Timeline for What is the geometric shape of the Monster sporadic group?
Current License: CC BY-SA 4.0
23 events
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| Sep 5, 2022 at 11:58 | comment | added | The Amplitwist |
The bit.ly link in a comment above points to a PDF, uploaded on ResearchGate, of the article Polytopes Derives from Sporadic Simple Groups by Michael I. Hartley and Alexander Hulpke (January 2010, Contributions to Discrete Mathematics, 5:106-118). Just posting this in case the URL shortener ends up breaking in the future.
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| Jul 27, 2022 at 6:55 | comment | added | S. Carnahan♦ | @DanielSebald I'm pretty sure the convex hull has many additional edges. To construct the graph, I am restricting my attention to those edges that come from certain central angles. | |
| Jul 26, 2022 at 17:52 | comment | added | Daniel Sebald | @S.Carnahan how do you know that the convex hull doesn’t have any additional edges? | |
| Dec 19, 2020 at 12:19 | comment | added | OzoneNerd | @Rudi_Birnbaum Unfortunately, the answer I provided was incorrect, but the feat is in fact possible. I have turned the post linked in my previous comment into an open forum for constructing simple polytopes with an automorphism group of $\mathrm M_{11} \hspace {-0.5pt} $, where "simple" can be interpreted in number of equally valid ways. | |
| Dec 18, 2020 at 1:34 | comment | added | OzoneNerd | @Rudi_Birnbaum I was inspired by your comment to construct a polytope with an automorphism group of $\mathrm M_{11} \hspace {-1pt}$. It consists of $66$ vectors in $\mathbb Z^{11} \! $. I found some truly interesting things along the way. I've posted a lengthy write-up here. | |
| Nov 2, 2020 at 5:12 | comment | added | OzoneNerd | @Rudi_Birnbaum I couldn't find that sentence in the linked article, but it does appear in this one. S. Carnahan is correct that the author is discussing regular polytopes, but they are also "abstract", which refers to a very different concept. Of course, many important lattices such as the Leech lattice include $\mathrm M_{11} \! $ as a subgroup of their automorphism group. | |
| May 24, 2020 at 22:29 | comment | added | S. Carnahan♦ | @Rudi_Birnbaum I cannot read the article because of the paywall. However, given the name of the article, I suspect the author is restricting his view to regular polytopes, and didn't bother to include the word "regular" in that particular sentence. | |
| May 24, 2020 at 20:13 | comment | added | Raphael J.F. Berger | I have found an interesting quote: "The Mathieu groups M$_{11}$ and M$_{22}$ are not automorphism groups of any polytopes.". Is this in contradiction to what has been written above? | |
| May 24, 2020 at 10:01 | comment | added | Raphael J.F. Berger | @S.Carnahan, yes exactly! | |
| May 23, 2020 at 21:45 | comment | added | S. Carnahan♦ | @Rudi_Birnbaum If such a symbol could be given, it would require 196882 elements. | |
| May 23, 2020 at 15:48 | comment | added | Raphael J.F. Berger | Can we give the Schläfli symbol for this polytope? | |
| May 13, 2020 at 15:10 | comment | added | Raphael J.F. Berger | @S.Carnahan, thanks for precision of the question. I just learned its called facets. So maybe then like that: Are the facets of the convex hull regular simplices? Maybe its worthwhile finding out. | |
| May 13, 2020 at 14:50 | comment | added | S. Carnahan♦ | @Rudi_Birnbaum Assuming you are asking whether the boundary of the convex hull decomposes into flat 196882-simplices joined to their neighbors at nonzero angles, I do not have an answer to that question. | |
| May 13, 2020 at 7:56 | comment | added | Raphael J.F. Berger | @S.Carnahan: Do these points form $196883-1$ dimensional simplexes? | |
| May 11, 2020 at 4:01 | comment | added | S. Carnahan♦ | @JamesEadon The pentagon was an analogy that was useful because of the link to star shapes. The points are very regularly spaced, in the sense that the shape has lots of symmetry. | |
| May 10, 2020 at 17:32 | comment | added | JamesEadon | That makes a lot of sense, thanks. Are the points regularly spaced, e.g. like a regular hexagon, or a dodecahedron, etc? Also, you mention a "pentagon". Does the number 5 turn up as a symmetry of the shape, or was "pentagon" just a pure analogy? | |
| May 7, 2020 at 5:15 | history | edited | S. Carnahan♦ | CC BY-SA 4.0 |
elaboration
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| May 7, 2020 at 4:38 | comment | added | S. Carnahan♦ | @JamesEadon Fundamentally, it is a collection of about $10^{20}$ points in 196883-dimensional space. If you had five points symmetrically arranged in the plane, you could choose to connect them like a pentagon, or like a star (or perhaps try some more exotic thing). Similarly, there are options here, but I do not have the vocabulary to describe them. Most people would choose the analog of "pentagon", i.e., the convex hull, because it is conceptually simpler. | |
| May 6, 2020 at 18:01 | comment | added | JamesEadon | This sounds fascinating, but, to a physics-educated layman, what is the nature of the shape of this "monster graph" / polytope? Conway referred to it as like a "star" you hang on your Christmas tree. Is there anyway of describing it in non-jargon? Is it a simple polytope in many dimensions, like a high-D tetrahedron? Or more intricate? He said, why is it there? He had something SPECIAL in mind. I can't visualise the shape from the jargon, I'm afraid, even though I do enjoy the jargon :) | |
| May 6, 2020 at 17:50 | vote | accept | JamesEadon | ||
| May 5, 2020 at 9:03 | comment | added | Adam P. Goucher | Ooh, I much prefer this construction! The minimal permutation representation of the Monster has the same number of elements (97239461142009186000) as there are vertices in your polytope, so your polytope is (in a certain sense) the 'simplest' polytope with Monster symmetry. Elegant! | |
| May 5, 2020 at 7:30 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added eudml link
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| May 5, 2020 at 6:25 | history | answered | S. Carnahan♦ | CC BY-SA 4.0 |