[math-fun] What are these polytopes called?
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
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
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
participants (3)
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asimovd＠aol.com -
John Conway -
Scott Huddleston