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114
_quarto.yml
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@ -9,15 +9,15 @@ website:
left:
- text: "Math"
menu:
- ./posts/polycount/index.qmd
- ./posts/pentagons/index.qmd
- ./posts/chebyshev/index.qmd
- ./posts/stereo/index.qmd
- ./posts/permutations/index.qmd
- ./posts/type-algebra/index.qmd
- ./posts/number-number/index.qmd
- ./posts/finite-field/index.qmd
- ./posts/misc/index.qmd
- ./posts/math/polycount/index.qmd
- ./posts/math/pentagons/index.qmd
- ./posts/math/chebyshev/index.qmd
- ./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
right:
- ./about/index.qmd
- icon: github
@ -32,123 +32,123 @@ website:
contents:
- section: "Topics"
contents:
- ./posts/polycount/index.qmd
- ./posts/pentagons/index.qmd
- ./posts/chebyshev/index.qmd
- ./posts/stereo/index.qmd
- ./posts/permutations/index.qmd
- ./posts/type-algebra/index.qmd
- ./posts/number-number/index.qmd
- ./posts/finite-field/index.qmd
- ./posts/misc/index.qmd
- ./posts/math/polycount/index.qmd
- ./posts/math/pentagons/index.qmd
- ./posts/math/chebyshev/index.qmd
- ./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
- id: misc-sidebar
style: "floating"
contents:
- section: "Miscellaneous"
contents:
- ./posts/misc/platonic-volume/index.qmd
- ./posts/misc/infinitesimals/index.qmd
- ./posts/math/misc/platonic-volume/index.qmd
- ./posts/math/misc/infinitesimals/index.qmd
- id: polycount-sidebar
style: "floating"
contents:
- section: "Polynomial Counting"
href: ./posts/polycount/index.qmd
href: ./posts/math/polycount/index.qmd
contents:
- text: "Part 1: A primer"
href: ./posts/polycount/1/index.qmd
href: ./posts/math/polycount/1/index.qmd
- text: "Part 2: Binary and beyond"
href: ./posts/polycount/2/index.qmd
href: ./posts/math/polycount/2/index.qmd
- text: "Part 3: The third degree"
href: ./posts/polycount/3/index.qmd
href: ./posts/math/polycount/3/index.qmd
- text: "Part 4: Two twos"
href: ./posts/polycount/4/index.qmd
href: ./posts/math/polycount/4/index.qmd
contents:
- text: "Appendix"
href: ./posts/polycount/4/appendix/index.qmd
href: ./posts/math/polycount/4/appendix/index.qmd
- text: "Part 5: Pentamerous multiplication"
href: ./posts/polycount/5/index.qmd
href: ./posts/math/polycount/5/index.qmd
- section: 2D
contents:
- text: "Part 1: Lines, leaves, and sand"
href: ./posts/polycount/sand-1/index.qmd
href: ./posts/math/polycount/sand-1/index.qmd
- text: "Part 2: Reorienting Polynomials"
href: ./posts/polycount/sand-2/index.qmd
href: ./posts/math/polycount/sand-2/index.qmd
- id: pentagons-sidebar
style: "floating"
contents:
- section: "12 Pentagons"
href: ./posts/pentagons/index.qmd
href: ./posts/math/pentagons/index.qmd
contents:
- text: "Part 1"
href: ./posts/pentagons/1/index.qmd
href: ./posts/math/pentagons/1/index.qmd
- text: "Part 2"
href: ./posts/pentagons/2/index.qmd
href: ./posts/math/pentagons/2/index.qmd
- text: "Part 3"
href: ./posts/pentagons/3/index.qmd
href: ./posts/math/pentagons/3/index.qmd
- id: chebyshev-sidebar
style: "floating"
contents:
- section: "Generating Polynomials"
href: ./posts/chebyshev/index.qmd
href: ./posts/math/chebyshev/index.qmd
contents:
- text: "Part 1: Regular Constructability"
href: ./posts/chebyshev/1/index.qmd
href: ./posts/math/chebyshev/1/index.qmd
- text: "Part 2: Ghostly Chains"
href: ./posts/chebyshev/2/index.qmd
href: ./posts/math/chebyshev/2/index.qmd
- text: "Extra: Legendary"
href: ./posts/chebyshev/extra/index.qmd
href: ./posts/math/chebyshev/extra/index.qmd
- id: stereography-sidebar
style: "floating"
contents:
- section: "Algebraic Stereography"
href: ./posts/stereo/index.qmd
href: ./posts/math/stereo/index.qmd
contents:
- ./posts/stereo/1/index.qmd
- ./posts/stereo/2/index.qmd
- ./posts/math/stereo/1/index.qmd
- ./posts/math/stereo/2/index.qmd
- id: permutations-sidebar
style: "floating"
contents:
- section: "A Game of Permutations"
href: ./posts/permutations/index.qmd
href: ./posts/math/permutations/index.qmd
contents:
- text: "Part 1"
href: ./posts/permutations/1/index.qmd
href: ./posts/math/permutations/1/index.qmd
- text: "Part 2"
href: ./posts/permutations/2/index.qmd
href: ./posts/math/permutations/2/index.qmd
- text: "Part 3"
href: ./posts/permutations/3/index.qmd
href: ./posts/math/permutations/3/index.qmd
- text: "Appendix"
href: ./posts/permutations/appendix/index.qmd
href: ./posts/math/permutations/appendix/index.qmd
- id: type-algebra-sidebar
style: "floating"
contents:
- section: "Type Algebra and You"
href: ./posts/type-algebra/index.qmd
href: ./posts/math/type-algebra/index.qmd
contents:
- text: "Part 1: Basics"
href: ./posts/type-algebra/1/index.qmd
href: ./posts/math/type-algebra/1/index.qmd
- text: "Part 2: A Fixer-upper"
href: ./posts/type-algebra/2/index.qmd
href: ./posts/math/type-algebra/2/index.qmd
- text: "Part 3: Combinatorial Types"
href: ./posts/type-algebra/3/index.qmd
href: ./posts/math/type-algebra/3/index.qmd
- id: number-number-sidebar
style: "floating"
contents:
- section: "Numbering Numbers"
href: ./posts/number-number/index.qmd
href: ./posts/math/number-number/index.qmd
contents:
- text: "From 0 to ∞"
href: ./posts/number-number/1/index.qmd
href: ./posts/math/number-number/1/index.qmd
- text: "Ordering Obliquely"
href: ./posts/number-number/2/index.qmd
href: ./posts/math/number-number/2/index.qmd
- id: finite-field-sidebar
style: "floating"
@ -157,16 +157,16 @@ website:
href: ./posts/finite-field/index.qmd
contents:
- text: "Part 1: Preliminaries"
href: ./posts/finite-field/1/index.qmd
href: ./posts/math/finite-field/1/index.qmd
- text: "Part 2: Matrix Boogaloo"
href: ./posts/finite-field/2/index.qmd
href: ./posts/math/finite-field/2/index.qmd
contents:
- text: "Appendix"
href: ./posts/finite-field/2/extra/index.qmd
href: ./posts/math/finite-field/2/extra/index.qmd
- text: "Part 3: Roll a d20"
href: ./posts/finite-field/2/index.qmd
href: ./posts/math/finite-field/2/index.qmd
- text: "Part 5: The Power of Forgetting"
href: ./posts/finite-field/2/index.qmd
href: ./posts/math/finite-field/2/index.qmd
format:
html:

18
index.qmd
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@ -2,14 +2,14 @@
title: "Posts"
listing:
contents:
- posts/polycount/*/index.*
- posts/pentagons/*/index.*
- posts/chebyshev/*/index.*
- posts/stereo/*/index.*
- posts/permutations/*/index.*
- posts/type-algebra/*/index.*
- posts/number-number/*/index.*
- posts/finite-field/*/index.*
- posts/misc/*/index.*
- 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"
---

7
interactive/p-adics/index.qmd
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@ -27,9 +27,10 @@ Each term of the series is weighted by a geometrically decreasing coefficient *c
$$
[...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p}
+ c e^{2\pi i [d_1 d_0] / p^2}
+ c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
+ ... \\
+ c e^{2\pi i [d_1 d_0] / p^2}
+ c^2 e^{2\pi i [d_2 d_1 d_0] / p^2}
+ ...
\\
f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}}
$$

11
posts/math/chebyshev/1/index.qmd
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@ -38,7 +38,7 @@ from sympy.abc import z
```
[Recently](/posts/misc/platonic-volume), I used coordinate-free geometry to derive
[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
@ -742,16 +742,13 @@ 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/misc/platonic-volume).
[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.
<!--
Update: I have since rederived the Chebyshev polynomials without the complex exponential,
which you can read about in [this post]().
-->

4
posts/math/finite-field/2/extra/index.qmd
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@ -1,5 +1,5 @@
---
title: "Exploring Finite Fields, Part 2 (Extra)"
title: "Exploring Finite Fields, Part 2 Appendix"
description: |
Additional notes about polynomial evaluation.
format:
@ -280,7 +280,7 @@ $$
\end{gather*}
$$
The "[path swaps](/posts/permutations/1/)" shown commute only the adjacent elements.
The "[path swaps](/posts/math/permutations/1/)" shown commute only the adjacent elements.
By contrast, the permutation $(0 ~ 2)$ commutes *Ξ*~0~ past both *Ξ*~1~ and *Ξ*~2~.
But since we already know *Ξ*~0~ and *Ξ*~1~ commute by the above list,
we learn at this step that *Ξ*~0~ and *Ξ*~2~ commute.

2
posts/math/finite-field/3/index.qmd
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@ -281,7 +281,7 @@ If you've studied enough group theory, you know that there are two groups of ord
Since the former group has order-6 elements, but none of these matrices are of order 6,
the matrix group must be isomorphic to the latter.
Since the group is small, it's not too difficult to construct an isomorphism between the two.
Writing the elements of *S*~3~ in [cycle notation](/posts/permutations/1/), we have:
Writing the elements of *S*~3~ in [cycle notation](/posts/math/permutations/1/), we have:
$$
\begin{gather*}

18
posts/math/index.qmd
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@ -2,14 +2,14 @@
title: "Posts by topic"
listing:
contents:
- posts/polycount/index.*
- posts/pentagons/index.*
- posts/chebyshev/index.*
- posts/stereo/index.*
- posts/permutations/index.*
- posts/type-algebra/index.*
- posts/number-number/index.*
- posts/finite-field/index.*
- posts/misc/*/index.*
- /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: false
---

2
posts/math/misc/platonic-volume/index.qmd
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@ -361,7 +361,7 @@ It is half the length of the diagonal, so the ratio of a diagonal to a side is a
To make calculations easier, some conversions will be made to base *φ*, or phinary.
If you are not familiar already with phinary, I have already written at length about it [here](
/posts/polycount/1
/posts/math/polycount/1
).
To calculate the apothem, we can calculate the sagitta *s* and height *l* by similar triangles.

2
posts/math/number-number/1/index.qmd
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@ -689,7 +689,7 @@ Personally, I like this definition a bit better, if only because it matches othe
For example,
- In topology, it's common to show that this interval is homeomorphic to the entire real line
- It's similar to the [rational functions which appear in stereography](/posts/stereo/1/),
- It's similar to the [rational functions which appear in stereography](/posts/math/stereo/1/),
which continue to infinity instead of being periodic
- It showcases how the Stern-Brocot tree sorts rational numbers by complexity better

2
posts/math/permutations/3/index.qmd
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@ -89,7 +89,7 @@ At least our beloved paths are left untouched, since $L(P_n) = P_{n-1}$.
### Coxeter Diagrams
As it turns out, these restrictions are significant and produce some very deep mathematical objects,
known as *Coxeter diagrams* (named for the same Coxeter as in [Goldberg-Coxeter](/posts/pentagons/1)).
known as *Coxeter diagrams* (named for the same Coxeter as in [Goldberg-Coxeter](/posts/math/pentagons/1)).
In this domain, the aforementioned rules about vertices and edges apply:
each vertex corresponds to an order 2 element and each edge signifies
that the product of two elements has order 3.

2
posts/math/polycount/4/appendix/index.qmd
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@ -234,7 +234,7 @@ In fact, since the expansions are in binary or (balanced) ternary, the integers
be a subset of the 2-adics or 3-adics.
Still, I wanted to see what these numbers actually "look" like, so I whipped up an interactive diagram.
You should definitely see [this page](/interactive/adic/) for more information, but
You should definitely see [this page](/interactive/p-adics/) for more information, but
the gist is that *p*-adics can be sent into the complex plane in a fractal-like way.

4
posts/math/polycount/4/appendix/showAdicWithKappa.ojs
View File

@ -18,7 +18,7 @@ adicExpansionsQuotRem = FileAttachment(
"./cendree_QuotRem_count_1024_256_digits.csv"
).text().then(asIntegers);
import { expansions as oldExpansions } with { base as base } from "../../../../interactive/p-adics/showAdic.ojs";
import { expansions as oldExpansions } with { base as base } from "/interactive/p-adics/showAdic.ojs";
expansionsOrAdics = baseSelector == "b-adic"
? oldExpansions
@ -32,7 +32,7 @@ import { plot } with {
expansionsOrAdics as expansions,
embedBase as embedBase,
geometric as geometric,
} from "../../../../interactive/p-adics/showAdic.ojs";
} from "/interactive/p-adics/showAdic.ojs";
viewof baseSelector = Inputs.radio([
"b-adic",

4
posts/math/stereo/2/index.qmd
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@ -60,7 +60,7 @@ In an effort to document more interesting facts about this mathematical object
Chebyshev Polynomials
---------------------
[Previously](/posts/chebyshev/1), I derived the
[Previously](/posts/math/chebyshev/1), I derived the
[Chebyshev polynomials](https://en.wikipedia.org/wiki/Chebyshev_polynomials)
with the archetypal complex exponential.
These polynomials express the sines and cosines of a multiple of an angle from
@ -99,7 +99,7 @@ Presented this way with such a simple derivation, it appears as though these rel
are inherently trigonometric.
However, these polynomials actually have *nothing* to do with sine and cosine on their own.
For one, [they appear in graph theory](/posts/chebyshev/2), and for two,
For one, [they appear in graph theory](/posts/math/chebyshev/2), and for two,
the importance of the complex exponential is overstated.
$e^{i\theta}$ really just specifies a point on the complex unit circle.
This property is used on the second line to coax the equation into a quadratic in $e^{i\theta}$.