Wikipedia:Reference desk/Archives/Mathematics/2020 September 10

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September 10

Stellation of the 16-cell?

I'm trying to picture in my head why the Octahedron has a Stellation (Stellated octahedron) but the 16-cell, its 4 dimensional equivalent doesn't...Naraht (talk) 13:44, 10 September 2020 (UTC)

@Naraht: But it does, see this paper. The most obvious is the first stellation (which is a compound of two tesseracts). Double sharp (talk) 14:44, 10 September 2020 (UTC)

@Double sharp: I'm sort of surprised. Given that we have a great deal of information on the stellations of the 120 cell and 600 cell, I thought the lack of information on the 16-cell meant it didn't have one. The 5-cell and the hypercube obviously don't have them, I don't know about the 24-cell.Naraht (talk) 18:23, 10 September 2020 (UTC)

Yep, definitely a nice find. There's probably something worth adding to the article here; I'll try to take a closer look later. –Deacon Vorbis (carbon • videos) 18:57, 10 September 2020 (UTC)

@Naraht: Well, surely the 600-cell has more stellations than what we show, if the icosahedron is anything to go by. The 24-cell definitely has stellations (the compound of 3 16-cells and the compound of 3 tesseracts are among them). Double sharp (talk) 03:02, 11 September 2020 (UTC)

It seems to me that it should be possible to devise an algorithm for enumerating the stellations of a given regular polytope. To get this right would be hard work, but it would be slightly surprising to me if no one has yet undertaken the effort.  --Lambiam 07:51, 11 September 2020 (UTC)

@Lambiam: It's been done by Robert Webb of the software Stella for the Platonic and Archimedean solids. The trouble is that in some cases the stellation counts get very high (well over a trillion). Nothing for 4D AFAICS. Double sharp (talk) 08:11, 11 September 2020 (UTC)

Stellar n-Space, Beyond the Final Frontier, Where No Algorithmician Has Gone Before. So 4D still has to be conquered. Might a combinatorial genius come up with a trick to count stellations rather than truly enumerate them? Also, presumably some (self-aggrandizements?) are more interesting than others.  --Lambiam 09:34, 11 September 2020 (UTC)