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170
_quarto.yml
170
_quarto.yml
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||||
type: website
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||||
|
||||
website:
|
||||
title: "Zenzicubi.co"
|
||||
favicon: "./logo-favicon.png"
|
||||
title: "zenzicubi.co"
|
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|
||||
repo-url: https://git.zenzicubi.co/cube/zenzicubi.co
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|
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repo-actions: [source]
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repo-link-target: "_blank"
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
- ./posts/math/polycount/index.qmd
|
||||
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|
||||
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|
||||
- ./posts/math/stereo/index.qmd
|
||||
- ./posts/math/permutations/index.qmd
|
||||
- ./posts/math/type-algebra/index.qmd
|
||||
- ./posts/math/number-number/index.qmd
|
||||
- ./posts/math/finite-field/index.qmd
|
||||
- ./posts/math/misc/index.qmd
|
||||
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|
||||
# - about.qmd
|
||||
- ./about/index.qmd
|
||||
- icon: git
|
||||
href: https://git.zenzicubi.co/cube/zenzicubi.co
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||||
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|
||||
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|
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||||
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|
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
- ./posts/math/pentagons/index.qmd
|
||||
- ./posts/math/chebyshev/index.qmd
|
||||
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|
||||
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|
||||
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|
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|
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||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
contents:
|
||||
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|
||||
- ./posts/math/misc/infinitesimals/index.qmd
|
||||
|
||||
- id: polycount-sidebar
|
||||
style: "floating"
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
- text: "Part 3: The third degree"
|
||||
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|
||||
- text: "Part 4: Two twos"
|
||||
href: ./posts/math/polycount/4/index.qmd
|
||||
contents:
|
||||
- text: "Appendix"
|
||||
href: ./posts/math/polycount/4/appendix/index.qmd
|
||||
- text: "Part 5: Pentamerous multiplication"
|
||||
href: ./posts/math/polycount/5/index.qmd
|
||||
- section: 2D
|
||||
contents:
|
||||
- text: "Part 1: Lines, leaves, and sand"
|
||||
href: ./posts/math/polycount/sand-1/index.qmd
|
||||
- text: "Part 2: Reorienting Polynomials"
|
||||
href: ./posts/math/polycount/sand-2/index.qmd
|
||||
|
||||
- id: pentagons-sidebar
|
||||
style: "floating"
|
||||
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|
||||
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|
||||
href: ./posts/math/pentagons/index.qmd
|
||||
contents:
|
||||
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|
||||
href: ./posts/math/pentagons/1/index.qmd
|
||||
- text: "Part 2"
|
||||
href: ./posts/math/pentagons/2/index.qmd
|
||||
- text: "Part 3"
|
||||
href: ./posts/math/pentagons/3/index.qmd
|
||||
|
||||
- id: chebyshev-sidebar
|
||||
style: "floating"
|
||||
contents:
|
||||
- section: "Generating Polynomials"
|
||||
href: ./posts/math/chebyshev/index.qmd
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
href: ./posts/math/chebyshev/extra/index.qmd
|
||||
|
||||
- id: stereography-sidebar
|
||||
style: "floating"
|
||||
contents:
|
||||
- section: "Algebraic Stereography"
|
||||
href: ./posts/math/stereo/index.qmd
|
||||
contents:
|
||||
- ./posts/math/stereo/1/index.qmd
|
||||
- ./posts/math/stereo/2/index.qmd
|
||||
|
||||
- id: permutations-sidebar
|
||||
style: "floating"
|
||||
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|
||||
- section: "A Game of Permutations"
|
||||
href: ./posts/math/permutations/index.qmd
|
||||
contents:
|
||||
- text: "Part 1"
|
||||
href: ./posts/math/permutations/1/index.qmd
|
||||
- text: "Part 2"
|
||||
href: ./posts/math/permutations/2/index.qmd
|
||||
- text: "Part 3"
|
||||
href: ./posts/math/permutations/3/index.qmd
|
||||
- text: "Appendix"
|
||||
href: ./posts/math/permutations/appendix/index.qmd
|
||||
|
||||
- id: type-algebra-sidebar
|
||||
style: "floating"
|
||||
contents:
|
||||
- section: "Type Algebra and You"
|
||||
href: ./posts/math/type-algebra/index.qmd
|
||||
contents:
|
||||
- text: "Part 1: Basics"
|
||||
href: ./posts/math/type-algebra/1/index.qmd
|
||||
- text: "Part 2: A Fixer-upper"
|
||||
href: ./posts/math/type-algebra/2/index.qmd
|
||||
- text: "Part 3: Combinatorial Types"
|
||||
href: ./posts/math/type-algebra/3/index.qmd
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
contents:
|
||||
- text: "From 0 to ∞"
|
||||
href: ./posts/math/number-number/1/index.qmd
|
||||
- text: "Ordering Obliquely"
|
||||
href: ./posts/math/number-number/2/index.qmd
|
||||
|
||||
- id: finite-field-sidebar
|
||||
style: "floating"
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
href: ./posts/math/finite-field/1/index.qmd
|
||||
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|
||||
href: ./posts/math/finite-field/2/index.qmd
|
||||
contents:
|
||||
- text: "Appendix"
|
||||
href: ./posts/math/finite-field/2/extra/index.qmd
|
||||
- text: "Part 3: Roll a d20"
|
||||
href: ./posts/math/finite-field/2/index.qmd
|
||||
- text: "Part 5: The Power of Forgetting"
|
||||
href: ./posts/math/finite-field/2/index.qmd
|
||||
|
||||
format:
|
||||
html:
|
||||
theme:
|
||||
light:
|
||||
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||||
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|
||||
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||||
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||||
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||||
63
about/index.qmd
Normal file
63
about/index.qmd
Normal file
@ -0,0 +1,63 @@
|
||||
---
|
||||
title: "About"
|
||||
---
|
||||
|
||||
This is my personal website (and the third iteration thereof).
|
||||
|
||||
The first version used Wordpress since it was quite easy to get into,
|
||||
didn't require much research, and web hosting services made it easy to set up.
|
||||
It lasted around three months near the end of 2020, after which I lost my posts because of
|
||||
hosting troubles and because I wasn't using proper backups.
|
||||
|
||||
The second version also used Wordpress, and lasted until the start of 2025
|
||||
(though the last post I had written up to that point was from the start of 2024).
|
||||
|
||||
This version uses [Quarto](https://quarto.org/), an open-source publishing platform that has
|
||||
some nice features like text-based configuration and Jupyter integration.
|
||||
As a bonus, it also produces static web pages.
|
||||
|
||||
|
||||
Why Quarto?
|
||||
-----------
|
||||
|
||||
I had a couple of reasons for switching platforms:
|
||||
|
||||
- Wordpress is either overkill or not enough.
|
||||
I don't need a block editor or multiple users, and I don't want to make custom content
|
||||
just for it to be specific to Wordpress.
|
||||
- I write a lot of code and LaTeX, which Wordpress relies on plugins for.
|
||||
Quarto uses (primarily) Pandoc-style Markdown, which allows for inlining of both out of the box.
|
||||
- Also, because of Jupyter integration, code cells can generate output for the page they're in.
|
||||
- Since pages are written in Markdown, everything can be edited locally and version-controlled in Git.
|
||||
|
||||
The last two are particularly nice in ensuring that the site is reproducibile,
|
||||
technically even without Quarto.
|
||||
Instead of articles that live in a Wordpress database or as scattered random files,
|
||||
I have the complete documents in a structure 1:1 with how the website is organized.
|
||||
|
||||
|
||||
Mathematics
|
||||
-----------
|
||||
|
||||
As of writing, all posts on this site are about math.
|
||||
In particular, they are dedicated to certain non-obvious insights I choose to investigate.
|
||||
Typically, although information about these subjects may exist online, it does not exist in a single,
|
||||
easily-accessible source.
|
||||
|
||||
I find writing math posts to be an excellent motivator when it comes to researching things.
|
||||
It also gives me a chance to learn new tools that otherwise I would not have a reason to use,
|
||||
not to mention being a good exercise in writing and diagram creation.
|
||||
|
||||
An example of this (and one that relates to the creation of the site) is when I was writing code
|
||||
for what would become the contents of [this post](/posts/polycount/5/).
|
||||
It was easy enough to learn a library for rendering images (or GIFs),
|
||||
but I didn't have a gallery to host them, nor a means to share the rationale which produced them.
|
||||
In a frenzy, I tried gathering my notes in a single text file before eventually putting them on a website.
|
||||
Along the way, I learned LaTeX to typeset the relevant equations.
|
||||
|
||||
I do my best to attribute the programs I use and direct sources I consult along the way,
|
||||
but extra information is frequently available on Wikipedia,
|
||||
which I may link to in order to give my explanation some grounding.
|
||||
|
||||
Unless otherwise stated, the figures and articles in this category are available under
|
||||
[CC BY-SA](https://creativecommons.org/licenses/by-sa/4.0/).
|
||||
13
index.qmd
13
index.qmd
@ -1,6 +1,15 @@
|
||||
---
|
||||
title: "Posts by topic"
|
||||
title: "Posts"
|
||||
listing:
|
||||
contents: posts/polycount/index.*
|
||||
contents:
|
||||
- posts/math/polycount/*/index.*
|
||||
- posts/math/pentagons/*/index.*
|
||||
- posts/math/chebyshev/*/index.*
|
||||
- posts/math/stereo/*/index.*
|
||||
- posts/math/permutations/*/index.*
|
||||
- posts/math/type-algebra/*/index.*
|
||||
- posts/math/number-number/*/index.*
|
||||
- posts/math/finite-field/*/index.*
|
||||
- posts/math/misc/*/index.*
|
||||
sort: "date desc"
|
||||
---
|
||||
|
||||
@ -27,9 +27,10 @@ Each term of the series is weighted by a geometrically decreasing coefficient *c
|
||||
|
||||
$$
|
||||
[...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p}
|
||||
+ c e^{2\pi i [d_1 d_0] / p^2}
|
||||
+ c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
|
||||
+ ... \\
|
||||
+ c e^{2\pi i [d_1 d_0] / p^2}
|
||||
+ c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
|
||||
+ ...
|
||||
\\
|
||||
f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}}
|
||||
$$
|
||||
|
||||
|
||||
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<path
|
||||
d="m 3108,7004 c -54,-29 -64,-93 -23,-142 l 25,-30 -35,-35 c -39,-39 -44,-70 -20,-116 13,-26 36,-40 92,-56 36,-10 99,13 127,46 39,47 36,93 -11,138 l -38,36 28,27 c 83,81 -37,191 -145,132 z m 102,-19 c 17,-9 30,-20 30,-24 0,-5 2,-16 6,-24 3,-8 -7,-28 -21,-45 -26,-31 -27,-31 -57,-16 -17,9 -42,27 -55,40 -20,20 -22,27 -13,44 20,38 65,48 110,25 z m -8,-204 c 25,-16 49,-39 53,-50 13,-42 -31,-81 -91,-81 -31,0 -84,50 -84,78 0,24 46,82 64,82 7,0 33,-13 58,-29 z"
|
||||
id="path1" />
|
||||
<path
|
||||
d="m 2736,6628 c -38,-36 -56,-78 -34,-78 7,0 19,9 25,19 18,29 61,34 116,13 l 49,-19 -102,-94 c -89,-82 -170,-174 -170,-193 0,-15 24,-3 50,24 15,17 37,30 47,30 11,0 32,-13 48,-30 33,-34 72,-39 114,-14 47,28 99,124 67,124 -8,0 -18,-9 -21,-21 -8,-23 -61,-59 -88,-59 -10,0 -41,7 -68,16 l -51,17 125,121 c 129,125 156,159 136,172 -7,4 -22,-7 -36,-25 -31,-39 -57,-39 -93,-1 -37,40 -71,39 -114,-2 z"
|
||||
id="path2" />
|
||||
</g>
|
||||
</svg>
|
||||
|
After Width: | Height: | Size: 2.3 KiB |
2
posts/.gitignore
vendored
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2
posts/.gitignore
vendored
Normal file
@ -0,0 +1,2 @@
|
||||
*.o
|
||||
*.hi
|
||||
3
posts/math/_metadata.yml
Normal file
3
posts/math/_metadata.yml
Normal file
@ -0,0 +1,3 @@
|
||||
toc: true
|
||||
toc-location: right
|
||||
toc-title: " "
|
||||
BIN
posts/math/chebyshev/1/central_angle_figures.png
(Stored with Git LFS)
Normal file
BIN
posts/math/chebyshev/1/central_angle_figures.png
(Stored with Git LFS)
Normal file
Binary file not shown.
BIN
posts/math/chebyshev/1/decagon_divisible.png
(Stored with Git LFS)
Normal file
BIN
posts/math/chebyshev/1/decagon_divisible.png
(Stored with Git LFS)
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754
posts/math/chebyshev/1/index.qmd
Normal file
754
posts/math/chebyshev/1/index.qmd
Normal file
@ -0,0 +1,754 @@
|
||||
---
|
||||
title: "Generating Polynomials, Part 1: Regular Constructibility"
|
||||
description: |
|
||||
What kinds of regular polygons are constructible with compass and straightedge?
|
||||
format:
|
||||
html:
|
||||
html-math-method: katex
|
||||
date: "2021-08-18"
|
||||
date-modified: "2025-06-17"
|
||||
categories:
|
||||
- geometry
|
||||
- generating functions
|
||||
- algebra
|
||||
- python
|
||||
---
|
||||
|
||||
<style>
|
||||
.figure-img {
|
||||
max-width: 512px;
|
||||
object-fit: contain;
|
||||
height: 100%;
|
||||
}
|
||||
|
||||
.figure-img.wide {
|
||||
max-width: 768px;
|
||||
}
|
||||
</style>
|
||||
|
||||
```{python}
|
||||
#| echo: false
|
||||
|
||||
from math import comb
|
||||
|
||||
from IPython.display import Markdown
|
||||
from tabulate import tabulate
|
||||
import sympy
|
||||
from sympy.abc import z
|
||||
```
|
||||
|
||||
|
||||
[Recently](/posts/math/misc/platonic-volume), I used coordinate-free geometry to derive
|
||||
the volumes of the Platonic solids, a problem which was very accessible to the ancient Greeks.
|
||||
On the other hand, they found certain problems regarding which figures can be constructed via
|
||||
compass and straightedge to be very difficult. For example, they struggled with problems
|
||||
like [doubling the cube](https://en.wikipedia.org/wiki/Doubling_the_cube)
|
||||
or [squaring the circle](https://en.wikipedia.org/wiki/Squaring_the_circle),
|
||||
which are known (through circa 19th century mathematics) to be impossible.
|
||||
However, before even extending planar geometry by a third dimension or
|
||||
calculating the areas of circles, a simpler problem becomes apparent.
|
||||
Namely, what kinds of regular polygons are constructible?
|
||||
|
||||
|
||||
Regular Geometry and a Complex Series
|
||||
-------------------------------------
|
||||
|
||||
When constructing a regular polygon, one wants a ratio between the length of a edge
|
||||
and the distance from a vertex to the center of the figure.
|
||||
|
||||
{.wide}
|
||||
|
||||
In a convex polygon, the total central angle is always one full turn, or 2π radians.
|
||||
The central angle of a regular *n*-gon is ${2\pi \over n}$ radians,
|
||||
and the green angle above (which we'll call *θ*) is half of that.
|
||||
This means that the ratio we're looking for is $\sin(\theta) = \sin(\pi / n)$.
|
||||
We can multiply by *n* inside the function on both sides to give
|
||||
$\sin(n\theta) = \sin(\pi) = 0$.
|
||||
Therefore, constructing a polygon is actually equivalent to solving this equation,
|
||||
and we can rephrase the question as how to express $\sin(n\theta)$ (and $\cos(n\theta)$).
|
||||
|
||||
|
||||
### Complex Recursion
|
||||
|
||||
Thanks to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula)
|
||||
and [de Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula),
|
||||
the expressions we're looking for can be phrased in terms of the complex exponential.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
e^{i\theta}
|
||||
&= \text{cis}(\theta) = \cos(\theta) + i\sin(\theta)
|
||||
& \text{ Euler's formula}
|
||||
\\
|
||||
\text{cis}(n \theta) = e^{i(n\theta)}
|
||||
&= e^{(i\theta)n} = {(e^{i\theta})}^n = \text{cis}(\theta)^n
|
||||
\\
|
||||
\cos(n \theta) + i\sin(n \theta)
|
||||
&= (\cos(\theta) + i\sin(\theta))^n
|
||||
& \text{ de Moivre's formula}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
De Moivre's formula for $n = 2$ gives
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{cis}(\theta)^2
|
||||
&= (\text{c} + i\text{s})^2
|
||||
\\
|
||||
&= \text{c}^2 + 2i\text{cs} - \text{s}^2 + (0 = \text{c}^2 + \text{s}^2 - 1)
|
||||
\\
|
||||
&= 2\text{c}^2 + 2i\text{cs} - 1
|
||||
\\
|
||||
&= 2\text{c}(\text{c} + i\text{s}) - 1
|
||||
\\
|
||||
&= 2\cos(\theta)\text{cis}(\theta) - 1
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
This can easily be massaged into a recurrence relation.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{cis}(\theta)^2
|
||||
&= 2\cos(\theta)\text{cis}(\theta) - 1
|
||||
\\
|
||||
\text{cis}(\theta)^{n+2}
|
||||
&= 2\cos(\theta)\text{cis}(\theta)^{n+1} - \text{cis}(\theta)^n
|
||||
\\
|
||||
\text{cis}((n+2)\theta)
|
||||
&= 2\cos(\theta)\text{cis}((n+1)\theta) - \text{cis}(n\theta)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Recurrence relations like this one are powerful.
|
||||
Through some fairly straightforward summatory manipulations,
|
||||
the sequence can be interpreted as the coefficients in a Taylor series,
|
||||
giving a [generating function](https://en.wikipedia.org/wiki/Generating_function).
|
||||
Call this function *F*. Then,
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\sum_{n=0}^\infty \text{cis}((n+2)\theta)x^n
|
||||
&= 2\cos(\theta) \sum_{n=0}^\infty \text{cis}((n+1)\theta) x^n
|
||||
- \sum_{n=0}^\infty \text{cis}(n\theta) x^n
|
||||
\\
|
||||
{F(x; \text{cis}(\theta)) - 1 - x\text{cis}(\theta) \over x^2}
|
||||
&= 2\cos(\theta) {F(x; \text{cis}(\theta)) - 1 \over x}
|
||||
- F(x; \text{cis}(\theta))
|
||||
\\[10pt]
|
||||
F - 1 - x\text{cis}(\theta)
|
||||
&= 2\cos(\theta) x (F - 1)
|
||||
- x^2 F
|
||||
\\
|
||||
F - 2\cos(\theta) x F + x^2 F
|
||||
&= 1 + x(\text{cis}(\theta) - 2\cos(\theta))
|
||||
\\[10pt]
|
||||
F(x; \text{cis}(\theta))
|
||||
&= {1 + x(\text{cis}(\theta) - 2\cos(\theta)) \over
|
||||
1 - 2\cos(\theta)x + x^2}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Since $\text{cis}$ is a complex function, we can separate *F* into real and imaginary parts.
|
||||
Conveniently, these correspond to $\cos(n\theta)$ and $\sin(n\theta)$, respectively.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\Re[ F(x; \text{cis}(\theta)) ]
|
||||
&= {1 + x(\cos(\theta) - 2\cos(\theta)) \over 1 - 2\cos(\theta)x + x^2}
|
||||
\\
|
||||
&= {1 - x\cos(\theta) \over 1 - 2\cos(\theta)x + x^2} = A(x; \cos(\theta))
|
||||
\\
|
||||
\Im[ F(x; \text{cis}(\theta)) ]
|
||||
&= {x \sin(\theta) \over 1 - 2\cos(\theta)x + x^2} = B(x; \cos(\theta))\sin(\theta)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
In this form, it becomes obvious that the even though the generating function *F* was originally
|
||||
parametrized by $\text{cis}(\theta)$, *A* and *B* are parametrized only by $\cos(\theta)$.
|
||||
Extracting the coefficients of *x* yields an expression for $\cos(n\theta)$ and $\sin(n\theta)$
|
||||
in terms of $\cos(\theta)$ (and in the latter case, a common factor of $\sin(\theta)$).
|
||||
|
||||
If $\cos(\theta)$ in *A* and *B* is replaced with the parameter *z*, then all trigonometric functions
|
||||
are removed from the equation, and we are left with only polynomials[^1].
|
||||
These polynomials are [*Chebyshev polynomials*](https://en.wikipedia.org/wiki/Chebyshev_polynomial)
|
||||
*of the first (A) and second (B) kind*.
|
||||
In actuality, the polynomials of the second kind are typically offset by 1
|
||||
(the x in the numerator of *B* is omitted).
|
||||
However, retaining this term makes indexing consistent between *A* and *B*
|
||||
(and will make things clearer later).
|
||||
|
||||
[^1]:
|
||||
This can actually be observed as early as the recurrence relation.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\text{cis}(\theta)^{n+2}
|
||||
&= 2\cos(\theta)\text{cis}(\theta)^{n+1} - \text{cis}(\theta)^n
|
||||
\\
|
||||
a_{n+2}
|
||||
&= 2 z a_{n+1} - a_n
|
||||
\\
|
||||
\Re[ a_0 ]
|
||||
&= 1,~~ \Im[ a_0 ] = 0
|
||||
\\
|
||||
\Re[ a_1 ]
|
||||
&= z,~~ \Im[ a_1 ] = 1 \cdot \sin(\theta)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
|
||||
We were primarily interested in $\sin(n\theta)$, so let's tabulate
|
||||
the first few polynomials of the second kind (at $z / 2$).
|
||||
|
||||
```{python}
|
||||
#| echo: false
|
||||
#| label: tbl-chebyshevu
|
||||
#| tbl-cap: "[OEIS A049310](http://oeis.org/A049310)"
|
||||
#| classes: plain
|
||||
|
||||
Markdown(tabulate(
|
||||
[
|
||||
[
|
||||
n,
|
||||
"$" + sympy.latex(poly) + "$",
|
||||
"$" + sympy.latex(sympy.factor(poly)) + "$",
|
||||
]
|
||||
for n in range(0, 11)
|
||||
for poly in [sympy.chebyshevu_poly(n - 1, z / 2) if n > 0 else sympy.sympify(0)]
|
||||
],
|
||||
headers=[
|
||||
"*n*",
|
||||
"$[x^n]B(x; z / 2) = U_{n - 1}(z / 2)$",
|
||||
"Factored",
|
||||
],
|
||||
numalign="left",
|
||||
stralign="left",
|
||||
))
|
||||
```
|
||||
|
||||
Evaluating the polynomials at $z / 2$ cancels the 2 in the denominator (and recurrence),
|
||||
making these expressions much simpler.
|
||||
This evaluation has an interpretation in terms of the previous diagram --
|
||||
recall we used *half* the length of a side as a leg of the right triangle.
|
||||
For a unit circumradius, the side length itself is then $2\sin( {\pi / n} )$.
|
||||
To compensate for this doubling, the Chebyshev polynomial must be evaluated at half its normal argument.
|
||||
|
||||
|
||||
### Back on the Plane
|
||||
|
||||
The constructibility criterion is deeply connected to the Chebyshev polynomials.
|
||||
In compass and straightedge constructions, one only has access to linear forms (lines)
|
||||
and quadratic forms (circles).
|
||||
This means that a figure is constructible if and only if the root can be expressed using
|
||||
normal arithmetic (which is linear) and square roots (which are quadratic).
|
||||
|
||||
|
||||
#### Pentagons
|
||||
|
||||
Let's look at a regular pentagon.
|
||||
The relevant polynomial is
|
||||
|
||||
$$
|
||||
[x^5]B ( x; z / 2 )
|
||||
= z^4 - 3z^2 + 1
|
||||
= (z^2 - z - 1) (z^2 + z - 1)
|
||||
$$
|
||||
|
||||
According to how we derived this series, when $z = 2\cos(\theta)$, the roots of this polynomial
|
||||
correspond to when $\sin(5\theta) / \sin(\theta) = 0$.
|
||||
This relation itself is true when $\theta = \pi / 5$, since $\sin(5 \pi / 5) = 0$.
|
||||
|
||||
One of the factors must therefore be the minimal polynomial of $2\cos(\pi / 5 )$.
|
||||
The former happens to be correct correct, since $2\cos( \pi / 5 ) = \varphi$, the golden ratio.
|
||||
Note that the second factor is the first evaluated at -*z*.
|
||||
|
||||
|
||||
#### Heptagons
|
||||
|
||||
An example of where constructability fails is for $2\cos( \pi / 7 )$.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
[x^7]B ( x; z / 2 )
|
||||
&= z^6 - 5 z^4 + 6 z^2 - 1
|
||||
\\
|
||||
&= ( z^3 - z^2 - 2 z + 1 ) ( z^3 + z^2 - 2 z - 1 )
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Whichever is the minimal polynomial (the former), it is a cubic, and constructing
|
||||
a regular heptagon is equivalent to solving it for *z*.
|
||||
But there are no (nondegenerate) cubics that one can produce via compass and straightedge,
|
||||
and all constructions necessarily fail.
|
||||
|
||||
|
||||
#### Decagons
|
||||
|
||||
One might think the same of $2\cos(\pi /10 )$
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
[x^{10}]B ( x; z / 2 )
|
||||
&= z^9 - 8 z^7 + 21 z^5 - 20 z^3 + 5 z
|
||||
\\
|
||||
&= z ( z^2 - z - 1 )( z^2 + z - 1 )( z^4 - 5 z^2 + 5 )
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
This expression also contains the polynomials for $2\cos( \pi / 5 )$.
|
||||
This is because a regular decagon would contain two disjoint regular pentagons,
|
||||
produced by connecting every other vertex.
|
||||
|
||||
|
||||
|
||||
The polynomial which actually corresponds to $2\cos( \pi / 10 )$ is the quartic,
|
||||
which seems to suggest that it will require a fourth root and somehow decagons are not constructible.
|
||||
However, it can be solved by completing the square...
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
z^4 - 5z^2 &= -5
|
||||
\\
|
||||
z^4 - 5z^2 + (5/2)^2 &= -5 + (5/2)^2
|
||||
\\
|
||||
( z^2 - 5/2)^2 &= {25 - 20 \over 4}
|
||||
\\
|
||||
( z^2 - 5/2) &= {\sqrt 5 \over 2}
|
||||
\\
|
||||
z^2 &= {5 \over 2} + {\sqrt 5 \over 2}
|
||||
\\
|
||||
z &= \sqrt{ {5 + \sqrt 5 \over 2} }
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
...and we can breathe a sigh of relief.
|
||||
|
||||
|
||||
The Triangle behind Regular Polygons
|
||||
------------------------------------
|
||||
|
||||
Preferring *z* to be halved in $B(x; z/2)$ makes something else more evident.
|
||||
Observe these four rows of the Chebyshev polynomials
|
||||
|
||||
```{python}
|
||||
#| echo: false
|
||||
#| classes: plain
|
||||
|
||||
Markdown(tabulate(
|
||||
[
|
||||
[
|
||||
n,
|
||||
"$" + sympy.latex(poly) + "$",
|
||||
k,
|
||||
int(poly.coeff(z, k)), # type: ignore
|
||||
]
|
||||
for n, k in zip(range(4, 8), range(3, -1, -1))
|
||||
for poly in [sympy.chebyshevu_poly(n - 1, z / 2) if n > 0 else sympy.sympify(0)]
|
||||
],
|
||||
headers=[
|
||||
"*n*",
|
||||
"$[x^n]B(x; z / 2)$",
|
||||
"*k*",
|
||||
"$[z^{k}][x^n]B(x; z / 2)$",
|
||||
],
|
||||
numalign="left",
|
||||
stralign="left",
|
||||
))
|
||||
```
|
||||
|
||||
The last column looks like an alternating row of Pascal's triangle
|
||||
(namely, ${n - \lfloor {k / 2} \rfloor - 1 \choose k}(-1)^k$).
|
||||
This resemblance can be made more apparent by listing the coefficients of the polynomials in a table.
|
||||
|
||||
```{python}
|
||||
#| echo: false
|
||||
#| classes: plain
|
||||
|
||||
rainbow_classes = [
|
||||
"",
|
||||
"red",
|
||||
"orange",
|
||||
"yellow",
|
||||
"green",
|
||||
"cyan",
|
||||
"aqua",
|
||||
"blue",
|
||||
"purple"
|
||||
"",
|
||||
"",
|
||||
]
|
||||
|
||||
rainbow_class = lambda x, color: f"<span style=\"color: {rainbow_classes[color]}\">{x}</span>"
|
||||
|
||||
Markdown(tabulate(
|
||||
[
|
||||
[
|
||||
n,
|
||||
*[" " for _ in range(1, 11 - n)], # offset for terms of the polynomial
|
||||
*[
|
||||
0 if k % 2 == 1
|
||||
else rainbow_class(
|
||||
comb(n - (k // 2) - 1, k // 2) * (-1)**(k // 2),
|
||||
n - (k // 2) - 1,
|
||||
)
|
||||
for k in range(n)
|
||||
]
|
||||
]
|
||||
for n in range(1, 11)
|
||||
],
|
||||
headers=[
|
||||
"n",
|
||||
*[f"$z^{nm}$" for nm in reversed(range(2, 10))],
|
||||
"$z$",
|
||||
"$1$",
|
||||
],
|
||||
numalign="right",
|
||||
stralign="right",
|
||||
))
|
||||
```
|
||||
|
||||
Though they alternate in sign, the rows of Pascal's triangle appear along diagonals,
|
||||
which I have marked in rainbow.
|
||||
Meanwhile, alternating versions of the naturals (1, 2, 3, 4...),
|
||||
the triangular numbers (1, 3, 6, 10...),
|
||||
the tetrahedral numbers (1, 4, 10, 20...), etc.
|
||||
are present along the columns, albeit spaced out by 0's.
|
||||
|
||||
The relationship of the Chebyshev polynomials to the triangle is easier to see if
|
||||
the coefficient extraction of $B(x; z / 2)$ is reversed.
|
||||
In other words, we extract *z* before extracting *x*.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
B(x; z / 2) &= {x \over 1 - zx + x^2}
|
||||
= {x \over 1 + x^2 - zx}
|
||||
= {x \over 1 + x^2}
|
||||
\cdot {1 \over {1 + x^2 \over 1 + x^2} - z{x \over 1 + x^2}}
|
||||
\\[10pt]
|
||||
[z^n]B(x; z / 2) &= {x \over 1 + x^2} [z^n] {1 \over 1 - z{x \over 1 + x^2}}
|
||||
= {x \over 1 + x^2} \left( {x \over 1 + x^2} \right)^n
|
||||
\\
|
||||
&= \left( {x \over 1 + x^2} \right)^{n+1}
|
||||
= x^{n+1} (1 + x^2)^{-n - 1}
|
||||
\\
|
||||
&= x^{n+1} \sum_{k=0}^\infty {-n - 1 \choose k}(x^2)^k
|
||||
\quad \text{Binomial theorem}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
While the use of the binomial theorem is more than enough to justify
|
||||
the appearance of Pascal's triangle (along with explaining the 0's),
|
||||
I'll simplify further to explicitly show the alternating signs.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
{(-n - 1)_k} &= (-n - 1)(-n - 2) \cdots (-n - k)
|
||||
\\
|
||||
&= (-1)^k (n + k)(n + k - 1) \cdots (n + 1)
|
||||
\\
|
||||
&= (-1)^k (n + k)_k
|
||||
\\
|
||||
\implies {-n - 1 \choose k}
|
||||
&= {n + k \choose k}(-1)^k
|
||||
\\[10pt]
|
||||
[z^n]B(x; z / 2)
|
||||
&= x^{n+1} \sum_{k=0}^\infty {n + k \choose k} (-1)^k x^{2k}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Squinting hard enough, the binomial coefficient is similar to the earlier
|
||||
which gave the third row of Pascal's triangle.
|
||||
If k is fixed, then this expression actually generates the antidiagonal entries
|
||||
of the coefficient table, which are the columns with uniform sign.
|
||||
The alternation instead occurs between antidiagonals (one is all positive,
|
||||
the next is 0's, the next is all negative, etc.).
|
||||
The initial $x^{n+1}$ lags these sequences so that they reproduce the triangle.
|
||||
|
||||
|
||||
### Imagined Transmutation
|
||||
|
||||
The generating function of the Chebyshev polynomials resembles other two term recurrences.
|
||||
For example, the Fibonacci numbers have generating function
|
||||
|
||||
$$
|
||||
\sum_{n = 0}^\infty \text{Fib}_n x^n = {1 \over 1 - x - x^2}
|
||||
$$
|
||||
|
||||
This resemblance can be made explicit with a simple algebraic manipulation.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
B(ix; -iz / 2)
|
||||
&= {1 \over 1 -\ (-i z)(ix) + (ix)^2}
|
||||
= {1 \over 1 -\ (-i^2) z x + (i^2)(x^2)}
|
||||
\\
|
||||
&= {1 \over 1 -\ z x -\ x^2}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
If $z = 1$, these two generating functions are equal.
|
||||
The same can be said for $z = 2$ with the generating function of the Pell numbers,
|
||||
and so on for higher recurrences (corresponding to metallic means) for higher integral *z*.
|
||||
|
||||
In terms of the Chebyshev polynomials, this series manipulation removes the alternation in
|
||||
the coefficients of $U_n$, restoring Pascal's triangle to its nonalternating form.
|
||||
Related to the previous point, it is possible to find the Fibonacci numbers (Pell numbers, etc.)
|
||||
in Pascal's triangle, which you can read more about
|
||||
[here](http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Pascal/Koshy2011.pdf).
|
||||
|
||||
|
||||
Manipulating the Series
|
||||
-----------------------
|
||||
|
||||
Look back to the table of $U_{n - 1}(z / 2)$ (@tbl-chebyshevu).
|
||||
When I brought up $U_{10 - 1}(z / 2)$ and decagons, I pointed out their relationship to pentagons
|
||||
as an explanation for why $U_{5 -\ 1}(z / 2)$ appears as a factor.
|
||||
Conveniently, $U_{2 -\ 1}(z / 2) = z$ is also a factor, and 2 is likewise a factor of 10.
|
||||
|
||||
This pattern is present throughout the table; $n = 6$ contains factors for
|
||||
$n = 2 \text{ and } 3$ and the prime numbers have no smaller factors.
|
||||
If this observation is legitimate, call the newest term $f_n(z)$
|
||||
and denote $p_n(z) = U_{n -\ 1}( z / 2 )$.
|
||||
|
||||
|
||||
### Factorization Attempts
|
||||
|
||||
The relationship between $p_n$ and the intermediate $f_d$, where *d* is a divisor of *n*,
|
||||
can be made explicit by a [Möbius inversion](https://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula).
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
p_n(z) &= \prod_{d|n} f_n(z)
|
||||
\\
|
||||
\log( p_n(z) )
|
||||
&= \log \left( \prod_{d|n} f_d(z) \right)
|
||||
= \sum_{d|n} \log( f_d(z) )
|
||||
\\
|
||||
\log( f_n(z) ) &= \sum_{d|n} { \mu \left({n \over d} \right)}
|
||||
\log( p_d(z) )
|
||||
\\
|
||||
f_n(z) &= \prod_{d|n} p_d(z)^{ \mu (n / d) }
|
||||
\\[10pt]
|
||||
f_6(z) = g_6(z)
|
||||
&= p_6(z)^{\mu(1)}
|
||||
p_3(z)^{\mu(2)}
|
||||
p_2(z)^{\mu(3)}
|
||||
\\
|
||||
&= {p_6(z) \over p_3(z) p_2(z)}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Unfortunately, it's difficult to apply this technique across our whole series.
|
||||
Möbius inversion over series typically uses more advanced generating functions such as
|
||||
[Dirichlet series](https://en.wikipedia.org/wiki/Dirichlet_series#Formal_Dirichlet_series)
|
||||
or [Lambert series](https://en.wikipedia.org/wiki/Lambert_series).
|
||||
However, naively reaching for these fails for two reasons:
|
||||
|
||||
- We built our series of polynomials on a recurrence relation, and these series
|
||||
are opaque to such manipulations.
|
||||
- To do a proper Möbius inversion, we need these kinds of series over the *logarithm*
|
||||
of each polynomial (*B* is a series over the polynomials themselves).
|
||||
|
||||
Ignoring these (and if you're in the mood for awful-looking math) you may note
|
||||
the Lambert equivalence[^2]:
|
||||
|
||||
[^2]:
|
||||
This equivalence applies to other polynomial series obeying the same factorization rule
|
||||
such as the [cyclotomic polynomials](https://en.wikipedia.org/wiki/Cyclotomic_polynomial).
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\log( p_n(z) )
|
||||
&= \sum_{d|n} \log( f_d(z) )
|
||||
\\
|
||||
\sum_{n = 1}^\infty \log( p_n ) x^n
|
||||
&= \sum_{n = 1}^\infty \sum_{d|n} \log( f_d ) x^n
|
||||
\\
|
||||
&= \sum_{k = 1}^\infty \sum_{m = 1}^\infty \log( f_m ) x^{m k}
|
||||
\\
|
||||
&= \sum_{m = 1}^\infty \log( f_m ) \sum_{k = 1}^\infty (x^m)^k
|
||||
\\
|
||||
&= \sum_{m = 1}^\infty \log( f_m ) {x^m \over 1 - x^m}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Either way, the number-theoretic properties of this sequence are difficult to ascertain
|
||||
without advanced techniques.
|
||||
If research has been done, it is not easily available in the OEIS.
|
||||
|
||||
|
||||
### Total Degrees
|
||||
|
||||
It can be also be observed that the new term is symmetric ($f(z) = f(-z)$), and is therefore
|
||||
either irreducible or the product of polynomial and its reflection (potentially negated).
|
||||
For example,
|
||||
|
||||
$$
|
||||
p_9(z) = \left\{
|
||||
\begin{matrix}
|
||||
(z - 1)(z + 1)
|
||||
& \cdot
|
||||
& (z^3 - 3z - 1)(z^3 - 3z + 1)
|
||||
\\
|
||||
\shortparallel && \shortparallel
|
||||
\\
|
||||
f_3(z)
|
||||
& \cdot
|
||||
& f_9(z)
|
||||
\\
|
||||
\shortparallel && \shortparallel
|
||||
\\
|
||||
g_3(z) \cdot g_3(-z)
|
||||
& \cdot
|
||||
& g_9(z) \cdot -g_9(-z)
|
||||
\end{matrix}
|
||||
\right.
|
||||
$$
|
||||
|
||||
These factor polynomials $g_n$ are the minimal polynomials of $2\cos( \pi / n )$.
|
||||
|
||||
Multiplying these minimal polynomials by their reflection can be observed in the Chebyshev polynomials
|
||||
for $n = 3, 5, 7, 9$, strongly implying that it occurs on the odd terms.
|
||||
Assuming this is true, we have
|
||||
|
||||
$$
|
||||
f_n(z) = \begin{cases}
|
||||
g_n(z) & \text{$n$ is even}
|
||||
\\
|
||||
g_n(z)g_n(-z)
|
||||
& \text{$n$ is odd and ${\deg(f_n) \over 2}$ is even}
|
||||
\\
|
||||
-g_n(z)g_n(-z)
|
||||
& \text{$n$ is odd and ${\deg(f_n) \over 2}$ is odd}
|
||||
\end{cases}
|
||||
$$
|
||||
|
||||
Without resorting to any advanced techniques, the degrees of $f_n$ are
|
||||
not too difficult to work out.
|
||||
The degree of $p_n(z)$ is $n -\ 1$, which is also the degree of $f_n(z)$ if *n* is prime.
|
||||
If *n* is composite, then the degree of $f_n(z)$ is $n -\ 1$ minus the degrees
|
||||
of the divisors of $n -\ 1$.
|
||||
This leaves behind how many numbers less than *n* are coprime to *n*.
|
||||
Therefore $\deg(f_n) = \phi(n)$, the
|
||||
[Euler totient function](https://en.wikipedia.org/wiki/Euler_totient_function) of the index.
|
||||
|
||||
The totient function can be used to examine the parity of *n*.
|
||||
If *n* is odd, it is coprime to 2 and all even numbers.
|
||||
The introduced factor of 2 to 2*n* removes the evens from the totient, but this is compensated by
|
||||
the addition of the odd multiples of old numbers coprime to *n* and new primes.
|
||||
This means that $\phi(2n) = \phi(n)$ for odd *n* (other than 1).
|
||||
|
||||
The same argument can be used for even *n*: there are as many odd numbers from 0 to *n* as there are
|
||||
from *n* to 2*n*, and there are an equal number of numbers coprime to 2*n* in either interval.
|
||||
Therefore, $\phi(2n) = 2\phi(n)$ for even *n*.
|
||||
|
||||
This collapses all cases of the conditional factorization of $f_n$ into one,
|
||||
and the degrees of $g_n$ are
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
\deg( g_n(z) )
|
||||
&= \begin{cases}
|
||||
\deg( f_n(z) )
|
||||
= \phi(n)
|
||||
& n \text{ is even} & \implies \phi(n) = \phi(2n) / 2
|
||||
\\
|
||||
\deg( f_n(z) ) / 2
|
||||
= \phi(n) / 2
|
||||
& n \text{ is odd} & \implies \phi(n) / 2 = \phi(2n) / 2
|
||||
\end{cases}
|
||||
\\
|
||||
&= \varphi(2n) / 2
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Though they were present in the earlier Chebyshev table,
|
||||
the $g_n$ themselves are presented again, along with the expression for their degree
|
||||
|
||||
```{python}
|
||||
#| echo: false
|
||||
#| classes: plain
|
||||
|
||||
def poly_to_rising_power_list(poly, var):
|
||||
"""
|
||||
Convert a polynomial to a list in rising powers.
|
||||
E.g., x^2 + x - 1 will be converted to [-1, 1, 1].
|
||||
"""
|
||||
ret = []
|
||||
for term in reversed(poly.as_ordered_terms()):
|
||||
deg = sympy.degree(term, var)
|
||||
if deg > len(ret):
|
||||
ret.extend(0 for _ in range(int(deg) - len(ret)))
|
||||
|
||||
if deg == 0:
|
||||
ret.append(term)
|
||||
else:
|
||||
ret.append(term.coeff(z**deg))
|
||||
|
||||
return ret
|
||||
|
||||
|
||||
Markdown(tabulate(
|
||||
[
|
||||
[
|
||||
n,
|
||||
sympy.totient(2*n) / 2,
|
||||
"$" + sympy.latex(g) + "$",
|
||||
str(poly_to_rising_power_list(g, z)),
|
||||
]
|
||||
for n in range(2, 11)
|
||||
for g in [
|
||||
factor # the first factor polynomial with matching degree and negative second term
|
||||
for factor in sympy.chebyshevu_poly(n - 1, z / 2).factor(z).as_ordered_factors() # type: ignore
|
||||
for second_term in [
|
||||
0 if len(factor.as_ordered_terms()) == 1 # if there's only one term, pass it through
|
||||
else factor.as_ordered_terms()[1].as_ordered_factors()[0] # type: ignore
|
||||
] if n == 2
|
||||
or (
|
||||
sympy.degree(factor) == sympy.totient(2*n) / 2
|
||||
and isinstance(second_term, sympy.Integer)
|
||||
and second_term < 0
|
||||
)
|
||||
]
|
||||
] + [[
|
||||
"-",
|
||||
"[OEIS A055034](http://oeis.org/A055034)",
|
||||
"-",
|
||||
"[OEIS A187360](http://oeis.org/A187360)",
|
||||
]],
|
||||
headers=[
|
||||
"n",
|
||||
"$\\varphi(2n)/2$",
|
||||
"$g_n(z)$",
|
||||
"Coefficient list, rising powers",
|
||||
],
|
||||
numalign="right",
|
||||
stralign="left",
|
||||
))
|
||||
```
|
||||
|
||||
|
||||
Closing
|
||||
-------
|
||||
|
||||
My initial jumping off point for writing this article was completely different.
|
||||
However, in the process of writing, its share of the article shrank and shrank until its
|
||||
introduction was only vaguely related to what preceded it.
|
||||
But alas, the introduction via geometric constructions flows better coming off my
|
||||
[post about the Platonic solids](/posts/math/misc/platonic-volume).
|
||||
Also, it reads better if I rely less on "if you search for this sequence of numbers"
|
||||
and more on how to interpret the definition.
|
||||
|
||||
Consider reading [the follow-up](../2) to this post if you're interested in another way
|
||||
one can obtain the Chebyshev polynomials.
|
||||
I have since rederived the Chebyshev polynomials without the complex exponential,
|
||||
which you can read about in [this post](/posts/math/stereo/2).
|
||||
|
||||
Diagrams created with GeoGebra.
|
||||
793
posts/math/chebyshev/2/index.qmd
Normal file
793
posts/math/chebyshev/2/index.qmd
Normal file
@ -0,0 +1,793 @@
|
||||
---
|
||||
title: "Generating Polynomials, Part 2: Ghostly Chains"
|
||||
description: |
|
||||
What do polygons without distance still know about planar geometry?
|
||||
format:
|
||||
html:
|
||||
html-math-method: katex
|
||||
date: "2021-08-19"
|
||||
date-modified: "2025-06-24"
|
||||
categories:
|
||||
- algebra
|
||||
- linear algebra
|
||||
- generating functions
|
||||
- graph theory
|
||||
- python
|
||||
---
|
||||
|
||||
<style>
|
||||
.figure-img {
|
||||
max-width: 512px;
|
||||
object-fit: contain;
|
||||
height: 100%;
|
||||
}
|
||||
|
||||
.figure-img.wide {
|
||||
max-width: 768px;
|
||||
}
|
||||
</style>
|
||||
|
||||
In the [previous post](../1), I tied the geometry regular polygons to a sequence of polynomials
|
||||
though some clever algebraic manipulation.
|
||||
But let's deign to ask a very basic question: what is a polygon?
|
||||
|
||||
|
||||
Loops without Distance
|
||||
----------------------
|
||||
|
||||
Fundamentally, a polygon is just a collection of vertices and edges.
|
||||
For polygons in a Euclidean setting, the position of points matters,
|
||||
as well as the lines connecting them -- a rectangle is different from a trapezoid or a kite.
|
||||
But at its simplest, this is just a tabulation of points and adjacencies.
|
||||
|
||||
|
||||
|
||||
Only examining these figures by their connectedness is precisely the kind of thing
|
||||
*graph theory* deals with.
|
||||
"Graph" is a potentially confusing term, since it has nothing to do with "graphs of functions",
|
||||
but the name is supposed to evoke the fact that they are "drawings".
|
||||
For the graphs we're interested in, there's some additional terminology:
|
||||
|
||||
- Vertices themselves are sometimes instead called *nodes*
|
||||
- Edge in the graph have no direction in how they connect nodes, so the graph is called *undirected*.
|
||||
- If the nodes in a graph can be arranged so that no edges appear to intersect,
|
||||
the graph is *planar*.
|
||||
- For example, lower-right figure in the above diagram appears to have intersecting edges,
|
||||
but the nodes can be rearranged to look like the other graphs, so it is planar.
|
||||
|
||||
It's easiest to study families of graphs, rather than isolated examples.
|
||||
If the graph is a simple loop of nodes, it is called a
|
||||
[*cycle graph*](https://en.wikipedia.org/wiki/Cycle_graph).
|
||||
They are denoted by $C_n$, where *n* is the number of nodes.
|
||||
In a cycle graph, since all nodes are identical to each other (they all connect to two edges)
|
||||
and all edges are identical to each other (they connect identical vertices),
|
||||
the best geometric interpretation is a shape which is
|
||||
|
||||
- Regular, so that each edge and each angle (vertex) are of equal measure
|
||||
- Convex, so that no edge meets another without creating a vertex (or node)
|
||||
|
||||
In other words, $C_3$ is analogous to an equilateral triangle, $C_4$ is analogous to a square, and so on.
|
||||
|
||||
|
||||
### Encoding Graphs
|
||||
|
||||
There are two primary ways to store information about a graph.
|
||||
The first is by labelling each node (for example, with integers), then recording the edges as
|
||||
a list of pairs of connected nodes.
|
||||
In the case of an undirected graph, these are unordered pairs.
|
||||
While such a list is convenient, it doesn't convey a lot of information about the graph
|
||||
besides the number of edges.
|
||||
|
||||
Alternatively, these pairs can also be interpreted as addresses in a square matrix,
|
||||
called an *adjacency matrix*.
|
||||
Each column and row correspond to a specific node, and an entry is 1
|
||||
when the nodes of a row and column of are joined by an edge (and 0 otherwise).
|
||||
For undirected graphs, these matrices are symmetric, since it is possible
|
||||
to traverse an edge in either direction.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
C_3 := \begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 0)
|
||||
]\end{matrix} & \cong
|
||||
\begin{pmatrix}
|
||||
0 & 1 & 1 \\
|
||||
1 & 0 & 1 \\
|
||||
1 & 1 & 0
|
||||
\end{pmatrix}
|
||||
\\ \\
|
||||
C_4 := \begin{matrix} [
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 3), \\
|
||||
(3, 0)
|
||||
]\end{matrix} & \cong
|
||||
\begin{pmatrix}
|
||||
0 & 1 & 0 & 1 \\
|
||||
1 & 0 & 1 & 0 \\
|
||||
0 & 1 & 0 & 1 \\
|
||||
1 & 0 & 1 & 0
|
||||
\end{pmatrix}
|
||||
\\ \\
|
||||
C_5 := \begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 3), \\
|
||||
(3, 4), \\
|
||||
(4, 0)
|
||||
]\end{matrix} &\cong
|
||||
\begin{pmatrix}
|
||||
0 & 1 & 0 & 0 & 1 \\
|
||||
1 & 0 & 1 & 0 & 0 \\
|
||||
0 & 1 & 0 & 1 & 0 \\
|
||||
0 & 0 & 1 & 0 & 1 \\
|
||||
1 & 0 & 0 & 1 & 0
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Swapping the labels on two nodes is will exchange two rows and two columns
|
||||
of the adjacency matrix.
|
||||
Just one of these swaps would flip the sign of its determinant, but since they occur in pairs,
|
||||
the determinant is invariant of the labelling (equally, a graph invariant).
|
||||
|
||||
|
||||
Prismatic Recurrence
|
||||
--------------------
|
||||
|
||||
The determinant of a matrix is also the product of its eigenvalues, which are another matrix invariant.
|
||||
The set of eigenvalues is also called its *spectrum*, and the study of the spectra of graphs is called
|
||||
[*spectral graph theory*](https://en.wikipedia.org/wiki/Spectral_graph_theory)[^1],
|
||||
|
||||
[^1]: It is also among the most mystifying names in math to read without any context
|
||||
|
||||
Eigenvalues are the roots of the characteristic polynomial of a matrix.
|
||||
The matrix $C_5$ is sufficiently large enough to generalize to $C_n$, and its characteristic polynomial by
|
||||
[Laplace expansion](https://en.wikipedia.org/wiki/Laplace_expansion) is:
|
||||
|
||||
$$
|
||||
\begin{gather*}
|
||||
Ax = \lambda x \implies (\lambda I - A)x = 0
|
||||
\\ \\
|
||||
c_5(\lambda) = |\lambda I - C_5|
|
||||
= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 & -1 \\
|
||||
-1 & \lambda & -1 & 0 & 0 \\
|
||||
0 & -1 & \lambda & -1 & 0 \\
|
||||
0 & 0 & -1 & \lambda & -1 \\
|
||||
-1 & 0 & 0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
= \lambda m_{1,1}
|
||||
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 2 \ }}^\text{sign} m_{1, 2}
|
||||
+ \overbrace{(-1)}^\text{entry}\overbrace{(-1)^{1 + 5 \ }}^\text{sign} m_{1, 5}
|
||||
\end{gather*}
|
||||
$$
|
||||
|
||||
Note that every occurrence of "5" generalizes to higher *n*.
|
||||
The first [minor](https://en.wikipedia.org/wiki/Matrix_minor)
|
||||
is easily expressed in terms of *another* matrix's characteristic polynomial.
|
||||
|
||||
$$
|
||||
m_{1, 1}[C_5]
|
||||
= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 \\
|
||||
-1 & \lambda & -1 & 0 \\
|
||||
0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
= |\lambda I - P_4| = p_{5 - 1}(\lambda)
|
||||
$$
|
||||
|
||||
We will come to the meaning of the $P_n$ in a moment.
|
||||
The other minors require extra expansions, but ones that (thankfully) quickly terminate.
|
||||
|
||||
$$
|
||||
\begin{matrix}
|
||||
m_{1, 2}[C_5]
|
||||
&= \left |
|
||||
\begin{matrix}
|
||||
-1 & -1 & 0 & 0 \\
|
||||
0 & \lambda & -1 & 0 \\
|
||||
0 & -1 & \lambda & -1 \\
|
||||
-1 & 0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
&=& (-1)
|
||||
\left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 \\
|
||||
-1 & \lambda & -1 \\
|
||||
0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
&+& (-1)(-1)^{1 + 4}
|
||||
\left |
|
||||
\begin{matrix}
|
||||
-1 & 0 & 0 \\
|
||||
\lambda & -1 & 0 \\
|
||||
-1 & \lambda & -1
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&&=& (-1)|\lambda I - P_3|
|
||||
&+& (-1)\overbrace{(-1)^{5}(-1)^{5 - 2}}^{\text{even power, even when $\scriptsize n \neq 5$}}
|
||||
\\
|
||||
&&=& (-1)p_{5 - 2}(\lambda) &+& (-1)
|
||||
\\
|
||||
&&=& -(p_{5 - 2}(\lambda) &+& 1)
|
||||
\end{matrix}
|
||||
$$
|
||||
|
||||
The "1 + 4" exponent when evaluating this minor comes from the address of the lower-left -1, (i.e., (1, 4)).
|
||||
This entry exists for all $C_n$.
|
||||
The determinant of the rightmost matrix is just the product of the -1's on the diagonal, so it will always
|
||||
have a power of the same parity as *n*, which cancels out with the sign of the minor.
|
||||
Meanwhile, another $P$-type matrix appears in the other term, this time of two lower orders.
|
||||
|
||||
$$
|
||||
\begin{matrix}
|
||||
\\ \\
|
||||
m_{1, 5}[C_5] &=
|
||||
\left |
|
||||
\begin{matrix}
|
||||
-1 & \lambda & -1 & 0 \\
|
||||
0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & -1 & \lambda \\
|
||||
-1 & 0 & 0 & -1
|
||||
\end{matrix}
|
||||
\right |
|
||||
&=& (-1)
|
||||
\left |
|
||||
\begin{matrix}
|
||||
-1 & \lambda & -1 \\
|
||||
0 & -1 & \lambda \\
|
||||
0 & 0 & -1
|
||||
\end{matrix}
|
||||
\right |
|
||||
&+& (-1)(-1)^{5 - 2}
|
||||
\left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 \\
|
||||
-1 & \lambda & -1 \\
|
||||
0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&&=& (-1)(-1)^{5-2} &+& (-1)(-1)^{5 - 2}|\lambda I - P_3|
|
||||
\\
|
||||
&&=& (-1)^{5-1}((-1)(-1) &+& (-1)(-1)p_{5 - 2}(\lambda))
|
||||
\\
|
||||
&&=& (-1)^{5-1}(1 &+& p_{5 - 2}(\lambda))
|
||||
\end{matrix}
|
||||
$$
|
||||
|
||||
A third $P$-type matrix appears, just like the other minor.
|
||||
Unfortunately, this minor *does* depend on the parity of *n*.
|
||||
|
||||
All together, this produces a characteristic polynomial in terms of the polynomials $p_n(\lambda)$:
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
&& c_5(\lambda) &= \lambda p_{5 - 1}
|
||||
+ (-1)(p_{5 - 2} + 1)
|
||||
+ (-1)\overbrace{(-1)^{5 - 1} (-1)^{5 - 1}}^{\text{even, even when $\scriptsize n \neq 5$}}(p_{5 - 2} + 1)
|
||||
\\
|
||||
&&&= \lambda p_{5 - 1}
|
||||
- (p_{5 - 2} + 1)
|
||||
- (p_{5 - 2} + 1)
|
||||
\\
|
||||
&&&= \lambda p_{5 - 1}
|
||||
- 2(p_{5 - 2} + 1)
|
||||
\\
|
||||
&& \implies c_n(\lambda) &= \lambda p_{n - 1}(\lambda)
|
||||
- 2(p_{n - 2}(\lambda) + 1)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Fortunately, the minor whose determinant depended on the parity of *n* cancels with $(-1)^{1 + 5}$,
|
||||
and the resulting expression seems to generically apply across all *n*.
|
||||
Further, this resembles a recurrence relation, which is great for building a rule.
|
||||
But it is meaningless without knowing $p_n(\lambda)$.
|
||||
|
||||
|
||||
Powerful Chains
|
||||
---------------
|
||||
|
||||
The various $P_n$ are in fact the adjacency matrices of a path on *n* nodes.
|
||||
|
||||
{.wide}
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
P_2 &:=
|
||||
\begin{matrix}[
|
||||
(0, 1)
|
||||
]\end{matrix}
|
||||
\cong \begin{pmatrix}
|
||||
0 & 1 \\
|
||||
1 & 0
|
||||
\end{pmatrix}
|
||||
\\
|
||||
P_3 &:=
|
||||
\begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2)
|
||||
]\end{matrix}
|
||||
\cong \begin{pmatrix}
|
||||
0 & 1 & 0 \\
|
||||
1 & 0 & 1 \\
|
||||
0 & 1 & 0
|
||||
\end{pmatrix}
|
||||
\\
|
||||
P_4 &:=
|
||||
\begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 3)
|
||||
]\end{matrix}
|
||||
\cong \begin{pmatrix}
|
||||
0 & 1 & 0 & 0 \\
|
||||
1 & 0 & 1 & 0 \\
|
||||
0 & 1 & 0 & 1 \\
|
||||
0 & 0 & 1 & 0
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
These matrices are similar to the ones for cycle graphs, but lack the entries in bottom-left
|
||||
and upper-right corners.
|
||||
Consequently, the characteristic polynomials of $P_n$ are much easier to solve for.
|
||||
|
||||
$$
|
||||
\begin{gather*}
|
||||
p_4(\lambda) = |\lambda I - P_4|
|
||||
= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 \\
|
||||
-1 & \lambda & -1 & 0 \\
|
||||
0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\ \\
|
||||
= \lambda \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 \\
|
||||
-1 & \lambda & -1 \\
|
||||
0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right | + (-1)(-1)^{1+2} \left |
|
||||
\begin{matrix}
|
||||
-1 & -1 & 0 \\
|
||||
0 & \lambda & -1 \\
|
||||
0 & -1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\ \\
|
||||
= \lambda |\lambda I - P_3| + \left (
|
||||
(-1) \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 \\
|
||||
-1 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
+ (-1)(-1) \left |
|
||||
\begin{matrix}
|
||||
0 & -1 \\
|
||||
0 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\right)
|
||||
\\ \\
|
||||
= \lambda |\lambda I - P_3| - |\lambda I - P_2|
|
||||
\\
|
||||
= \lambda p_{4 - 1}(\lambda) - p_{4 - 2}(\lambda)
|
||||
\\
|
||||
\implies p_{n}(\lambda) = \lambda p_{n - 1}(\lambda) - p_{n - 2}(\lambda)
|
||||
\end{gather*}
|
||||
$$
|
||||
|
||||
While the earlier equation for $c_n$ in terms of $p_n$ reminded of a recurrence relation,
|
||||
*this* actually is one (and it should look familiar).
|
||||
|
||||
Since the recurrence has order 2, it requires two initial terms: $p_0$ and $p_1$.
|
||||
The graph corresponding to $p_1$ is a single node, not connected to anything.
|
||||
Therefore, its adjacency matrix is a 1x1 matrix with 0 as its only entry,
|
||||
and its characteristic polynomial is $\lambda$.
|
||||
By the recurrence, $p_2 = \lambda p_1 -\ p_0 = \lambda^2 -\ p_0$.
|
||||
Equating terms with the characteristic polynomial of $P_2$, it is obvious that
|
||||
|
||||
$$
|
||||
|\lambda I - P_2|
|
||||
= \begin{pmatrix}
|
||||
\lambda & -1 \\
|
||||
-1 & \lambda
|
||||
\end{pmatrix}
|
||||
= \lambda^2 - 1 = \lambda p_1 - p_0 \\
|
||||
\implies p_0 = 1
|
||||
$$
|
||||
|
||||
which makes sense, since $p_0$ should have degree zero.
|
||||
Therefore, the sequence of polynomials $p_n(\lambda)$ is:
|
||||
|
||||
$$
|
||||
\begin{gather*}
|
||||
p_0(\lambda) &=&& && 1
|
||||
\\
|
||||
p_1(\lambda) &=&& && \lambda
|
||||
\\
|
||||
p_2(\lambda) &=&& \lambda \lambda - 1
|
||||
&=& \lambda^2 - 1
|
||||
\\
|
||||
p_3(\lambda) &=&& \lambda (\lambda^2 - 1) - \lambda
|
||||
&=& \lambda(\lambda^2 - 2)
|
||||
\\
|
||||
p_4(\lambda) &=&& \lambda (\lambda(\lambda^2 - 2)) - (\lambda^2 - 1)
|
||||
&=& \lambda^4 - 3\lambda^2 + 1
|
||||
\\
|
||||
\vdots & && \vdots && \vdots
|
||||
\end{gather*}
|
||||
$$
|
||||
|
||||
But wait, we've seen these before (if you read the previous post, that is).
|
||||
These are just the Chebyshev polynomials of the second kind, evaluated at $\lambda / 2$.
|
||||
Indeed, their recurrence relations are identical, so the characteristic polynomial of $P_n$ is $U_n(\lambda / 2)$.
|
||||
Effectively, this connects an *n*-path to a regular *n+1*-gon.
|
||||
|
||||
|
||||
### Back to Cycles
|
||||
|
||||
Since the generating function of $U_n$ is known, the generating function for the $c_n$
|
||||
(which prompted this) is also easily determined.
|
||||
For ease of use, let
|
||||
|
||||
$$
|
||||
P(x; \lambda) = {B(x; \lambda / 2) \over x} = {1 \over 1 - \lambda x +\ x^2}
|
||||
$$
|
||||
|
||||
Discarding the initial $c_0$ and $c_1$ by setting them to zero[^2], the generating function is
|
||||
|
||||
[^2]: It's a good idea to ask why we can do this.
|
||||
Try examining $c_2$ and $c_3$.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
c_{n+2}(\lambda) &= \lambda p_{n+1}(\lambda) - 2(p_n(\lambda) + 1)
|
||||
\\[14pt]
|
||||
{C(x; \lambda) - c_0(\lambda) - x c_1(\lambda) \over x^2}
|
||||
&= \lambda \left( {P(x; \lambda) - 1 \over x} \right)
|
||||
- 2\left( P(x; \lambda) + {1 \over 1 - x} \right)
|
||||
\\
|
||||
C &= x \lambda (P - 1) -\
|
||||
2x^2\left( P + {1 \over 1 - x} \right)
|
||||
\\
|
||||
C{(1 - x) \over P} &= x \lambda \left(1 - {1 \over P} \right)(1 - x) -\
|
||||
2x^2\left( (1 - x) + {1 \over P} \right)
|
||||
\\
|
||||
&= x^4 \lambda - 2 x^4 - x^3 \lambda^2 + x^3 \lambda
|
||||
+ 2 x^3 + x^2 \lambda^2 - 4 x^2
|
||||
\\
|
||||
&= x^2 (\lambda - 2) (x^2 - \lambda x - x + \lambda + 2)
|
||||
\\[14pt]
|
||||
C(x; \lambda) &= x^2 (\lambda - 2)
|
||||
{(x^2 - (\lambda + 1) x + \lambda + 2)
|
||||
\over (1 - x)(1 - \lambda x + x^2)}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
While the numerator is considerably more complicated than the one for P,
|
||||
the factor $\lambda - 2$ drops out of the entire series.
|
||||
This pleasantly informs that 2 is an eigenvalue of all $C_n$.
|
||||
|
||||
|
||||
Off the Beaten Path
|
||||
-------------------
|
||||
|
||||
When we use Laplace expansion on the adjacency matrices, we were very fortunate that the minors
|
||||
*also* looked like adjacency matrices undergoing expansion.
|
||||
This let us terminate early and recurse.
|
||||
From the perspective of the graph, Laplace expansion almost looks like removing a node,
|
||||
but requires special treatment for the nodes connected to the one being removed.
|
||||
For example, in cycle graphs, the first stage of expansion had three minors:
|
||||
|
||||
- The node itself, on the main diagonal
|
||||
- Being on the main diagonal, this immediately produced another adjacency matrix.
|
||||
- Either neighbor connected to it, which are on opposite sides of
|
||||
a path after the node is removed
|
||||
- Both of these nodes required second expansion to get the *λ*s back on the main diagonal.
|
||||
|
||||
For "good enough" graphs that are nearly paths (including paths themselves),
|
||||
this gives a second-order recurrence relation.
|
||||
|
||||
|
||||
### Trees
|
||||
|
||||
Another simple family of graphs are [*trees*](https://en.wikipedia.org/wiki/Tree_%28graph_theory%29).
|
||||
In some sense, they are the opposite of cycle graphs, since by definition they contain no cycles.
|
||||
|
||||
Paths are degenerate trees, but we can make them slightly more interesting by instead adding
|
||||
exactly one node and edge to (the middle of) a path.
|
||||
|
||||
{.wide}
|
||||
|
||||
In this notation, the subscripts denote the consituent paths if the "added" node and the
|
||||
one it is connected to are both removed.
|
||||
It's easy to see that $T_{a,b} \cong T_{a,b}$, since this just swaps the arms.
|
||||
Also, $T_{a, 0} \cong T_{0, a} \cong P_{a + 2}$.
|
||||
|
||||
Let's try dissecting some of the larger trees.
|
||||
The adjacency matrices for $T_{1,3}$ and $T_{2,2}$ are:
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
T_{1,3} := \begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 3), \\
|
||||
(3, 4), \\
|
||||
(1, 5)
|
||||
]\end{matrix} &\cong
|
||||
\begin{pmatrix}
|
||||
0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
1 & 0 & 1 & 0 & 0 & 1 \\
|
||||
0 & 1 & 0 & 1 & 0 & 0 \\
|
||||
0 & 0 & 1 & 0 & 1 & 0 \\
|
||||
0 & 0 & 0 & 1 & 0 & 0 \\
|
||||
0 & 1 & 0 & 0 & 0 & 0
|
||||
\end{pmatrix}
|
||||
\\ \\
|
||||
T_{2,2} := \begin{matrix}[
|
||||
(0, 1), \\
|
||||
(1, 2), \\
|
||||
(2, 3), \\
|
||||
(3, 4), \\
|
||||
(2, 5)
|
||||
]\end{matrix} &\cong
|
||||
\begin{pmatrix}
|
||||
0 & 1 & 0 & 0 & 0 & 0 \\
|
||||
1 & 0 & 1 & 0 & 0 & 0 \\
|
||||
0 & 1 & 0 & 1 & 0 & 1 \\
|
||||
0 & 0 & 1 & 0 & 1 & 0 \\
|
||||
0 & 0 & 0 & 1 & 0 & 0 \\
|
||||
0 & 0 & 1 & 0 & 0 & 0
|
||||
\end{pmatrix}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Starting with $T_{1,3}$, its characteristic polynomial is:
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
|I \lambda - T_{1,3}|
|
||||
&= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 & 0 & 0 \\
|
||||
-1 & \lambda & -1 & 0 & 0 & -1 \\
|
||||
0 & -1 & \lambda & -1 & 0 & 0 \\
|
||||
0 & 0 & -1 & \lambda & -1 & 0 \\
|
||||
0 & 0 & 0 & -1 & \lambda & 0 \\
|
||||
0 & -1 & 0 & 0 & 0 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&= \lambda (-1)^{6 + 6} m_{6,6} + (-1) (-1)^{2 + 6} m_{2,6}
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
It's easy to see that $m_{6,6}$ is just $p_5(\lambda)$, since the rest of the graph
|
||||
other than the additional node is a 5-path.
|
||||
But the other minor is trickier.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
m_{2,6}
|
||||
&= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 & 0 \\
|
||||
0 & -1 & \lambda & -1 & 0 \\
|
||||
0 & 0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & 0 & -1 & \lambda \\
|
||||
0 & -1 & 0 & 0 & 0
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&= (-1) (-1)^{5 + 2}
|
||||
\left |
|
||||
\begin{matrix}
|
||||
\lambda & 0 & 0 & 0 \\
|
||||
0 & \lambda & -1 & 0 \\
|
||||
0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & -1 & \lambda \\
|
||||
\end{matrix}
|
||||
\right |
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Through one extra expansion, the determinant of this final matrix can be written as
|
||||
a product of $\lambda$ and $p_3(\lambda)$.
|
||||
|
||||
Before making any conjectures, let's do the same thing to $T_{2,2}$.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
|I \lambda - T_{2,2}|
|
||||
&= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 & 0 & 0 \\
|
||||
-1 & \lambda & -1 & 0 & 0 & 0 \\
|
||||
0 & -1 & \lambda & -1 & 0 & -1 \\
|
||||
0 & 0 & -1 & \lambda & -1 & 0 \\
|
||||
0 & 0 & 0 & -1 & \lambda & 0 \\
|
||||
0 & 0 & -1 & 0 & 0 & \lambda
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&= (-1)^{6 + 6} \lambda m_{6,6} + (-1) (-1)^{3 + 6} m_{3,6}
|
||||
\\ \\
|
||||
m_{3,6}
|
||||
&= \left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 & 0 \\
|
||||
-1 & \lambda & -1 & 0 & 0 \\
|
||||
0 & 0 & -1 & \lambda & -1 \\
|
||||
0 & 0 & 0 & -1 & \lambda \\
|
||||
0 & 0 & -1 & 0 & 0
|
||||
\end{matrix}
|
||||
\right |
|
||||
\\
|
||||
&= (-1) (-1)^{5 + 3}
|
||||
\left |
|
||||
\begin{matrix}
|
||||
\lambda & -1 & 0 & 0 \\
|
||||
-1 & \lambda & 0 & 0 \\
|
||||
0 & 0 & \lambda & -1 \\
|
||||
0 & 0 & -1 & \lambda \\
|
||||
\end{matrix}
|
||||
\right |
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Here we get something similar: a combination of $p_5(\lambda)$ and an extra term.
|
||||
In this case, the final determinant can be written as $p_2(\lambda)^2$.
|
||||
|
||||
Now it can be observed that the extra terms are the polynomials corresponding
|
||||
to $P_a$ and $P_b$ (recall that $p_1(\lambda) = \lambda$, after all).
|
||||
In both cases, the second expansion was necessary to get rid of the symmetric -1
|
||||
entries added to the matrix.
|
||||
The sign of this extra term is always negative, since the -1 entries cancel
|
||||
and one of the signs of the minors along the two expansions must be negative.
|
||||
|
||||
Therefore, the expression for these characteristic polynomials should be:
|
||||
|
||||
$$
|
||||
t_{a,b}(\lambda) = \lambda p_{a + b + 1}(\lambda) - p_a(z) p_b(\lambda)
|
||||
$$
|
||||
|
||||
Note that if *b* is 0, this agrees with the recurrence for $p_n(\lambda)$.
|
||||
|
||||
|
||||
### Examining Small Trees
|
||||
|
||||
Due to the subscript of the first term of the RHS, this recurrence is harder to turn into
|
||||
a generating function.
|
||||
Instead, let's look at a few smaller trees to see what kind of polynomials they build.
|
||||
We'll also change the variable of the polynomial to *z* for simplicity.
|
||||
|
||||
The first tree of note is $T_{1,1}$.
|
||||
This has characteristic polynomial
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
t_{1,1}(z) &= z p_{3}(z) - p_1(z) p_1(z)
|
||||
\\
|
||||
&= z (z^3 - z^2) - z^2
|
||||
\\
|
||||
&= z^2 (z^2 - 3)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Next, we have both $T_{2,1}$ and $T_{1,2}$.
|
||||
By symmetry, these are the same graph, so we have characteristic polynomial
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
t_{1,2}(z) &= z p_{4}(z) - p_1(z) p_2(z)
|
||||
\\
|
||||
&= z (z^4 - 3z^2 + 2) - z \cdot (z^2 - 1)
|
||||
\\
|
||||
&= z (z^4 - 4z^2 + 2)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Finally, let's look at $T_{1,3}$ and $T_{2,2}$, the trees we used to derive the rule.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
t_{1,3}(z) &= z p_{5}(z) - p_1(z) p_3(z)
|
||||
\\
|
||||
&= z (z^5 - 4z^3 + 3z) - z \cdot (z^3 - 2z)
|
||||
\\
|
||||
&= z^2 (z^4 - 5z^2 + 5)
|
||||
\\[10pt]
|
||||
t_{2,2}(z) &= z p_{5}(z) - p_2(z) p_2(z)
|
||||
\\
|
||||
&= z (z^5 - 4z^3 + 3z) - ( z^2 - 1 )^2
|
||||
\\
|
||||
&= (z^2 - 1)(z^4 - 4z^2 + 1)
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Many of these expressions factor surprisingly nicely.
|
||||
Further, some of these might seem familiar.
|
||||
From the last post, we saw that $z^4 - 5z^2 + 5$ is a factor of $p_9(z)$, from which we know
|
||||
it is the minimal polynomial of $2 \cos(\pi / 10)$.
|
||||
|
||||
This is also true for:
|
||||
|
||||
- In $t_{1,2}$, the factor $z^4 - 4z^2 + 2$, $p_7(z)$, and $2 \cos(\pi / 8)$, respectively
|
||||
- In $t_{2,2}$, the factor $z^4 - 4z^2 + 1$, $p_11(z)$, and $2 \cos(\pi / 12)$, respectively
|
||||
|
||||
We established that the subscripts of the tree (*a* and *b*) indicate constituent *n*-paths,
|
||||
which we know to correspond to *n+1*-gons.
|
||||
But these trees also seem to "know" about higher polygons.
|
||||
|
||||
|
||||
### Some Extra Trees
|
||||
|
||||
$T_{2,3}$ is the first tree not to partition two equal paths or a path and a single node.
|
||||
In this regard, the next such tree is $T_{2,4}$.
|
||||
|
||||
These graphs turn out to have characteristic polynomials whose factors we haven't seen before.
|
||||
|
||||
$$
|
||||
\begin{align*}
|
||||
t_{2,3}(z)
|
||||
&= z (z^{6} - 6 z^{4} + 9 z^{2} - 3)
|
||||
\\
|
||||
t_{2,4}(z)
|
||||
&= z^{8} - 7 z^{6} + 14 z^{4} - 8 z^{2} + 1
|
||||
\end{align*}
|
||||
$$
|
||||
|
||||
Searching the OEIS for the coefficients of $t_{2,4}$ returns sequence
|
||||
[A228786](http://oeis.org/A228786), which informs that it is the minimal polynomial
|
||||
of $2\sin( \pi/15 )$.
|
||||
This sequence also informs that the unknown factor in the other polynomial is
|
||||
the minimal polynomial of $2\sin( \pi / 9 )$:
|
||||
|
||||
In fact, both of these polynomials show up in factorizations of Chebyshev polynomials
|
||||
of the *first* kind (specifically, $2T_15(z / 2)$ and $2T_9(z / 2)$).
|
||||
Perhaps this is not surprising since we were already working with those of the second kind.
|
||||
However, it is interesting to see them appear from the addition of a single node.
|
||||
|
||||
|
||||
Closing
|
||||
-------
|
||||
|
||||
Regardless of whether chains or polygons are more fundamental, it is certainly interesting
|
||||
that they are just an algebraic stone's (a *calculus*'s?) toss away from one another.
|
||||
Perhaps Euler skipped such stones from the bridges of Koenigsberg which inspired him
|
||||
to initiate graph theory.
|
||||
|
||||
Trees are certainly more complicated than either, and we only investigated those removed
|
||||
from a path by a single node.
|
||||
Regardless, they still related to Chebyshev polynomials, albeit through their factors.
|
||||
|
||||
In fact, I was initially prompted to look into them due to a remarkable correspondence between
|
||||
certain trees and Platonic solids.
|
||||
I have since reorganized these thoughts, as from the perspective of this article, the relationship
|
||||
is tangential at best.
|
||||
BIN
posts/math/chebyshev/2/path_graphs.png
(Stored with Git LFS)
Normal file
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posts/math/chebyshev/2/path_graphs.png
(Stored with Git LFS)
Normal file
Binary file not shown.
BIN
posts/math/chebyshev/2/quadrilaterals.png
(Stored with Git LFS)
Normal file
BIN
posts/math/chebyshev/2/quadrilaterals.png
(Stored with Git LFS)
Normal file
Binary file not shown.
BIN
posts/math/chebyshev/2/tree_graphs.png
(Stored with Git LFS)
Normal file
BIN
posts/math/chebyshev/2/tree_graphs.png
(Stored with Git LFS)
Normal file
Binary file not shown.
4
posts/math/chebyshev/_metadata.yml
Normal file
4
posts/math/chebyshev/_metadata.yml
Normal file
@ -0,0 +1,4 @@
|
||||
# freeze computational output
|
||||
freeze: auto
|
||||
|
||||
sidebar: chebyshev-sidebar
|
||||
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Reference in New Issue
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