4 changed files with 630 additions and 725 deletions
--- a/posts/math/stereo/1/index.qmd
+++ b/posts/math/stereo/1/index.qmd
@ -485,7 +485,7 @@ $$
 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 *j8 be *anticommutative* (meaning that $ij = -ji$) then $stij$ cancels with $stji$.
+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
@ -533,7 +533,7 @@ If you know a little group theory, you might know there are only two nonabelian
 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 of order 2 which commutes with all others.
+    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.
--- a/posts/math/stereo/3/index.qmd
+++ b/posts/math/stereo/3/index.qmd
@ -1,5 +1,5 @@
 ---
-title: "Stereographic Hyperspheres"
+title: "Stereography, Algebraic, and Hyperspheres"
 description: |
  TODO
 format:
@ -9,129 +9,299 @@ jupyter: python3
 date: "2026-09-11"
 categories:
  - algebra
  - topology
  - geometric algebra
 draft: true
 ---
 ```{python}
 #| echo: false
-In [the first post of this series](../1/), we explored the application of the quaternions
+import sympy
-  to rotation in three dimensions.
+from IPython.display import Markdown
-Rotation is one problem, but the quaternions and the characterization of the sphere given
+from tabulate import tabulate
  correspond to another: how do we parameterize higher-dimensional spheres?
-It's relatively easy to describe spheres implicitly using coordinates.
+x, x1, z = sympy.symbols("x x_1 z")
-The natural definition is the locus of points which all have the same distance to the origin
+```
  in Euclidean space.
 In other words, a point $(x_0, x_1, x_2, ... x_n)$ is on a unit hypersphere
  in *n*+1-dimensional space if
-$$
+The Algebra Part
-x_0^2 + x_1^2 + x_2^2 + ... + x_n^2 = 1
+----------------
 $$
 Though I alluded to the ability of the stereoscopic circle to generate the Chebyshev polynomials,
  there is an important caveat which differs their use from typical spheres.
-Guidance from Lower Dimensions
+To review, the stereoscopic definition of the circle is:
 ------------------------------
 Because points on the sphere are constrained by an equation, there is one fewer degree of freedom
  than a general point in the space they occupy.
 Hence, a sphere in *n+1*-dimensional space is itself *n*-dimensional, and is termed an *n*-sphere.
 As a basic example, the complex unit circle is a 1-sphere in the 2-dimensional complex plane:
 $$
 o(t) = {1 + it \over 1 - it}
 = {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2}
 $$
 The explicit map for the 2-sphere is similar; we have two parameters and
  have two "nonreal"s *i* and *j*, which turned out to be quaternions.
 $$
 o_2(s, t) = {1 + is + jt \over 1 - is - jt}
 = {1 - s^2 - t^2 \over 1 + s^2 + t^2} + i{2s \over 1 + s^2} + j{2t \over 1 + t^2}
 $$
 These are valid constructions because division works for both complex numbers and quaternions.
 ### Topological Insights
 In above equation for a circle, we assign values to *t* from a number line,
  a 1-dimensional Euclidean space.
 More precisely, the line is the imaginary axis $it$ in the numerator.
 We also include an extra point "at infinity".
 This same point is approached regardless of whether *t* is negative or positive,
  and "closes" the circle.
 $$
 \begin{align*}
-  o(\infty) &\approx {1 + i\infty \over 1 - i\infty}
+  o_1(t) &= {1 + it \over 1 - it}
-  \approx {-\infty \over \infty}
+\\
-  \approx -1
+  &= c_1 + i s_1
-  \\
+  = {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2}
  o(-\infty) &\approx {1 - i\infty \over 1 + i\infty}
  \approx {\infty \over -\infty}
  \approx -1
 \end{align*}
 $$
-For (2-)spheres, a similar statement holds true.
+The second line decomposes the first into real and nonreal terms, which are each rational functions.
-Rather than a line, we range over the imaginary plane $is + jt$, a 2-dimensional Euclidean space.
+We can further define terms for the numerator and denominator:
-If one or both of the parameters *s* or *t* has a value of "infinity",
+
-  then they seem to describe the same point.
+$$
 \begin{gather*}
  x_1 = 1 - t^2
  \qquad
  y_1 = 2t
  \qquad
  d = 1 + t^2 = 2 - x_1
 \\
  o_1 = {z_1 \over d} = {x_1 \over d} + i{y_1 \over d}
 \end{gather*}
 $$
 We have a recurrence relation for *o*, but it does not obey same relations as *z*:
 $$
 \begin{align*}
-  o_2(s, \infty) &\approx {1 + is + j\infty \over 1 - is - j\infty}
+  o_{n+2} &= 2c_1 o_{n+1} - o_n
-  \approx {\infty \over -\infty}
+\\
-  \approx -1
+  z_{n+2} &\stackrel{✗}{=} 2x_1 z_{n+1} - z_n
  \\
  o_2(\infty, t) &\approx {1 + i\infty + jt \over 1 - i\infty - jt}
  \approx {\infty \over -\infty}
  \approx -1
 \end{align*}
 $$
-In both expressions, the point at infinity contains no "nonreals" like *i* or *j*.
+Fortunately, the correction is simple.
-In another sense, the real space is the extra dimension into which the sphere extends as a surface.
+The denominator term $d^{n+2}$ can be multiplied through the top equation to produce:
 Topologically, this description of the resulting space is called the
  [one-point compactification](https://mathworld.wolfram.com/One-PointCompactification.html).
 In other words, the circle is the one-point compactification of the line,
  and in general, an *n* sphere is the one-point compactification of Euclidean *n*-space.
 $$
-\mathbb{E}^{n} \cup \{ \infty \} \cong S^n
+z_{n+2} = 2x_1 z_{n+1} - z_n d^2
 $$
-![]()
+The only term that changes is the term lagging two terms behind, so the generating function *Z* is:
 $$
 \begin{align*}
  O(x; o_1)
  &= {1 + x(o_1 - 2 c_1) \over 1 - 2 c_1 x + x^2}
 \\[10pt]
  Z(x; z_1)
  &= {1 + x(z_1 - 2 x_1) \over 1 - 2 x_1 x + \textcolor{red}{d^2} x^2}
 \end{align*}
 $$
 We can express *d* in terms of $x_1$, so the terms of the series, like the one for *F*, have
 - A real component which is a polynomial in $x_1$ (cf. $c_1$)
 - An imaginary component which is the product of $y_1$ (cf. $s_1$) and a polynomial in $x_1$
 $$
 Z(x; z_1) = X(x; x_1) + i y_1 Y(x; x_1)
 $$
 Surprisingly, the polynomials in *Y* still factor cleanly,
  like the [Chebyshev *U* polynomials](../../chebyshev/1/#tbl-chebyshevu).
 ```{python}
 #| code-fold: true
 #| label: tbl-newupolynomials
 #| tbl-cap: "Table of numerator polynomials"
 #| classes: plain
 # cosine series
 X = ( 1 - x1*x ) / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
 # sine series
 Y = x / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
 def factor_sequence(polys, offset=0, symbol_name="p"):
    ret = []
    symbols = []
    for i, poly in enumerate(polys):
        new_poly = poly.copy()
        old_factor = 1
        for old, symbol in zip(ret, symbols):
            q, r = sympy.div(new_poly, old)
            if r == 0:
                new_poly = q
                old_factor *= symbol
        if new_poly != 1:
            ret.append(new_poly)
            symbols.append(sympy.symbols(f"{symbol_name}_{i + offset}"))
        yield poly, old_factor*new_poly.factor()
 Markdown(tabulate(
  [
    [ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
    for n, (unfactored, poly) in enumerate(
      factor_sequence(
          sorted(
              [
                  i.subs(x,1).subs(x1, z).expand().factor()
                  for i in Y.series(x, n=11).args
              ][:-1],
              key=lambda x: sympy.degree(x, z)
          ),
          1
      )
    )
  ],
  headers=[ "*n*", "$[x^n]Y(x; z) = p_n(z)$", "Coefficients (descending powers)" ],
  numalign="left",
  stralign="left",
 ))
 ```
-### Invariance of Dimension
+Unfortunately, the sequence formed by the coefficients of the polynomials
  does not appear in the OEIS.
 Their factorizations appear to have the following traits:
-The topological definition seems to imply that our construction shouldn't care about
+- Like the Chebyshev *U* polynomials, they have "cyclotomic factoring" --
-  how many dimensions are in the space.
+  for the new term of index *n*, the factors can be separated into old factors
-In fact, when constructing the 2-sphere, all we cared about was that *i* and *j* anti-commute
+  at indices of factors of *n* and new factors.
-  to get cancellation.
+- If the index is even, then there is only one new monic, irreducible factor.
 - If the index is odd, then there are two new irreducible factors
  - If the index is prime or a prime power, the new factors are a monic and a non-monic
    whose leading coefficient is that prime.
  - Otherwise, the new factors are both monic.
-[Geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra) gives some tools to generalize
+The characterization of the leading terms of the new factor corresponds to
-  this argument to higher dimensions.
+  [OEIS A014963](https://oeis.org/A014963), which is related to cyclotomic polynomials.
 In an *n*-dimensional algebra, we have unit vectors $e_0, e_1, ..., 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$
  - 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$
 - 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.
+### Similar Sequences
-This can be seen by arranging the components of the product after distributing as a square:
+
 In fact, for a polynomial $q(z)$, it seems to be the case that the terms of
 $$
 Y(x; z) = {x \over 1 - 2 z x + q(z) x^2}
 $$
 tend to factor similarly.
 Naturally, the *U* polynomials are the choice where *q = 1* and the new polynomials
  are the choice when $q = (2 - z)^2$.
 One can also write down a series for $z^n - 1$, which factor as the cyclotomic polynomials,
  and also end up being generated by an order-2 recurrence.
 $$
 \begin{align*}
 N(x; z) &= \sum_n (z^n - 1)x^n = {x(z - 1) \over 1 - (z + 1)x + zx^2}
 \\
  &= \sum_n \left ( x^n \prod_{d | n} \Phi_d(z) \right )
 \end{align*}
 $$
 Actually, this shouldn't be terribly surprising.
 For example, a simple result from generating functions tells us
  that a series for the integers is:
 $$
 \begin{align*}
  F(z) &= {1 \over 1 - z}
  = \sum_n z^n
  && \text{All coefficients equal 1}
 \\
  F'(z) &= {1 \over ( 1 - z )^2 }
  = \sum_n n z^{n - 1}
  && \text{Integers}
 \\
  z F'(z) &= {z \over 1 - 2z + z^2}
  = \sum_n n z^n
  && \text{Integers matching powers}
 \end{align*}
 $$
 The denominator being a quadratic polynomial means that the series terms *n*,
  the integers, obey an order-2 recurrence, and factor in a similar way.
 Obviously, the integers factor into primes by the fundamental theorem of arithmetic.
 There's still an important distinction to be made about the polynomials, though.
 Factoring a composite like 6 into 2 and 3 leaves an empty product behind, but
  for polynomials, nonprime indices end up accumulate an "extra" factor.
 Additionally (or rather, probably because of this), *all* factors of the index correspond
  to a factor in the factorization, rather than pairing off as in integers.
 For example, factoring 12 once gives either 3 and 4 or 2 and 6,
  but the polynomial at index 12 includes polynomials at indices of all factors: 2, 3, 4, 6, and 12.
 ### Higher-order Recurrences
 The integers also obey an order-3 recurrence:
 $$
 \begin{align*}
  F(x) &= {x \over 1 - 2x + x^2} = {x(1 - x) \over (1 - 2x + x^2)(1 - x)}
  \\
  &= {x - x^2 \over 1 - 3x + 3x^2 - x^3}
  \\[10pt]
  &\equiv a_{n+3} = 3a_{n+2} - 3a_{n+1} + a_n
 \end{align*}
 $$
 Another sequence that obeys similar factoring rules rules to the integers is
 $$
 G(x; z) = {x - x^2 \over 1 - z x + z x^2 - x^3}
 $$
 ```{python}
 #| code-fold: true
 #| tbl-cap: "Table of G polynomials"
 #| classes: plain
 G = (x - x**2) / ( 1 - z*x + z*x**2 - x**3 )
 Markdown(tabulate(
  [
    [ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
    for n, (unfactored, poly) in enumerate(
      factor_sequence(
          sorted(
              [
                  i.subs(x,1).subs(x1, z).expand().factor()
                  for i in G.series(x, n=11).args
              ][:-1],
              key=lambda x: sympy.degree(x, z)
          ),
          1,
          "o"
      )
    )
  ],
  headers=[ "*n*", "$[x^n]G(x; z) = o_n(z)$", "Coefficients (descending powers)" ],
  numalign="left",
  stralign="left",
 ))
 ```
 There are a couple of things to note here.
 Some of the factor polynomials here are the
  [minimal polynomials of cosine](/posts/math/chebyshev/1/#tbl-cosinepolynomials)
 The row where *n* = 5 is somewhat interesting.
 Namely, $x^2 - x - 1$ has $\varphi$ (the golden ratio) as a root.
 The other polynomial, $x^2 - 3x + 1$, has $\varphi^2$ as a root.
 This seems to indicate that the other polynomial has roots which are
  an algebraic expression of the other's, but I haven't bothered attempting a proof of this.
 Unfortunately, peppering polynomials into the denominator and hoping that the same factorization
  occurs isn't as easy as in the order-2 recurrence.
 In fact, higher-order recurrences become more and more restrictive as $x^\bullet$ terms are added to
  the numerator and denominator.
 If there is a rule to determine what relation must be obeyed between the coefficients
  for factorization to occur, it is not obvious, especially as the order grows.
 Higher-dimensional Spheres
 --------------------------
 We derived an explicit map for the 2-sphere (or rather, the 3-sphere, since the description ended up matching the quaternions)
  in [the first post in this series](../1/).
 It's quite easy to generalize the argument to higher dimensions.
 In *n* dimensions, assume that we have unit vectors $e_0 ... e_{n-1}$.
 Placing these vectors within a [geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra)
  gives some promising properties:
 - Vectors can be multiplied like ordinary numbers, and even added to ordinary numbers
 - The product of a vector with itself can be chosen among -1, 0, or 1
 - The product of two vectors anticommutes (e.g., $e_0 e_1 = - e_1 e_0$)
  - Consequently, the square of a 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$)
 The square of a general vector with components $x_k e_k$ is a scalar, its norm.
 $$
 \begin{align*}
@ -143,377 +313,88 @@ $$
  &= \underset{\diagdown}{\sum_k^{n-1} e_k^2 x_k^2}
  + \underset{◥}{ \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l }
  + \underset{◣}{ \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l }
  \\
  &= \diagdown + ◥
  + \sum_k^{n-1} \sum_{l > k} e_l e_k x_k x_l
  = \diagdown + ◥
  - \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l
  \\
  &= \diagdown + ◥ - ◥
  = \diagdown
 \end{align*}
 $$
-Due to anticommutativity, we can cancel the upper and lower triangles, leaving only the diagonal.
+For simplicity, assume that $e_k^2 = -1$ so that ${\bm v}^2 = - ||{\bm v}||$,
  the sum of squares of the extent in each basis.
 Despite multiplication between two vectors being defined, dividing one vector by another is not.
 Ignoring this, consider the expression
 $$
 \begin{align*}
-  ◣ &= \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l
+  {1 + {\bm v} \over 1 - {\bm v}}
-  = \sum_k^{n-1} \sum_{k < l} e_l e_k x_l x_k
+  &= \left( {1 + {\bm v} \over 1 - {\bm v}} \right)
  \left( {1 + {\bm v} \over 1 + {\bm v}} \right)
  = {(1 + {\bm v})^2 \over (1 - {\bm v})(1 + {\bm v})}
  \\
-  &= - \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l
+  &= {1 + 2{\bm v} + {\bm v}^2 \over 1 - {\bm v}^2}
-  = - ◥
+  \\
-  \\[10pt]
+  &= {1 - ||{\bm v}|| \over 1 + ||{\bm v}||} + {2{\bm v} \over 1 + ||{\bm v}||}
  &\implies \diagdown + ◥ + ◣ = \diagdown + ◥ - ◥ = \diagdown
 \end{align*}
 $$
-To align with the prior examples *i* and *j*, we'll assume that $e_k^2 = -1$ for all *k*.
+Similarly to quaternions, this is the sum of a vector and a scalar.
-This means that ${\bm v}^2 = - ||{\bm v}||$, the sum of squares of the extent
+If the scalar component is considered the extent in a new dimension,
-  in each basis (or Euclidean norm).
+  then the norm of the resulting vector is
 ### Being Hyperrational
 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 $a + {\bm v}$.
 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
 $$
-||a + {\bm v}|| = a^2 + ||{\bm v}|| = a^2 - {\bm v}^2 = 1
+a^2 + ||u|| = a^2 - u^2 = (a + u)(a - u)
 $$
-As a vector ***u*** ranges over *n*-dimensional space, the expression...
+This is actually an inductive hypothesis.
 This forces us to choose two things: the norm we use is Euclidean, and each new unit vector squares to -1.
 For the sphere, focusing just on the numerator
 $$
-o_n({\bm u}) = {1 + {\bm u} \over 1 - {\bm u}}
+(1 - ||v||)^2 - u^2 = (1 - ||v||)^2 - (2v)^2 = 1 - 2||v|| + ||v||^2 - 4v^2 = 1 - 2||v|| + ||v||^2 + 4||v|| = 1 + 2||v|| + ||v||^2 = (1 + ||v||)^2
 $$
-...seems to be a ratio between two distinct quantities with the same norm,
+This is the denominator, so the dubious step of dividing by a vector has been backed up with pure algebra.
  since $1^2 - {\bm u}^2 = 1^2 - (-{\bm u})^2$.
 This expression is actually ill-defined since there is a vector in the denominator,
  but we can use a conjugation trick to clear it:
 $$
 \begin{align*}
  {1 + {\bm u} \over 1 - {\bm u}}
  &= \left( {1 + {\bm u} \over 1 - {\bm u}} \right)
  \left( {1 + {\bm u} \over 1 + {\bm u}} \right)
  = {(1 + {\bm u})^2 \over (1 - {\bm u})(1 + {\bm u})}
  \\
  &= {1 + 2{\bm u} + {\bm u}^2 \over 1 - {\bm u}^2}
  \\
  &= {1 - ||{\bm u}|| \over 1 + ||{\bm u}||} + {2{\bm u} \over 1 + ||{\bm u}||}
  = a + {\bm v}
 \end{align*}
 $$
-The quantity in the denominator of both components is always a scalar and greater than zero,
+### Degree Maps
  so there are no concerns about the validity of division.
 We can also show that the norm of this expression is 1, as desired:
-$$
+If $\bm u$ is a vector with norm 1.
 \begin{align*}
  a^2 - {\bm v}^2 &= 1
 \\
  \implies
  \stackrel{\text{Numerator of } a}{(1 + {\bm u}^2)^2}
  - \stackrel{\text{Numerator of } \bm v}{(2{\bm u})^2}
  &= \stackrel{\text{Common denominator}}{1 - {\bm u}^2}
 \end{align*}
 $$
-This is true no matter how many dimensions ***v*** has[^1], justifying our earlier abuse of notation.
+The first coordinate expresses the condition that if **v** lies on the unit (hyper)sphere
  in the dimension below, then the coordinate is zero, and such points lie on an equator.
-[^1]: Technically, this should only hold for finitely many dimensions.
+In the case of the 2-sphere, doubling the map by squaring quaternions produces a more interesting feature:
-  The $\infty$-sphere, composed of vectors with only finitely many nonzero components,
+  all **v** lying on the unit (hyper)sphere in the dimension below get mapped to the same point, antipodal to 0.
-  is probably also valid under this construction, but I haven't proven this.
+In other words, the sphere in the next dimension is obtained by considering all points on the boundary of a ball to be the same.
 Inducing an Alternative
 -----------------------
 The previous topological description of spheres lacks a couple of things:
 - It does not make reference to lower-dimensional spheres
 - "Points at infinity", while intuitive, are logically suspect
 Fortunately, topology has an alternate description.
 The 0-dimensional sphere is a little bit special.
 On a number line, there are two points equidistant to the origin,
  and these comprise the 0-sphere $S^0$.
 This can (topologically) be turned into a 1-sphere $S^1$ (the circle) by an operation called
  [suspension](https://en.wikipedia.org/wiki/Suspension_%28topology%29), which connects
  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 $S^2$.
 In general,
 $$
 \text{Susp}(S^{n-1}) = S^n
 $$
 ![]()
 ### Algebraic Dual, Part 2
 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,
  the space looks two discrete points; to wit, a 0-dimensional sphere.
 The remaining two points 1 and -1 are in some sense "new" to the space.
 ![]()
 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.
 ![]()
 If we create a 2D vector with components in the real and imaginary parts of *o*,
  we can immediately create an expression for the 2-sphere:
 $$
 \begin{align*}
  {\bm w}_1(t) &= {1 - t^2 \over 1 + t^2} e_0 + {2t \over 1 + t^2} e_1
  \\[10pt]
  \varsigma_2(s,t) &= {1 + {\bm w}_1(t)s \over 1 - {\bm w}_1(t)s}
  = {1 - s^2 \over 1 + s^2} + {2s \over 1 + s^2} {\bm w}_1(t)
 \end{align*}
 $$
 Note that if *s* is exchanged with *-s*, then the scalar part remains the same,
  but the vector part, which corresponds to latitudinal circles, is negated.
 In effect, this means that if *s* is allowed to range over negative numbers,
  we will produce two duplicate circles.
 This process can be continued indefinitely -- at each stage,
  $\varsigma_k$[^2] describes a *k*-dimensional unit sphere.
 It be converted to a pure vector ${\bm w}_k$ by multiplying the scalar component
  with a new unit vector $e_k$.
 In this form, ${\bm w}_k^2 = -1$ for any *k*-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".
 [^3]: This should sound familiar from the first post -- it matches the "unit quaternions".
 $$
 \begin{align*}
  {\bm w}_k(x_0, x_1, ..., x_{k-1})
  &= \text{Scalar}(\varsigma_k(x_0, x_1, ..., x_{k-1}))e_k
  \\
  &+ \text{Vector}(\varsigma_k(x_0, x_1, ..., x_{k-1}))
  \\
  \varsigma_{k+1}(x_0, x_1, ..., x_{k-1}, x_k)
  &= {1 + {\bm w}_k(x_0, x_1, ..., x_{k-1})x_k \over 1 - {\bm w}_k(x_0, x_1, ..., k_{k-1})x_k}
 \end{align*}
 $$
 When the new parameter $x_k$ is 0 or $\infty$, the vector part collapses,
  and we get either 1 or -1, the "new points" of the suspension.
 $$
 \begin{align*}
  \varsigma_{k+1}(..., 0)
  &= {1 + {\bm w}_k(...)\cdot 0 \over 1 - {\bm w}_k(...) \cdot 0}
  - {1 \over 1} = 1
 \\
  \varsigma_{k+1}(..., \infty)
  &\approx {1 + {\bm w}_k(...)\cdot \infty \over 1 - {\bm w}_k(...) \cdot \infty}
  \approx {\infty \over -\infty} \approx -1
 \end{align*}
 $$
 The phenomenon of duplicate latitudinal spheres is a recurring one in dimensions greater than 1.
 This can be seen from the 0-sphere being unique among spheres --
  as two discrete, disconnected points, negative numbers are needed 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
 ------------------
 One feature of the complex rational circle mentioned in [the previous article](../2/)
  was that its powers correspond to going around multiple times.
 Conveniently, a similar fact holds for *n*-spheres in general.
 Starting with the scalar/vector form of the sphere, we can square the sphere and apply
  the fact that the difference of squares of each part is constant:
 $$
 \begin{align*}
  o_n &= a + {\bm v}
  \\
  o_n^2 &= (a + {\bm v})^2 = a^2 + {\bm v}^2 + 2a{\bm v}
  \\
  &= a^2 + {\bm v}^2 + \textcolor{red}{(a^2 - {\bm v}^2 - 1 = 0)} + 2a{\bm v}
  \\
  &= 2a^2 - 1 + 2a{\bm v} = 2a(a + {\bm v}) - 1 = 2a o_n - 1
 \end{align*}
 $$
 This gives the familiar recurrence relation...
 $$
 o_n^{m+2} = 2a o_n^{m+1} - o_n^m
 $$
 ...and thus a sphere can be wrapped around itself any number of times, as given by:
 $$
 o_n^m = T_m(a) + U_{m-1}(a){\bm u}
 $$
 *T* and *U* are the standard Chebyshev polynomials.
 ### Negative Indices and Beyond
 The topological equivalent to this statement is
 $$
 H_n(S^n) = \Z
 $$
 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
  which correspond to negative degrees.
 This seems to align with the behavior of the exponent in $o_n^m$.
 However, we've only defined *m* over positive integers;
  after all, $o_n$ contains a vector, so we can't really divide by it.
 Fortunately, it's pretty easy to make sense of this.
 Since we have a recurrence relation for the powers of the sphere, we
  can extend it backwards to define it over negative indices[^4].
 $$
 \begin{align*}
  o_n^1 &= 2a o_n^{0} - o_n^{-1}
  \\
  a + {\bm v} &= 2a - o_n^{-1}
  \\
  o_n^{-1} &= a - {\bm v}
 \end{align*}
 $$
 [^4]: The same argument holds for the Chebyshev polynomials.
  In general, $T_{-n}(x) = T_n(x)$ and $U_{-1} = 0$, $U_{-n}(x) = -U_{n-2}(x)$ for
    the standard indexing of *U*.
  If anything, this is another argument that this indexing of *U* isn't very well-suited,
    since if $U_0 \stackrel{\Delta}{=} 0$, it follows that $U_{-n}(x) = -U_{n}(x)$.
 This actually aligns with what we'd expect according to adding powers, since:
 $$
 o_n^1 \cdot o_n^{-1} = (a + {\bm v})(a - {\bm v}) = a^2 - {\bm v}^2 = 1 = o_n^0
 $$
 ### Degrees and Induction
 In the inductive case, since ${\bm w}^2 = -1$, we can pull a similar trick.
 Replacing *i* with ***w***, the only thing that needs changing from the previous article is
  replacing "real" with "scalar" and "nonreal" with "vector".
 $$
 \varsigma_n^m
 = ( c + s \cdot { {\bm w}_{n-1}} )^m
 = T_m(c) + s U_m(c) \cdot {\bm w}_{n-1}
 $$
 Similarly,
 $$
 \begin{align*}
 \varsigma_n^{-1} &= ( c - s \cdot { {\bm w}_{n-1}} )
 \\
 \varsigma_n^{1} \cdot \varsigma_n^{-1}
  &= ( c + s \cdot {{\bm w}_{n-1}} )( c - s \cdot {{\bm w}_{n-1}} )
  \\
  &= c^2 - s^2 {\bm w}_{n-1}^2 = c^2 + s^2
  \\
  &= 1 = \varsigma_n^0
 \end{align*}
 $$
 Degree 2 and the Equator
 ------------------------
 The prior discussion about degree gives us the tools to address "points at infinity".
 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 former wraps around once $-\infty$ to $\infty$, while the latter wraps around once from -1 to 1.
 Conveniently, -1 and 1 are both the points at which the real part of the degree 1 map becomes 0.
 Points in this range lie on the semicircle bounded by these two points (which form a 0-sphere).
 ![]()
 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.
 $$
 o_n({\bm u}_n)
 = {1 - ||{\bm u_n}|| \over 1 + ||{\bm u_n}||}
 + {2{\bm u_n} \over 1 + ||{\bm u_n}||}
 = {1 - 1 \over 1 + 1} + {2 \over 1 + 1}{\bm u_n}
 = {\bm u}_n
 $$
 Let *n* = 2 so we can plot it.
 Then we're actually talking about place in which the unit sphere in 3D space looks like the unit circle.
 Specifically, this unit circle can be considered an equator bounding the hemisphere around the scalar 1.
 ![]()
 In general ${\bm u}_n$ has a norm of 1 exactly when it's a point on the equatorial *n-1*-sphere.
 Conveniently, under the degree 2 map, this equatorial sphere collapses to a single point.
 $$
 o_n({\bm u}_n)^2
 = {\bm u}_n^2 = -1
 $$
 This point was formerly the image of the point at infinity,
  eliminating its necessity in the description of the *n*-sphere.
 ### Closing the Sphere
 Of course, this 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.
 $$
 { D^n / \partial D^n } = S^n
 $$
-This exactly aligns with the behavior of the equator when going from the degree 1 to the degree 2 map.
+This applies generally.
-If ${\bm u}_n$ has a norm of less than or equal to 1, then it lies within a unit disc.
+This can be made more topological by doubling the sphere, but we don't know how to do that generally,
 This unit disc gets sent by $o_n$ to the aforementioned "hemisphere around the scalar 1",
  and when fed to $o_n^2$, it produces the *n*-sphere.
 A classical homotopy result informs
-Closing
+$$
-------
+\pi_n(S^n) = \Z
 $$
-There are a couple of things that I still want to explore here.
+Telling us we can wrap an *n* sphere around itself, backwards and forwards any number of times.
 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.
+It's annoying to do this explicitly without a generic way to describe higher dimensional spheres.
-One of these is the degree of the antipodal map.
+
-Another is constructing explicit homotopies purely from algebra.
+The relation seems to be:
 $$
 _n o_m = \left( T_m(o_{1,0}), o_{1,[1:n]}U(o_{1,0}) \right)
 $$
 Just like in the circle. This is extraordinarily convenient.
--- a/posts/math/stereo/3/inductive.qmd
+++ b/posts/math/stereo/3/inductive.qmd
@ -0,0 +1,310 @@
 ---
 title: "Stereography, Algebraic, and Hyperspheres"
 description: |
  TODO
 format:
  html:
    html-math-method: katex
 jupyter: python3
 date: "2026-09-11"
 categories:
  - algebra
 ---
 ```{python}
 #| echo: false
 import sympy
 from IPython.display import Markdown
 from tabulate import tabulate
 x, x1, z = sympy.symbols("x x_1 z")
 ```
 Topological Spheres
 -------------------
 In topology, hyperspheres are some of the primary spaces of interest.
 Spheres have a natural geometric definition: the locus of points which all have
  the same distance to the origin.
 Topologically, however, they're better-described inductively.
 First, notice that the equation for the circle depends on a single parameter *t*
  which ranges over the entire number line.
 There is also a point on the circle "at infinity", which "closes" the circle.
 Topologically, the resulting space is called the [https://mathworld.wolfram.com/One-PointCompactification.html](one-point compactification).
 In other words, the circle is the one-point compactification of the (open) line.
 For spheres, the same thing holds true, but the notion of "one-point" starts to become relevant.
 We map a 2-dimensional plane to the sphere, so there are two variables.
 But if one or both of these variables has a value of "infinity", then they are all said to describe the same point.
 Concretely, this gives the topological relation
 $$
 \mathbb{E}^{n} \cup \{ \infty \} \cong S^n
 $$
 ### Algebraic Dual
 As a locus of points, the *n*-sphere exists within *n+1* dimensional space.
 But since the sphere is *n*-dimensional, a point on it is described by *n* coordinates,
  just like a point in *n*-dimensional space.
 We need a way to augment an *n*-dimensional vector with an extra dimension
 Fortunately, we have some direction from [the first post in this series](../1/),
  in which we derived an explicit map for the 2-, and 3-spheres.
 Namely, the result for 2-spheres was derived by the assertion
 $$
 o = {1 + {\bm v} \over 1 - {\bm v}} = a + {\bm u},
 \quad {\bm v} = is + jt,
 \quad i^2 = j^2 = -1
 \quad ij = -ji
 $$
 *i* and *j* are quaternions, which form a division algebra.
 This makes this expression legitimate, but not easy to generalize to higher dimensions[^1].
 [^1]: The limited number of division algebras is typically proved using algebraic topology
  through arguments that depend on spheres and quotient spaces thereof.
 Fortunately, the argument can be adjusted a little.
 [Geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra) gives some tools to generalize
  this argument to higher dimensions.
 In such an algebra, we have unit vectors $e_0 ... e_{n-1}$ 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., $e_0 e_1 = - e_1 e_0$
  - Consequently, the square of the product is the negative of the product of the squares
    - e.g., $e_0 e_1 e_0 e_1 = - e_0 e_1 e_1 e_0 = - e_0^2 e_1^2$
 The square of a general vector ***v*** with components $x_k e_k$ is a scalar[^2].
 $$
 \begin{align*}
  {\bm v} &= e_0 x_0 + e_1 x_1 + ... e_{n-1} x_{n-1} = \sum_k e_k x_k
  \\
  {\bm v}^2 &= (\sum_k^{n-1} e_k x_k) (\sum_l^{n-1} e_l x_l)
  = \sum_k^{n-1} \sum_l^{n-1} e_k e_l x_k x_l
  \\
  &= \underset{\diagdown}{\sum_k^{n-1} e_k^2 x_k^2}
  + \underset{◥}{ \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l }
  + \underset{◣}{ \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l }
  \\
  &= \diagdown + ◥
  + \sum_k^{n-1} \sum_{l > k} e_l e_k x_k x_l
  = \diagdown + ◥
  - \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l
  \\
  &= \diagdown + ◥ - ◥
  = \diagdown
 \end{align*}
 $$
 For simplicity, assume that $e_k^2 = -1$ so that ${\bm v}^2 = - ||{\bm v}||$, its Euclidean norm,
  the sum of squares of the extent in each basis.
 Consider the expression
 $$
 \begin{align*}
  {1 + {\bm v} \over 1 - {\bm v}}
  &= \left( {1 + {\bm v} \over 1 - {\bm v}} \right)
  \left( {1 + {\bm v} \over 1 + {\bm v}} \right)
  = {(1 + {\bm v})^2 \over (1 - {\bm v})(1 + {\bm v})}
  \\
  &= {1 + 2{\bm v} + {\bm v}^2 \over 1 - {\bm v}^2}
  \\
  &= {1 - ||{\bm v}|| \over 1 + ||{\bm v}||} + {2{\bm v} \over 1 + ||{\bm v}||}
  = a + {\bm u}
 \end{align*}
 $$
 Similarly to quaternions, this is the sum of a vector and a scalar.
 If the scalar component is considered the extent in a new dimension,
  then the norm of the resulting vector is
 $$
 a^2 + ||{\bm u}|| = a^2 - {\bm u}^2
 $$
 According to this definition, we started with the ratio of two expressions with the same norm.
 This means that our resulting expression should have a norm of 1.
 $$
 ||1 + {\bm v}|| = 1^2 - {\bm v}^2
 = 1^2 - ({\bm -v})^2 = ||1 - {\bm v}||
 $$
 Consequently,
 $$
 \begin{align*}
  a^2 - {\bm u}^2 &= 1
 \\
  \implies
  \stackrel{\text{Numerator of } a}{(1 + {\bm v}^2)^2}
  - \stackrel{\text{Numerator of } \bm u}{(2{\bm v})^2}
  &= \stackrel{\text{Common denominator}}{1 - {\bm v}^2}
 \end{align*}
 $$
 The final expression is always valid, no matter how many dimensions ***v*** has.
 Not only that, since *a* and ***u*** contain no vectors in the denominator, there are no concerns with the validity of division.
 The only reason we started from an expression which did contain vectors was to justify the requirement for anticommutativity.
 This is the denominator, so the dubious step of dividing by a vector has been backed up with pure algebra.
 Despite multiplication between two vectors being defined, dividing one vector by another is not.
 Inductivity
 -----------
 The previous topological description of spheres lacks a couple of things:
 - It does not make reference to lower-dimensional spheres
 - "Points at infinity", while intuitive, are logically suspect
 Fortunately, topology has an alternate description.
 The the 1-dimensional sphere is a little bit special.
 On a number line, there are two points equidistant to the origin,
  and these comprise the 0-sphere $S^0$.
 This can be turned into a 1-sphere $S^1$ (the circle) through a topological operation called
  [suspension](https://en.wikipedia.org/wiki/Suspension_%28topology%29), which connects
  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 $S^2$.
 In general,
 $$
 \text{Susp}(S^{n-1}) = S^n
 $$
 ### Algebraic Dual, Part 2
 First, let's look at the first interesting case.
 We first definied the circle, or 1-dimensional sphere as
 $$
 {1 + it \over 1 - it}
 $$
 If the first coordinate remains fixed, then in most cases,
  the space looks two discrete points, or to wit, a 0-dimensional sphere.
 The remaining two points are in some sense "new" to the space.
 Examining the 2-dimensional sphere in the same way, at an intersecting plane,
  the space looks like a 1-dimensional sphere except at two points.
 This matches the inductive topological description of spheres one-for-one.
 Using the results of the previous section, we have a way to generalize *i* to any dimension.
 Coincidentally, this generalization is *also* inductive --
  if ***u*** is already a vector with norm 1, then the scalar component *a* must be 0.
 Further, this means that ${\bm u}^2 = -1$.
 This should sound familiar -- it matches the "unit quaternions"
 Intuitively, this means we can also describe a sphere by the equation:
 $$
 {1 + {\bm u}t \over 1 - {\bm u}t}
 = {1 - t^2 \over 1 + t^2} + {2t \over 1 + t^2}{\bm u}
 = a + b{\bm u}
 $$
 ### Degree Maps
 Since ${\bm u}^2 = -1$, there's an interesting trick we can pull again.
 We wrapped the circle around itself twice in [the previous article](../2/) by
  simply squaring the same expression from the last article.
 That argument is only contingent upon one thing: the split between
  real and nonreal components, and the squaring of the unit nonreal to -1.
 When going from *i* to ***u***, the only thing that needs changing is replacing
  "real" with "scalar" and "nonreal" with "vector".
 $$
 _n o^m
 = ( a + b \cdot { {}_{n-1} {\bm u}} )^m
 = T_m(a) + b U_m(a) \cdot { {}_{n-1} {\bm u}}
 $$
 *T* and *U* here are the standard Chebyshev polynomials.
 This amounts to wrapping the sphere around itself any number of times as desired, *m*.
 *m* here is only really defined over positive integers here, since the Chebyshev polynomials
  are only defined over positive indices.
 The topological equivalent to this statement is
 $$
 \pi_n(S^n) = \Z
 $$
 This states that a map from the *n*-sphere to itself can be characterized by an integer,
  the *degree*.
 Maps sharing the same integer are considered to be *homotopic* to one another,
  and composing maps can be composed in the same way that the integers add.
 Telling us we can wrap an *n* sphere around itself, backwards and forwards any number of times.
 "Backwards" comes from antipodal map
 Degree of antipodal map is negative only if *n* is even
 But this just means we can look at a map where we negate only the vector components.
 Equator
 -------
 By describing spheres purely in terms of other spheres, we've taken care of the first problem.
 We still have another -- if we let a vector *v* range over the entirety of Euclidean space,
  we still have to include a point at infinity.
 By virtue of degree, we're still in the clear.
 The trick is actually the same from the previous post when integrating.
 For the circle, a degree 1 map wraps around once from $-\infty$ to $\infty$,
  and a degree 2 map wraps around once from -1 to 1.
 The -1 to 1 range in the degree 1 map describes only a semicircle, whose boundary is
  two points (the 0-sphere).
 One dimension up, to continue the analogy, we have a 1-sphere which bounds a hemisphere
  in the degree 1 map.
 The 1-sphere is just the equator of the sphere.
 The hemisphere replaces the range "from -1 to 1"; instead, we have a unit disc --
  geometrically, this consists of all vectors whose norm is less than 1.
 In fact, this still agrees with the 1 dimensional case.
 This analogy continues inductively to all *n*-dimensional spheres.
 Another way of seeing this is by looking at the equators of *n*-spheres.
 Using the inductive algebraic definition of the sphere, if the scalar component is 0, then
  the vector component is just the *n-1*-sphere.
 This can be thought of as the equator.
 The first coordinate expresses the condition that if **v** lies on the unit (hyper)sphere
  in the dimension below, then the coordinate is zero, and such points lie on an equator.
 :::
 USE THE ABOVE AS JUSTIFICATION FOR THIS SHIT.
 MAYBE START EARLIER, WHEN INDUCTION WAS BEING DISCUSSED?
 :::
 In the case of the 2-sphere, doubling the map by squaring quaternions produces a more interesting feature:
  all **v** lying on the unit (hyper)sphere in the dimension below get mapped to the same point, antipodal to 0.
 In other words, the sphere in the next dimension is obtained by considering all points on the boundary of a ball to be the same.
 $$
 { D^n / \partial D^n } = S^n
 $$
--- a/posts/math/stereo/4/index.qmd
+++ b/posts/math/stereo/4/index.qmd
@ -1,286 +0,0 @@
 ---
 title: "Stereography, Algebraic, and Hyperspheres"
 description: |
  TODO
 format:
  html:
    html-math-method: katex
 jupyter: python3
 date: "2026-09-11"
 categories:
  - algebra
 draft: true
 ---
 ```{python}
 #| echo: false
 import sympy
 from IPython.display import Markdown
 from tabulate import tabulate
 x, x1, z = sympy.symbols("x x_1 z")
 ```
 The Algebra Part
 ----------------
 Though I alluded to the ability of the stereoscopic circle to generate the Chebyshev polynomials,
  there is an important caveat which differs their use from typical spheres.
 To review, the stereoscopic definition of the circle is:
 $$
 \begin{align*}
  o_1(t) &= {1 + it \over 1 - it}
 \\
  &= c_1 + i s_1
  = {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2}
 \end{align*}
 $$
 The second line decomposes the first into real and nonreal terms, which are each rational functions.
 We can further define terms for the numerator and denominator:
 $$
 \begin{gather*}
  x_1 = 1 - t^2
  \qquad
  y_1 = 2t
  \qquad
  d = 1 + t^2 = 2 - x_1
 \\
  o_1 = {z_1 \over d} = {x_1 \over d} + i{y_1 \over d}
 \end{gather*}
 $$
 We have a recurrence relation for *o*, but it does not obey same relations as *z*:
 $$
 \begin{align*}
  o_{n+2} &= 2c_1 o_{n+1} - o_n
 \\
  z_{n+2} &\stackrel{✗}{=} 2x_1 z_{n+1} - z_n
 \end{align*}
 $$
 Fortunately, the correction is simple.
 The denominator term $d^{n+2}$ can be multiplied through the top equation to produce:
 $$
 z_{n+2} = 2x_1 z_{n+1} - z_n d^2
 $$
 The only term that changes is the term lagging two terms behind, so the generating function *Z* is:
 $$
 \begin{align*}
  O(x; o_1)
  &= {1 + x(o_1 - 2 c_1) \over 1 - 2 c_1 x + x^2}
 \\[10pt]
  Z(x; z_1)
  &= {1 + x(z_1 - 2 x_1) \over 1 - 2 x_1 x + \textcolor{red}{d^2} x^2}
 \end{align*}
 $$
 We can express *d* in terms of $x_1$, so the terms of the series, like the one for *F*, have
 - A real component which is a polynomial in $x_1$ (cf. $c_1$)
 - An imaginary component which is the product of $y_1$ (cf. $s_1$) and a polynomial in $x_1$
 $$
 Z(x; z_1) = X(x; x_1) + i y_1 Y(x; x_1)
 $$
 Surprisingly, the polynomials in *Y* still factor cleanly,
  like the [Chebyshev *U* polynomials](../../chebyshev/1/#tbl-chebyshevu).
 ```{python}
 #| code-fold: true
 #| label: tbl-newupolynomials
 #| tbl-cap: "Table of numerator polynomials"
 #| classes: plain
 # cosine series
 X = ( 1 - x1*x ) / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
 # sine series
 Y = x / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
 def factor_sequence(polys, offset=0, symbol_name="p"):
    ret = []
    symbols = []
    for i, poly in enumerate(polys):
        new_poly = poly.copy()
        old_factor = 1
        for old, symbol in zip(ret, symbols):
            q, r = sympy.div(new_poly, old)
            if r == 0:
                new_poly = q
                old_factor *= symbol
        if new_poly != 1:
            ret.append(new_poly)
            symbols.append(sympy.symbols(f"{symbol_name}_{i + offset}"))
        yield poly, old_factor*new_poly.factor()
 Markdown(tabulate(
  [
    [ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
    for n, (unfactored, poly) in enumerate(
      factor_sequence(
          sorted(
              [
                  i.subs(x,1).subs(x1, z).expand().factor()
                  for i in Y.series(x, n=11).args
              ][:-1],
              key=lambda x: sympy.degree(x, z)
          ),
          1
      )
    )
  ],
  headers=[ "*n*", "$[x^n]Y(x; z) = p_n(z)$", "Coefficients (descending powers)" ],
  numalign="left",
  stralign="left",
 ))
 ```
 Unfortunately, the sequence formed by the coefficients of the polynomials
  does not appear in the OEIS.
 Their factorizations appear to have the following traits:
 - Like the Chebyshev *U* polynomials, they have "cyclotomic factoring" --
  for the new term of index *n*, the factors can be separated into old factors
  at indices of factors of *n* and new factors.
 - If the index is even, then there is only one new monic, irreducible factor.
 - If the index is odd, then there are two new irreducible factors
  - If the index is prime or a prime power, the new factors are a monic and a non-monic
    whose leading coefficient is that prime.
  - Otherwise, the new factors are both monic.
 The characterization of the leading terms of the new factor corresponds to
  [OEIS A014963](https://oeis.org/A014963), which is related to cyclotomic polynomials.
 ### Similar Sequences
 In fact, for a polynomial $q(z)$, it seems to be the case that the terms of
 $$
 Y(x; z) = {x \over 1 - 2 z x + q(z) x^2}
 $$
 tend to factor similarly.
 Naturally, the *U* polynomials are the choice where *q = 1* and the new polynomials
  are the choice when $q = (2 - z)^2$.
 One can also write down a series for $z^n - 1$, which factor as the cyclotomic polynomials,
  and also end up being generated by an order-2 recurrence.
 $$
 \begin{align*}
 N(x; z) &= \sum_n (z^n - 1)x^n = {x(z - 1) \over 1 - (z + 1)x + zx^2}
 \\
  &= \sum_n \left ( x^n \prod_{d | n} \Phi_d(z) \right )
 \end{align*}
 $$
 Actually, this shouldn't be terribly surprising.
 For example, a simple result from generating functions tells us
  that a series for the integers is:
 $$
 \begin{align*}
  F(z) &= {1 \over 1 - z}
  = \sum_n z^n
  && \text{All coefficients equal 1}
 \\
  F'(z) &= {1 \over ( 1 - z )^2 }
  = \sum_n n z^{n - 1}
  && \text{Integers}
 \\
  z F'(z) &= {z \over 1 - 2z + z^2}
  = \sum_n n z^n
  && \text{Integers matching powers}
 \end{align*}
 $$
 The denominator being a quadratic polynomial means that the series terms *n*,
  the integers, obey an order-2 recurrence, and factor in a similar way.
 Obviously, the integers factor into primes by the fundamental theorem of arithmetic.
 There's still an important distinction to be made about the polynomials, though.
 Factoring a composite like 6 into 2 and 3 leaves an empty product behind, but
  for polynomials, nonprime indices end up accumulate an "extra" factor.
 Additionally (or rather, probably because of this), *all* factors of the index correspond
  to a factor in the factorization, rather than pairing off as in integers.
 For example, factoring 12 once gives either 3 and 4 or 2 and 6,
  but the polynomial at index 12 includes polynomials at indices of all factors: 2, 3, 4, 6, and 12.
 ### Higher-order Recurrences
 The integers also obey an order-3 recurrence:
 $$
 \begin{align*}
  F(x) &= {x \over 1 - 2x + x^2} = {x(1 - x) \over (1 - 2x + x^2)(1 - x)}
  \\
  &= {x - x^2 \over 1 - 3x + 3x^2 - x^3}
  \\[10pt]
  &\equiv a_{n+3} = 3a_{n+2} - 3a_{n+1} + a_n
 \end{align*}
 $$
 Another sequence that obeys similar factoring rules rules to the integers is
 $$
 G(x; z) = {x - x^2 \over 1 - z x + z x^2 - x^3}
 $$
 ```{python}
 #| code-fold: true
 #| tbl-cap: "Table of G polynomials"
 #| classes: plain
 G = (x - x**2) / ( 1 - z*x + z*x**2 - x**3 )
 Markdown(tabulate(
  [
    [ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
    for n, (unfactored, poly) in enumerate(
      factor_sequence(
          sorted(
              [
                  i.subs(x,1).subs(x1, z).expand().factor()
                  for i in G.series(x, n=11).args
              ][:-1],
              key=lambda x: sympy.degree(x, z)
          ),
          1,
          "o"
      )
    )
  ],
  headers=[ "*n*", "$[x^n]G(x; z) = o_n(z)$", "Coefficients (descending powers)" ],
  numalign="left",
  stralign="left",
 ))
 ```
 There are a couple of things to note here.
 Some of the factor polynomials here are the
  [minimal polynomials of cosine](/posts/math/chebyshev/1/#tbl-cosinepolynomials)
 The row where *n* = 5 is somewhat interesting.
 Namely, $x^2 - x - 1$ has $\varphi$ (the golden ratio) as a root.
 The other polynomial, $x^2 - 3x + 1$, has $\varphi^2$ as a root.
 This seems to indicate that the other polynomial has roots which are
  an algebraic expression of the other's, but I haven't bothered attempting a proof of this.
 Unfortunately, peppering polynomials into the denominator and hoping that the same factorization
  occurs isn't as easy as in the order-2 recurrence.
 In fact, higher-order recurrences become more and more restrictive as $x^\bullet$ terms are added to
  the numerator and denominator.
 If there is a rule to determine what relation must be obeyed between the coefficients
  for factorization to occur, it is not obvious, especially as the order grows.