Paul Bennett
1,025 followers -
(Mostly) unappreciated supergenius.
(Mostly) unappreciated supergenius.

1,025 followers
Paul's posts
Post has shared content
Post has shared content
It's surprisingly hard to convince people of these truths. ﻿
Post has shared content
It's funny. "The Logical Song" is now the anthem for non-leftists.﻿
Post has shared content
Mathematicians prove the darnedest things ...﻿
Desargues’ concertina

Suppose you draw two regular polygons with the same number of vertices, sharing the same centre, one polygon larger than the other. Allow for the possibility that one or both of these polygons are star polygons, where you draw edges not between consecutive vertices, but after skipping some fixed number of points.

The graph you get by joining up the vertices of the inner and outer polygons is known as an “I-graph”, and if the outer polygon is a normal polygon, it is known as a “Generalised Petersen graph”.

In 2012, three mathematicians proved that almost every I-graph, and every generalised Petersen graph, is a unit-distance graph: you can find a way to draw the graph so that all the edges have distance 1.

Žitnik, Arjana; Horvat, Boris; Pisanski, Tomaž, "All generalized Petersen graphs are unit-distance graphs", J. Korean Math. Soc. 49 (2012), No. 3, pp. 475–491

http://basilo.kaist.ac.kr/mathnet/thesis_file/JKMS-49-3-475-491.pdf

One example of a generalised Petersen graph is known as the Desargues graph. Here, the outer polygon is a decagon, while the inner polygon is a 10-pointed star where each vertex is joined to the one you get by adding 3, in a counter-clockwise numbering of the vertices. This turns out to be very easy to draw as a unit-distance graph: you just make the radii of the inner and outer polygons equal to the small and large golden ratios, which differ by 1. In most other examples, you need to introduce a twist between the rings of vertices, but here that isn't necessary.

You can read much more about the Desargues graph in this page by John Baez (which describes its construction in terms of relationships between subsets of a set of 5 elements):

http://math.ucr.edu/home/baez/networks/networks_14.html

The unit-distance version of the Desargues graph described above is not rigid, and in fact there is an 8-parameter family (up to overall rotations and translations) of ways of drawing the graph while maintaining the same edge lengths. The image below shows one highly symmetrical, 1-parameter family of unit-distance versions of the graph.

﻿
Post has shared content
"...with our admiration for the music you have given to this world."

Carl Sagan and Ann Druyan wrote this letter to Chuck Berry 30 years ago. Ms. Druyan was the creative director of NASA's Voyager Interstellar Message Project, the golden discs affixed to both the Voyager 1 and Voyager 2 spacecraft. She also co-wrote the 1980 PBS documentary series Cosmos, hosted by Carl Sagan, whom she married in 1981.

#RIPChuckBerry
﻿
Post has shared content
Looking for something creative to do

I have a great swath of creativity-enabling software. Blender, POV-ray, Fractint, Corel Graphics Suite, FL Studio, AVI-Synth, VirtualDub, and a ton of ways to get them to talk to each other — I'm also a hyperpowered ninja programmer. There's gotta be something I can do with all that. I just can't think what.﻿
Post has shared content
Post has shared content