with another topological analogue.
 Another description of the *n*-sphere is by taking the boundary of an *n*-dimensional disc
   and collapsing its boundary to a single point.
 
@@ -499,21 +611,40 @@ $$
 { D^n / \partial D^n } = S^n
 $$
 
-This exactly aligns with the behavior of the equator when going from the degree 1 to the degree 2 map.
+This exactly aligns with the behavior of the equator when going from the degree-1 to the degree-2 map.
-If ${\bm u}_n$ has a norm of less than or equal to 1, then it lies within a unit disc.
+If ${\vec u}_n$ has a norm of less than or equal to 1, then it lies within a unit disc.
-This unit disc gets sent by $o_n$ to the aforementioned "hemisphere around the scalar 1",
+This unit disc gets sent by $\bm o_n$ to the aforementioned "hemisphere around the scalar 1",
-  and when fed to $o_n^2$, it produces the *n*-sphere.
+  and when fed to $\bm o_n^2$, it produces the *n*-sphere.
 
 
 Closing
 -------
 
-There are a couple of things that I still want to explore here.
+There's still a lot worth discussing here.
-One is the composition of one-point spheres and inductive spheres.
-This structure should describe an (unbounded) lattice, where every "one-point" parametrization
-  has no lesser element, but has a greater element as an element in an inductive parametrization.
-Parametrizations should be considered the same up to a symmetric transformation of coordinates.
 
-There's also a lot of interesting topological arguments to nail down.
+For spheres themselves, one-point spheres provide an base-case in any dimension
-One of these is the degree of the antipodal map.
+  for inductive spheres.
-Another is constructing explicit homotopies purely from algebra.
+This, combined with the choice of degree at each level of induction,
+  grants the potential for many interesting descriptions,
+  which get more numerous in higher dimensions.
+For example, while there's only one degree-4 map for the 1-sphere,
+  there are four for the 2-sphere (depending on choice of bounds).
+
+For topology, I find that these constructions do a lot to nail down its typically abstract nature.
+There are still a lot of interesting arguments to nail down,
+  such as the degree of the antipodal map, or describing explicit, purely algebraic homotopies.
+
+Finally there's the geometric algebra itself.
+Choosing anything but vectors with the expected properties results in surfaces other than spheres.
+This can get even more complicated when considering product of vectors as non-scalar components
+  of the "sphere".
+It's difficult to imagine what these look like in higher dimensions, or what interesting
+  propositions they connect to.
+
+The most convenient part of these constructions is the complexity they manage.
+The alternative is attempting to come up with complicated polynomials
+  in way too many variables to keep track of individually,
+  all while managing equalities between them.
+Instead, algebra serves algebra while also significantly benefitting geometry and topology.
+
+Diagrams created with Geogebra, Sympy and Matplotlib.