Regular polytopes--math challenge!?

Question

There are 5 regular Platonic solids in 3D space. How many regular polytopes are there in complex 3D space? A complex 3D space is where 3D coordinates (x,y,z) are of complex numbers x, y, z.

You can try to find the answer to this one on the internet, give it your best shot.

You can also buy a book on this subject, but even on Amazon, it'll cost you way over $200 even for an used one.

Brashion, that's right, I find it fascinating how a lot of mathematical objects cannot exist except in "small d", as you say, and the list includes things like electromagnetic waves. Makes you wonder if that's why we exist in 3D+T? Complex 3D spaces play a central role in string theory, and Coxeter has gong a long way suggesting a link between such poiytopes and physical laws. Google "E8", which some say might unite a lot of mathematics behind physics.

Answer 1

Neat stuff you put on here. Couldn't access Shepherd or Coxeter on line, but Google books includes the Handbook of Discrete and Computational Geometry (Goodman & O'Rourke, 2004, chapter by Schulte).

Looks like the answer is 8. There are the three that carry over from R^6: simplex, hypercube, and cross-polytope. Then there are five more in C^3:
generalized complex hypercube & its dual, gen. comp. cross-polytope (which exist in every C^d);
one self-daul with Schlaefli 3{3}3{3}3{3}, 27 vertices, 27 facets; and
one with Schlaefli 3{3}3{4}2, 72 vertices, 54 facets and its dual (reverse Schlaefli symbol, swap vertices & facets).

Interesting how with C^d there are regular polytopes that exist only for small d (lots for d = 2, above for d = 3, one self-dual for d = 4) and just the 5 infinite families for higher dimensions (three from R^2d, complex hypercube & cross-polytope), analogous to the situation for R^d (only 3 for each d > 4) -- suggests that somehow polytopes are a low dimensional sort of object, and makes me wonder what the right questions are for higher dimensions.