Barycentric Subdivision - Barycentric Subdivision of A Convex Polytope

Barycentric Subdivision of A Convex Polytope

Another way of defining the BCS of a simplex is to associate each part to a sequence of faces of, with increasing dimensions, such that is a facet of, for from 0 to . Then each vertex of the corresponding piece is the barycenter of face .

This alternative definition can be extended to the BCS of an arbitrary -dimensional convex polytope into a number of -simplices. Thus the BCS of a pentagon, for example, has 10 triangles: each triangle is associated to three elements of — respectively, a corner of, a side of incident to that corner, and itself.

Similarly the BCS of a cube consists of 48 tetrahedra, each of them associated to a sequence of nested elements — a vertex, an edge, a face, and the whole cube. Note that there are 8 choices for, 3 for (given ), and 2 for (given ).