Schläfli symbol for the 4d hypercube

[Kit McEvoy Gould]

Hi all,

Just a brief question. As I understand it, the Schläfli symbol (such as  {4,3,3} ) uses the number of facets around a vertex? (So a cube has 3 faces around a corner, so it's second number in the  Schläfli symbol is "3".) For a tesseract, the description {4,3,3} is given, "4" for number of sides on the polygon faces- square, "3" for faces around a cube's corner, but am I mistaken to think that a corner of a tesseract is adjacent to 4 cubes, and so should be {4,3,4}? I realise also that the first number just showing the number of sides on the shape not following my "rule" for following numbers, so I'm further thrown. I'm sure I am misunderstanding what the Schläfli symbol is actually showing, or something else, but I'd appreciate someone taking the time to explain it!

Regards,

Kit

[Nan Ma]

The last number of a schlafli symbol for 4D polytopes is the number of cells around an edge. There are 3 cubes around an edge in a tesseract, therefore it’s {4,3,3}.

Nan

Sent from my iPhone

On Jan 25, 2021, at 5:31 AM, Kit McEvoy Gould <kmcevo...@gmail.com> wrote:



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[Roice Nelson]

Yep, the last number of a Schläfli symbol tells you how many cells surround a "peak", which are vertices for tilings, edges for honeycombs (4D polytopes), etc.

As far as how the cubes arrange around a vertex, the Schläfli symbol for the hypercube describes that as well. The last two numbers denote the "vertex figure", {3,3} in this case. This means cubes meet each vertex in a tetrahedral pattern, which is why you get four of them. (Btw, you can read more about the Schläfli symbol and see pictures of general {p,q,r} honeycombs in this paper.)

Also interesting is that the Schläfli symbol for the hypercube is not unique. Other possibilities...

{4,3}x{} (cubical prism)

{4}x{4} (square duoprism)

probably more I'm not remembering :)

Cheers,

Roice

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[David Reens]

That’s a beautiful paper, thanks!! Has it been published outside of the archive?

On Jan 25, 2021, at 11:47 AM, Roice Nelson <roi...@gmail.com> wrote:



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[Roice Nelson]

Thanks! Yes, it was published in JMA (Journal of Mathematics and the Arts). That's here, but is likely not accessible to everyone, so I tend to use the arxiv link when sharing.

Best,

Roice

[Melinda Green]

Speaking of peaks, I recently offered a $100 prize to the first person to find a polyhedron with regular faces and containing at least one vertex surrounded by two pentagons and a square, or to the first person to prove that none exist. See the challenge page here. Related to peaks as Conway called these acrohedra as the vertices in question are like little peaks. Please share.

Thanks!
-Melinda

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