401 lines
12 KiB
Plaintext
401 lines
12 KiB
Plaintext
---
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title: "Stereography, Algebraic, and Hyperspheres"
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description: |
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TODO
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format:
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html:
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html-math-method: katex
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jupyter: python3
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date: "2026-09-11"
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categories:
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- algebra
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---
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```{python}
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#| echo: false
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import sympy
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from IPython.display import Markdown
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from tabulate import tabulate
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x, x1, z = sympy.symbols("x x_1 z")
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```
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The Algebra Part
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----------------
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Though I alluded to the ability of the stereoscopic circle to generate the Chebyshev polynomials,
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there is an important caveat which differs their use from typical spheres.
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To review, the stereoscopic definition of the circle is:
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$$
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\begin{align*}
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o_1(t) &= {1 + it \over 1 - it}
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\\
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&= c_1 + i s_1
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= {1 - t^2 \over 1 + t^2} + i{2t \over 1 + t^2}
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\end{align*}
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$$
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The second line decomposes the first into real and nonreal terms, which are each rational functions.
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We can further define terms for the numerator and denominator:
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$$
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\begin{gather*}
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x_1 = 1 - t^2
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\qquad
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y_1 = 2t
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\qquad
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d = 1 + t^2 = 2 - x_1
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\\
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o_1 = {z_1 \over d} = {x_1 \over d} + i{y_1 \over d}
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\end{gather*}
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$$
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We have a recurrence relation for *o*, but it does not obey same relations as *z*:
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$$
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\begin{align*}
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o_{n+2} &= 2c_1 o_{n+1} - o_n
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\\
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z_{n+2} &\stackrel{✗}{=} 2x_1 z_{n+1} - z_n
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\end{align*}
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$$
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Fortunately, the correction is simple.
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The denominator term $d^{n+2}$ can be multiplied through the top equation to produce:
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$$
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z_{n+2} = 2x_1 z_{n+1} - z_n d^2
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$$
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The only term that changes is the term lagging two terms behind, so the generating function *Z* is:
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$$
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\begin{align*}
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O(x; o_1)
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&= {1 + x(o_1 - 2 c_1) \over 1 - 2 c_1 x + x^2}
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\\[10pt]
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Z(x; z_1)
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&= {1 + x(z_1 - 2 x_1) \over 1 - 2 x_1 x + \textcolor{red}{d^2} x^2}
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\end{align*}
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$$
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We can express *d* in terms of $x_1$, so the terms of the series, like the one for *F*, have
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- A real component which is a polynomial in $x_1$ (cf. $c_1$)
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- An imaginary component which is the product of $y_1$ (cf. $s_1$) and a polynomial in $x_1$
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$$
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Z(x; z_1) = X(x; x_1) + i y_1 Y(x; x_1)
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$$
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Surprisingly, the polynomials in *Y* still factor cleanly,
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like the [Chebyshev *U* polynomials](../../chebyshev/1/#tbl-chebyshevu).
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```{python}
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#| code-fold: true
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#| label: tbl-newupolynomials
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#| tbl-cap: "Table of numerator polynomials"
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#| classes: plain
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# cosine series
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X = ( 1 - x1*x ) / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
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# sine series
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Y = x / ( 1 - 2*x1*x + (2 - x1)**2*x**2 )
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def factor_sequence(polys, offset=0, symbol_name="p"):
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ret = []
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symbols = []
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for i, poly in enumerate(polys):
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new_poly = poly.copy()
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old_factor = 1
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for old, symbol in zip(ret, symbols):
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q, r = sympy.div(new_poly, old)
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if r == 0:
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new_poly = q
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old_factor *= symbol
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if new_poly != 1:
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ret.append(new_poly)
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symbols.append(sympy.symbols(f"{symbol_name}_{i + offset}"))
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yield poly, old_factor*new_poly.factor()
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Markdown(tabulate(
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[
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[ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
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for n, (unfactored, poly) in enumerate(
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factor_sequence(
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sorted(
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[
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i.subs(x,1).subs(x1, z).expand().factor()
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for i in Y.series(x, n=11).args
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][:-1],
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key=lambda x: sympy.degree(x, z)
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),
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1
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)
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)
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],
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headers=[ "*n*", "$[x^n]Y(x; z) = p_n(z)$", "Coefficients (descending powers)" ],
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numalign="left",
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stralign="left",
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))
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```
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Unfortunately, the sequence formed by the coefficients of the polynomials
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does not appear in the OEIS.
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Their factorizations appear to have the following traits:
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- Like the Chebyshev *U* polynomials, they have "cyclotomic factoring" --
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for the new term of index *n*, the factors can be separated into old factors
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at indices of factors of *n* and new factors.
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- If the index is even, then there is only one new monic, irreducible factor.
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- If the index is odd, then there are two new irreducible factors
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- If the index is prime or a prime power, the new factors are a monic and a non-monic
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whose leading coefficient is that prime.
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- Otherwise, the new factors are both monic.
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The characterization of the leading terms of the new factor corresponds to
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[OEIS A014963](https://oeis.org/A014963), which is related to cyclotomic polynomials.
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### Similar Sequences
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In fact, for a polynomial $q(z)$, it seems to be the case that the terms of
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$$
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Y(x; z) = {x \over 1 - 2 z x + q(z) x^2}
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$$
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tend to factor similarly.
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Naturally, the *U* polynomials are the choice where *q = 1* and the new polynomials
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are the choice when $q = (2 - z)^2$.
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One can also write down a series for $z^n - 1$, which factor as the cyclotomic polynomials,
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and also end up being generated by an order-2 recurrence.
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$$
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\begin{align*}
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N(x; z) &= \sum_n (z^n - 1)x^n = {x(z - 1) \over 1 - (z + 1)x + zx^2}
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\\
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&= \sum_n \left ( x^n \prod_{d | n} \Phi_d(z) \right )
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\end{align*}
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$$
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Actually, this shouldn't be terribly surprising.
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For example, a simple result from generating functions tells us
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that a series for the integers is:
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$$
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\begin{align*}
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F(z) &= {1 \over 1 - z}
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= \sum_n z^n
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&& \text{All coefficients equal 1}
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\\
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F'(z) &= {1 \over ( 1 - z )^2 }
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= \sum_n n z^{n - 1}
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&& \text{Integers}
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\\
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z F'(z) &= {z \over 1 - 2z + z^2}
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= \sum_n n z^n
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&& \text{Integers matching powers}
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\end{align*}
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$$
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The denominator being a quadratic polynomial means that the series terms *n*,
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the integers, obey an order-2 recurrence, and factor in a similar way.
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Obviously, the integers factor into primes by the fundamental theorem of arithmetic.
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There's still an important distinction to be made about the polynomials, though.
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Factoring a composite like 6 into 2 and 3 leaves an empty product behind, but
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for polynomials, nonprime indices end up accumulate an "extra" factor.
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Additionally (or rather, probably because of this), *all* factors of the index correspond
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to a factor in the factorization, rather than pairing off as in integers.
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For example, factoring 12 once gives either 3 and 4 or 2 and 6,
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but the polynomial at index 12 includes polynomials at indices of all factors: 2, 3, 4, 6, and 12.
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### Higher-order Recurrences
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The integers also obey an order-3 recurrence:
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$$
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\begin{align*}
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F(x) &= {x \over 1 - 2x + x^2} = {x(1 - x) \over (1 - 2x + x^2)(1 - x)}
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\\
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&= {x - x^2 \over 1 - 3x + 3x^2 - x^3}
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\\[10pt]
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&\equiv a_{n+3} = 3a_{n+2} - 3a_{n+1} + a_n
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\end{align*}
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$$
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Another sequence that obeys similar factoring rules rules to the integers is
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$$
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G(x; z) = {x - x^2 \over 1 - z x + z x^2 - x^3}
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$$
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```{python}
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#| code-fold: true
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#| tbl-cap: "Table of G polynomials"
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#| classes: plain
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G = (x - x**2) / ( 1 - z*x + z*x**2 - x**3 )
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Markdown(tabulate(
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[
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[ n+1, "$" + sympy.latex(poly) + "$", sympy.Poly(unfactored, z).as_list() ]
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for n, (unfactored, poly) in enumerate(
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factor_sequence(
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sorted(
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[
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i.subs(x,1).subs(x1, z).expand().factor()
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for i in G.series(x, n=11).args
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][:-1],
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key=lambda x: sympy.degree(x, z)
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),
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1,
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"o"
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)
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)
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],
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headers=[ "*n*", "$[x^n]G(x; z) = o_n(z)$", "Coefficients (descending powers)" ],
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numalign="left",
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stralign="left",
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))
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```
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There are a couple of things to note here.
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Some of the factor polynomials here are the
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[minimal polynomials of cosine](/posts/math/chebyshev/1/#tbl-cosinepolynomials)
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The row where *n* = 5 is somewhat interesting.
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Namely, $x^2 - x - 1$ has $\varphi$ (the golden ratio) as a root.
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The other polynomial, $x^2 - 3x + 1$, has $\varphi^2$ as a root.
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This seems to indicate that the other polynomial has roots which are
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an algebraic expression of the other's, but I haven't bothered attempting a proof of this.
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Unfortunately, peppering polynomials into the denominator and hoping that the same factorization
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occurs isn't as easy as in the order-2 recurrence.
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In fact, higher-order recurrences become more and more restrictive as $x^\bullet$ terms are added to
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the numerator and denominator.
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If there is a rule to determine what relation must be obeyed between the coefficients
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for factorization to occur, it is not obvious, especially as the order grows.
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Higher-dimensional Spheres
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--------------------------
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We derived an explicit map for the 2-sphere (or rather, the 3-sphere, since the description ended up matching the quaternions)
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in [the first post in this series](../1/).
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It's quite easy to generalize the argument to higher dimensions.
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In *n* dimensions, assume that we have unit vectors $e_0 ... e_{n-1}$.
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Placing these vectors within a [geometric algebra](https://en.wikipedia.org/wiki/Geometric_algebra)
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gives some promising properties:
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- Vectors can be multiplied like ordinary numbers, and even added to ordinary numbers
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- The product of a vector with itself can be chosen among -1, 0, or 1
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- The product of two vectors anticommutes (e.g., $e_0 e_1 = - e_1 e_0$)
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- Consequently, the square of a product is the negative of the product of the squares (e.g., $e_0 e_1 e_0 e_1 = - e_0 e_1 e_1 e_0 = - e_0^2 e_1^2$)
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The square of a general vector with components $x_k e_k$ is a scalar, its norm.
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$$
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\begin{align*}
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{\bm v} &= e_0 x_0 + e_1 x_1 + ... e_{n-1} x_{n-1} = \sum_k e_k x_k
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\\
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{\bm v}^2 &= (\sum_k^{n-1} e_k x_k) (\sum_l^{n-1} e_l x_l)
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= \sum_k^{n-1} \sum_l^{n-1} e_k e_l x_k x_l
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\\
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&= \underset{\diagdown}{\sum_k^{n-1} e_k^2 x_k^2}
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+ \underset{◥}{ \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l }
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+ \underset{◣}{ \sum_k^{n-1} \sum_{l < k} e_k e_l x_k x_l }
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\\
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&= \diagdown + ◥
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+ \sum_k^{n-1} \sum_{l > k} e_l e_k x_k x_l
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= \diagdown + ◥
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- \sum_k^{n-1} \sum_{l > k} e_k e_l x_k x_l
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\\
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&= \diagdown + ◥ - ◥
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= \diagdown
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\end{align*}
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$$
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For simplicity, assume that $e_k^2 = -1$ so that ${\bm v}^2 = - ||{\bm v}||$,
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the sum of squares of the extent in each basis.
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Despite multiplication between two vectors being defined, dividing one vector by another is not.
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Ignoring this, consider the expression
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$$
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\begin{align*}
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{1 + {\bm v} \over 1 - {\bm v}}
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&= \left( {1 + {\bm v} \over 1 - {\bm v}} \right)
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\left( {1 + {\bm v} \over 1 + {\bm v}} \right)
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= {(1 + {\bm v})^2 \over (1 - {\bm v})(1 + {\bm v})}
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\\
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&= {1 + 2{\bm v} + {\bm v}^2 \over 1 - {\bm v}^2}
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\\
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&= {1 - ||{\bm v}|| \over 1 + ||{\bm v}||} + {2{\bm v} \over 1 + ||{\bm v}||}
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\end{align*}
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$$
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Similarly to quaternions, this is the sum of a vector and a scalar.
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If the scalar component is considered the extent in a new dimension,
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then the norm of the resulting vector is
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$$
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a^2 + ||u|| = a^2 - u^2 = (a + u)(a - u)
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$$
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This is actually an inductive hypothesis.
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This forces us to choose two things: the norm we use is Euclidean, and each new unit vector squares to -1.
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For the sphere, focusing just on the numerator
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$$
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(1 - ||v||)^2 - u^2 = (1 - ||v||)^2 - (2v)^2 = 1 - 2||v|| + ||v||^2 - 4v^2 = 1 - 2||v|| + ||v||^2 + 4||v|| = 1 + 2||v|| + ||v||^2 = (1 + ||v||)^2
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$$
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This is the denominator, so the dubious step of dividing by a vector has been backed up with pure algebra.
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### Degree Maps
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If $\bm u$ is a vector with norm 1.
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The first coordinate expresses the condition that if **v** lies on the unit (hyper)sphere
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in the dimension below, then the coordinate is zero, and such points lie on an equator.
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In the case of the 2-sphere, doubling the map by squaring quaternions produces a more interesting feature:
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all **v** lying on the unit (hyper)sphere in the dimension below get mapped to the same point, antipodal to 0.
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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.
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$$
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{ D^n / \partial D^n } = S^n
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$$
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This applies generally.
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This can be made more topological by doubling the sphere, but we don't know how to do that generally,
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A classical homotopy result informs
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$$
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\pi_n(S^n) = \Z
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$$
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Telling us we can wrap an *n* sphere around itself, backwards and forwards any number of times.
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It's annoying to do this explicitly without a generic way to describe higher dimensional spheres.
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The relation seems to be:
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$$
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_n o_m = \left( T_m(o_{1,0}), o_{1,[1:n]}U(o_{1,0}) \right)
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$$
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Just like in the circle. This is extraordinarily convenient.
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