On Wed, 13 Aug 2003 asimovd@aol.com wrote:
Given a regular n-simplex, it contains (n+1)-choose-(k+1) k-simplices.
Define T(n,k) as the convex hull of the barycenters of all these k-simplices. These polytopes T(n,k) are highly regular, but I haven't heard them referred to. Do they have standard names? ... Define a polytope of any dimension to be "Archimedean" if its isometry group carries any vertex into any other. Then clearly each T(n,k) is Archimedean in this sense.
QUESTION: For sufficiently high n, what are the (non-regular) Archimedean polytopes of dimension n other than all the T(n,k)'s ?
Most of the Archimedean polytopes are nicely characterized as ringed Coxeter-Dynkin diagrams of finite reflection groups. Exceptions in 3-D include snubs, prisms, and diprisms. Are there other exceptions in any dimension?
(A guess: All others are obtained from the n-cube and its dual (the "n-cross-polytope") in a manner like (but not the same as) the way the T(n,k)'s are obtained from the n-simplex.)
Not quite. Your T(n,k)'s (JHCs ambo-simplices) are just the singly-ringed simplicial polytopes. In 3-D, two or three-ringed Archimedean simplicial polytopes include the truncated tetrahedron, cuboctahedron, and truncated octahedron. The latter two also display cubic symmetry, but that's a 3-D artifact. There are at least two Coxeter-Dynkin diagrams with "cubic" symmetry which generate some distinct Archimedean polytopes, C_n and D_n. C_n is the straight cube. I intuit D_n as the symmetry group of a "semi-cube", the Voronoi region of a black cell in a checkerboard cubic coloring. Correction/comment/elaboration welcomed. I think the finite B_n and C_n reflection groups give rise to the same Archimedean polytopes, but the infinite B_n and C_n groups yield some distinct Archimedean tesselations. - Scott