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? E.g., T(3,0) ~ T(3,2) is a tetrahedron; T(3,1) is an octahedron. Combinatorially, T(n,k) should be identical to T(n,n-k-1) by duality. Clearly any lower-dimensional face of an T(n,k) is an T(m, j) itself for m < n. First interesting case seems to be T(4,1) ~ T(4,2), bounded by 10 vertices, 30 edges, 30 triangles, and 10 3-cells (5 tetrahedra and 5 octahedra). 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 ? (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.) --Dan