Why is the 24-cell (also called Icositetrachoron or Hyperdiamond) the unique regular convex polychoron which has no direct three-dimensional analog?

Question

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog.

http://mathworld.wolfram.com/24-Cell.html

I don't understand why that is true. Why isn't his three-dimensional analog the octahedron (which is a Platonic solid)?

Edit : Bump.

Answer 1

There are 5 "regular" convex polytopes in 3-dimensions but there are 6 "regular" convex polytopes in 4-dimensions. In all higher dimensions than 4, there are only three "regular" convex polytopes. These are the analogues of the regular tetrahedron, the regular cube, and the regular octahedron. Here are two places which might help you understand what is going on here: http://en.wikipedia.org/wiki/List_of_regular_polytopes#Four-dimensional_regular_polytopes and http://math.ucr.edu/home/baez/platonic.html