in higher order polytopes. Dorman Luke construction For a uniform polyhedron, the face of the dual polyhedron may be found from the original polyhedron's vertex figure using the "Dorman Luke" construction. 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}. 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}. For a regular 4-polytope 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. An example vertex figure of a honeycomb 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. 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 coxeter diagram uniform polytopes will have a single edge type. In general, a uniform polytope can have as many edge types 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 4-polytope {p,q,r} is the polygon {r}. Less trivially, the truncated cubic