4 - Amalgamation

Summary

A main thrust in the theory of regular polytopes is that of the amalgamation of polytopes of lower rank. Traditionally, the regular convex polytopes are constructed inductively, beginning with the regular polygons in the plane. The geometry of the ambient space considerably restricts the number of ways in which two regular convex n-polytopes P₁ and P₂ can occur as facets and vertex-figures, respectively, of a regular convex (n + 1)-polytope Q. Even when the simple necessary condition is satisfied that the vertex-figures of P₁ are isomorphic to the facets of P₂, the polytope Q need not exist in general. However, if we allow Q to be an infinite regular tessellation and the ambient space to be hyperbolic, then any two regular convex n-polytopes P₁ and P₂ can be “amalgamated” to form either a finite regular convex (n + 1)-polytope or an infinite regular tessellation of euclidean or hyperbolic n-space.

This amalgamation problem generalizes readily to abstract regular polytopes. Now, in the absence of an ambient geometry, obstructions to amalgamation must necessarily come from the combinatorics of the polytopes P₁ and P₂. Also, as a new phenomenon, if there does exist an abstract regular (n + 1)-polytope Q with facets P₁ and vertex-figures P₂, then in fact there can be many such polytopes, and all these are covered by a single polytope denoted {P₁, P₂}, and called the universal polytope.