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?
I have been calling this "the kth ambo-simplex" for many years - in particular, in print about 10 years ago. In general, if P is a sufficiently regular convex polytope, then "ambo(P)" is the convex hull of the mid-points of the edges of P, "second ambo(P)" that of the mid-points of the 2-dimensional faces, and so on. [These things are defined for non-convex polytopes too, but the definitions are a bit harder then.] I chose the prefix "ambo-" in view of the rather nice pun that in Greek it means "rim", and the vertices of the kth ambo-P may be said to be on the k-dimensional "rim" of P, while in Latin it means "both", and the sequence of ambo-polytopes of a given P has that relationship both to P and (in reverse order) to its dual polytope Q. Also, it sounds a bit like "rhombi", and we have the equalities ambo-cuboctahedron = rhombicuboctahedron ambo-icosidodecahedron = rhombicosidodecahedron. John Conway