computing the coordinates of vertices of convex regular polyhedra and 4-polytopes

Question

I am considering what I understand to be generalizations of the platonic solids. In the plane one can easily obtain the vertices of a convex regular k-gon by computing roots of unity, and this can be done for k any integer number of vertices. In 3-space the task is more arduous and appears to require a little creativity, although instead of infinitely many convex regular polygons we need only consider the five Platonic solids (tetrahedron, cube, octohedron, dodacahedron, and icosahedron). In 4 dimensions we have 6 sets of vertices to compute (4-simplex, 4-cube, 4-orthoplex, 24-cell, 120-cell, and the 600-cell), and in higher dimensions there are only the n-simplex, n-cube, and n-orthoplex, which all seem fairly straight forward (I may be mistaken). I believe it is the case that any convex regular polytope will have vertices on the unit n-sphere, but what those locations might be are a mystery to me. What general method does one use to compute the location of the vertices of these polyhedra/polytopes?

Answer 1

Some years ago I have written Haskell source code (which may have bitrotted by now) for numerically calculating the symmetry group (in terms of $4 \times 4$ matrices) of 4D polytopes starting from their Schläfli symbols. The reference I used to implement it is the Schläfli determinant in HSM Coxeter's Regular Polytopes, but unfortunately I no longer have a copy to hand for a more precise citation. I believe the determinant method works for arbitrary dimension, though I only tried it in 3D and 4D.

Once you have the symmetry matrices you can apply them all in turn to a vector to get a set of coordinates, depending which vector you choose you can construct variously truncated polytopes. I wrote a blog post in 2009 about it, but it's quite light on detail (though it does have a video).