Vertex figure

Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex.

For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint.

This construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices (see below), but not its precise geometry; it may be generalized to convex polytopes in any dimension.

The cut surface or vertex figure is thus a spherical polygon marked on this sphere.

Uniform polytopes, for instance, can have star polygons for faces and/or for vertex figures.

If the polytope is isogonal, the vertex figure will exist in a hyperplane surface of the n-space.

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}.

The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid.

One octahedron and four truncated cubes meet at each vertex form a space-filling tessellation.

[3] Edge figures are useful for expressing relations between the elements within regular and uniform polytopes.

Regular and single-ringed coxeter diagram uniform polytopes will have a single edge type.