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