Polytopes in Five or More Dimensions

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 2n vertices, and it has 2n 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 2n vertices and 2n 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.

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