Vertex Figure - General Properties

General Properties

A vertex figure for an n-polytope is an (n−1)-polytope. For example, a vertex figure for a polyhedron is a polygon figure, and the vertex figure for a polychoron is a polyhedron figure.

By considering the connectivity of these neighboring vertices an (n−1)-polytope, the vertex figure, can be constructed for each vertex of a polytope:

• Each vertex of the vertex figure coincides with a vertex of the original polytope.
• Each edge of the vertex figure exists on or inside of a face of the original polytope connecting two alternate vertices from an original face.
• Each face of the vertex figure exists on or inside a cell of the original n-polytope (for n > 3).
• ... and so on to higher order elements in higher order polytopes.

Vertex figures are the most useful for uniform polytopes because one vertex figure can imply the entire polytope.

For polyhedra, the vertex figure can be represented by a vertex configuration notation, by listing the faces in sequence around a vertex. For example 3.4.4.4 is a vertex with one triangle and three squares, and it represents the rhombicuboctahedron.

If the polytope is vertex-transitive, the vertex figure will exist in a hyperplane surface of the n-space. In general the vertex figure need not be planar.

Also nonconvex polyhedra, the vertex figure may also be nonconvex. Uniform polytopes can have either star polygon faces and vertex figures for instance.