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John Baez (Dec 01 2020 at 00:47):There are 6 regular polytopes in 4 dimensions - 4d analogues of the Platonic solids. The 600-cell is one with 120 vertices. The 24-cell is one with 24 vertices:
A group of high school students recently prove something cool about ways of inscribing the 24-cell in the 600-cell:
John Baez (Dec 01 2020 at 05:59):Namely, there are 25 ways to inscribe a 24-cell in a 600-cell, and we can list them in a 5 5 grid such that each row and each column gives a way of partitioning the vertices of the 600-cell into the vertices of 5 24-cells.
Morgan Rogers (he/him) (Dec 01 2020 at 09:53):This reminds me of the time I watched this Numberphile video and found myself rather disappointed that there was no explanation of why there couldn't be more than 3 regular polytopes in higher dimensions.
John Baez (Dec 01 2020 at 16:33):I've never worked through the proof, but you could try to use an inductive approach.
The faces of an (n+1)-dimensional polytope must be some n-dimensional regular polytope. There are 6 regular polytopes in 4 dimensions, two of which have five-fold symmetry, so you'd have to show those two can't be faces of regular polytopes in 5 dimensions. It's also interesting to see why the 24-cell can't be the face of a regular polytope in 5 dimensions. Then you'd have to show that the only way to use an n-cube, n-simplex or n-orthoplex to be the face of an (n+1)-dimensional regular polytope is to create an (n+1)-cube, (n+1)-simplex or (n+1)-orthoplex, respectively.
That'd be the "naive" proof strategy. A fancier strategy would be to use the classification of Coxeter groups. That might be easier!
John Baez (Dec 01 2020 at 16:35):I've always been more interested in doing things with the regular polytopes we have than proving that others don't exist.