Vertex Figure - Regular Polytopes

Regular Polytopes

If a polytope is regular, it can be represented by a Schläfli symbol and both the cell and the vertex figure can be trivially extracted from this notation.

In general a regular polytope with Schläfli symbol {a,b,c,...,y,z} has cells as {a,b,c,...,y}, and vertex figures as {b,c,...,y,z}.

1. For a regular polyhedron {p,q}, the vertex figure is {q}, a q-gon.
   • Example, the vertex figure for a cube {4,3}, is the triangle {3}.
2. For a regular polychoron or space-filling tessellation {p,q,r}, the vertex figure is {q,r}.
   • Example, the vertex figure for a hypercube {4,3,3}, the vertex figure is a regular tetrahedron {3,3}.
   • Also the vertex figure for a cubic honeycomb {4,3,4}, the vertex figure is a regular octahedron {3,4}.

Since the dual polytope of a regular polytope is also regular and represented by the Schläfli symbol indices reversed, it is easy to see the dual of the vertex figure is the cell of the dual polytope. For regular polyhedra, this is a special case of the Dorman Luke construction.