diff --git a/_quarto.yml b/_quarto.yml index fd587fe..e8d373a 100644 --- a/_quarto.yml +++ b/_quarto.yml @@ -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: diff --git a/index.qmd b/index.qmd index b4aa728..c83e25d 100644 --- a/index.qmd +++ b/index.qmd @@ -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" --- diff --git a/interactive/p-adics/index.qmd b/interactive/p-adics/index.qmd index b99c183..0000b7d 100644 --- a/interactive/p-adics/index.qmd +++ b/interactive/p-adics/index.qmd @@ -27,9 +27,10 @@ Each term of the series is weighted by a geometrically decreasing coefficient *c $$ [...d_2 d_1 d_0]_p \mapsto e^{2\pi i [d_0] / p} - + c e^{2\pi i [d_1 d_0] / p^2} - + c^2 e^{2\pi i [d_2 d_1 d_0] / p^2} - + ... \\ + + c e^{2\pi i [d_1 d_0] / p^2} + + c^2 e^{2\pi i [d_2 d_1 d_0] / p^2} + + ... + \\ f_N(d; p) = \sum_{n = 0}^N c^n e^{2\pi i \cdot [d_{n:0}]_p / p^{n + 1}} $$ diff --git a/posts/math/chebyshev/1/index.qmd b/posts/math/chebyshev/1/index.qmd index 9f59527..0da1147 100644 --- a/posts/math/chebyshev/1/index.qmd +++ b/posts/math/chebyshev/1/index.qmd @@ -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. - - diff --git a/posts/math/finite-field/2/extra/index.qmd b/posts/math/finite-field/2/extra/index.qmd index f6002e0..eeb187c 100644 --- a/posts/math/finite-field/2/extra/index.qmd +++ b/posts/math/finite-field/2/extra/index.qmd @@ -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. diff --git a/posts/math/finite-field/3/index.qmd b/posts/math/finite-field/3/index.qmd index 228be8a..00662e3 100644 --- a/posts/math/finite-field/3/index.qmd +++ b/posts/math/finite-field/3/index.qmd @@ -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*} diff --git a/posts/math/index.qmd b/posts/math/index.qmd index 426c1f4..ac0a0c5 100644 --- a/posts/math/index.qmd +++ b/posts/math/index.qmd @@ -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 --- diff --git a/posts/math/misc/platonic-volume/index.qmd b/posts/math/misc/platonic-volume/index.qmd index 9db8fb9..4757c8a 100644 --- a/posts/math/misc/platonic-volume/index.qmd +++ b/posts/math/misc/platonic-volume/index.qmd @@ -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. diff --git a/posts/math/number-number/1/index.qmd b/posts/math/number-number/1/index.qmd index c6c0a7a..be8e98f 100644 --- a/posts/math/number-number/1/index.qmd +++ b/posts/math/number-number/1/index.qmd @@ -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 diff --git a/posts/math/permutations/3/index.qmd b/posts/math/permutations/3/index.qmd index f223a1e..235234a 100644 --- a/posts/math/permutations/3/index.qmd +++ b/posts/math/permutations/3/index.qmd @@ -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. diff --git a/posts/math/polycount/4/appendix/index.qmd b/posts/math/polycount/4/appendix/index.qmd index 621aab0..8f2bb5e 100644 --- a/posts/math/polycount/4/appendix/index.qmd +++ b/posts/math/polycount/4/appendix/index.qmd @@ -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. diff --git a/posts/math/polycount/4/appendix/showAdicWithKappa.ojs b/posts/math/polycount/4/appendix/showAdicWithKappa.ojs index 1b86a6f..8a24915 100644 --- a/posts/math/polycount/4/appendix/showAdicWithKappa.ojs +++ b/posts/math/polycount/4/appendix/showAdicWithKappa.ojs @@ -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", diff --git a/posts/math/stereo/2/index.qmd b/posts/math/stereo/2/index.qmd index 7631557..6ac7760 100644 --- a/posts/math/stereo/2/index.qmd +++ b/posts/math/stereo/2/index.qmd @@ -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 $e^{i\theta}$.