# Intuition: 5 regular polyhedra, 6 regular 4-polytopes, and then 3 regular d-polytopes

I have struggled to offer an intuitive explanation (to U.S. college students) why the number of regular polytopes in dimension $d$ is:

• $d=2$, number: $\infty$.
• $d=3$, number: $5$, the five Platonic solids.
• $d=4$, number: $6$, with the $24$-cell the polytope with no clear $\mathbb{R}^3$ analog.
• $d \ge 5$, number: $3$, the simplex, hypercube, and its dual the cross-polytope (or orthoplex).

The derivations are convincing without providing clear intuition. Is there some intuitive explanation, perhaps connected to the maximum volume of a unit-ball achieved in dimension $5$? Plot: Dave Richeson's blog, 2010.
(But see Bill Thurston's remarks on the unit-ball volume.)

• Cool question, mathematically, not just educationally. Is there an intuitive rationale for the number of crystallographic space groups by dimension. Wiki seems to say we don't even know how many there are at dimension 6 or higher. en.wikipedia.org/wiki/… – guest May 29 '18 at 7:55
• These numbers are obtained as: number of positive-integer solutions to a particular system of inequalities. See en.wikipedia.org/wiki/Schläfli_symbol – Gerald Edgar May 29 '18 at 13:26
• This would be a reasonable question if posted on the usual Math.SE. If you don't get a good answer here, maybe considering asking over there too. – Mike Pierce May 29 '18 at 20:22
• @MikePierce: Good point. I'll wait a while. – Joseph O'Rourke May 29 '18 at 21:45
• Among the "cartoon" T-shirts sold by the AMS is one where, in the middle of a computation, there's "and then a miracle happens". This is the only intuitive explanation I have for the existence of the 24-cell. I'd be very interested to learn of a better one. – Andreas Blass May 30 '18 at 1:10