PensionDan

Pension actuary & math buff. He/him. Passionate about polytopes, integer sequences, statistics & science. Also jogging!

Last Notes

2026-09-08 21:24:44 UTC

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So you’re saying AI can do time travel now

2026-09-05 21:28:39 UTC

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The Birkhoff polytope \(\Omega_{n}\) has \(n!\) vertices which are the \(n\times n\) permutation matrices; I.e., (0,1) matrices that have exactly one 1 in each row and column. It has dimension \( (n-1)^{2}\), with \n^{2}\) facets, given by \(a_{i,j} = 0\) for each position in the matrix.

2026-09-05 14:33:06 UTC

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I’ve actually been looking into these! The 4-demicube is a 4-d cross polytope ( a regular polytope).
I encountered these by looking into even & odd tridiagonal Birkhoff polytopes. That is, polytopes whose vertices are just the even permutations making up the tridiagonal Birkhoff vertices (or just the odd). (These are Birkhoff non-faces.)
Demicubes show up as faces of even (but not odd) tridiagonal Birkhoff polytopes with lots of vertices that are a power of 2.

2026-05-01 18:02:42 UTC

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And maybe tally their Crackpot Index score.
https://math.ucr.edu/home/baez/crackpot.html

2025-10-04 19:32:19 UTC

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I think you need to specify the limit on the amount of train track. If it’s something like total length <= k, then the solution would depend on k.

2025-09-26 14:15:34 UTC

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Edinburgh Edification?

2025-02-06 20:56:17 UTC

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I happened to be familiar with this particular Schlegel diagram, so I drew it in Paint.
https://en.wikipedia.org/wiki/Schlegel_diagram

2025-02-06 19:04:14 UTC

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Another way to visualize this four dimensional shape is with a Schlegel diagram. If you stare at the diagram, you'll see six triangular prisms, three with red triangle ends and three with turquoise triangle ends. The Schlegel diagram just shows how the facets fit together, not the 4-dimensional interior. And, there is distortion. Also, the "inside" of the outer triangular prism is the outside, with a containing sphere shrunk to a point.
https://media.mathstodon.xyz/media_attachments/files/113/958/455/952/510/219/original/45298fea2b41b1e1.png

2024-12-03 20:13:16 UTC

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I recall reading that Feynman thought of himself as not as brilliant as some other physicists, but able to contribute because he thought differently.

2024-10-24 19:23:37 UTC

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I gotta read more carefully...
I overlooked the word 'least' in your first sentence.