Vertex Figure - Edge Figure

Related to the vertex figure, an edge figure is the vertex figure of a vertex figure. Edge figures are useful for expressing relations between the elements within regular and uniform polytopes.

An edge figure will be a (n−2)-polytope, representing the arrangement of facets around a given edge. Regular and single-ringed uniform polytopes will have a single edge figure type, while in general, a uniform polytope can have as many edges as active mirrors in the construction, since each active mirror produces one edge in the fundamental domain.

Regular polytopes (and honeycombs) have a single edge figure which is also regular. For a regular polytope {p,q,r,s,...,z}, the edge figure is {r,s,...,z}.

In four dimensions, the edge figure of a 4-polytope or 3-honeycomb is a polygon representing the arrangement of a set of facets around an edge. For example, the edge figure for a regular cubic honeycomb {4,3,4} is a square, and for a regular polychoron {p,q,r} is the polygon {r}.

Less trivially, the truncated cubic honeycomb t_0,1{4,3,4}, has a square pyramid vertex figure, with truncated cube and octahedron cells. Here there are two types of edge figures. One is a square edge figure at the apex of the pyramid. This represents the four truncated cubes around an edge. The other four edge figures are isosceles triangles on the base vertices of the pyramid. These represent the arrangement of two truncated cubes and one octahedron around the other edges.