From 7b14ea318d197b8d894576a851fb524cce4850c7 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Wed, 25 Jun 2025 18:35:00 -0500 Subject: [PATCH] add new image; fix labelling in old image --- posts/chebyshev/2/index.qmd | 23 +++++++++++++---------- posts/chebyshev/2/path_graphs.png | Bin 23724 -> 19347 bytes posts/chebyshev/2/tree_graphs.png | Bin 0 -> 40590 bytes 3 files changed, 13 insertions(+), 10 deletions(-) create mode 100644 posts/chebyshev/2/tree_graphs.png diff --git a/posts/chebyshev/2/index.qmd b/posts/chebyshev/2/index.qmd index 5bd569e..188ab4f 100644 --- a/posts/chebyshev/2/index.qmd +++ b/posts/chebyshev/2/index.qmd @@ -1,12 +1,12 @@ --- title: "Generating Polynomials, Part 2: Ghostly Chains" description: | - "Do polygons without distance still know about planar geometry?" + "What do polygons without distance still know about planar geometry?" format: html: html-math-method: katex date: "2021-08-19" -date-modified: "2025-06-20" +date-modified: "2025-06-24" categories: - geometry - algebra @@ -290,7 +290,6 @@ Powerful Chains The various $P_n$ are in fact the adjacency matrices of a path on *n* nodes. - ![ Example path graphs of orders 2, 3, and 4 ](./path_graphs.png) @@ -506,7 +505,10 @@ In some sense, they are the opposite of cycle graphs, since by definition they c 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. - +![ + Nondegenerate tree graphs based on 3-, 4-, 5-, and 6-paths +](./tree_graphs.png) + $$ \begin{align*} T_{1,1} := \begin{matrix}[ @@ -774,12 +776,13 @@ $$ 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 other factor is the minimal polynomial of $2\sin( \pi / 9 )$: +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 the factorization - of Chebyshev polynomials of the *first* kind. -Perhaps this is not surprising, given how $2\cos(\pi / n)$ also appear in the spectra of graphs. -However, it is immensely interesting to see them pop out from the addition of a single node. +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. 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