The above constructions are in no way special for four-space. In
every dimension there is a self-dual simplex, with *n* + 1
vertices when the dimension is *n*. Also in every dimension is an
analogue of the cube. In *n*-dimensional space, the *n*-cube has
2^{n} vertices, and it has 2*n* highest-dimensional faces
of dimension *n* - 1. There will always be a third regular
polytope in *n*-space, the dual polytope to the *n*-cube, with 2*n*
vertices and 2^{n} highest-dimensional faces, which are
simplices of dimension *n* - 1. These constructions will become
clearer when we introduce coordinates in Chapter 8.

As it happens, for *n* larger than four, this is all we get. In
*n*-space, there are exactly three regular *n*-dimensional
polytopes, the *n*-simplex, the *n*-cube, and the
*n*-dimensional cube-dual. There are no further regular
polytopes.

The Regular 600-Cell and Its Dual | ||

Table of Contents | ||

The Hypercube Dual or Sixteen-Cell |