Abstract polytope

Summary

An abstract polytope is a partially ordered set that behaves like the set of faces of a polytope ordered by inclusion. It keeps only the incidence pattern of vertices, edges and higher faces and forgets all geometry. The axioms ask for a grading by dimension, a connectedness condition, and the diamond condition. The diamond condition says that between two faces whose dimensions differ by two there are exactly two faces. Abstract polytopes include the face lattices of convex polytopes, but also tilings and objects such as the hemicube that no convex polytope realises.