Abstract polytope

An abstract polytope is an abstract representation of the structure of a polytope.

A geometrical polytope with the structure of a given abstract polytope is called a realization of that polytope. Not every abstract polytope has a realization. Some abstract polytopes have two uniform realizations. They are called conjugate polytopes.

Abstract regular polytope
An abstract polytope is regular if it meets the criteria of a regular polytope. Not every abstract regular polytope has a regular realization. For example, there is an infinite family of abstract polytopes that are the comb products of polygons and represent looped portions of the square tiling. They can be realized as non-regular toroids in 3-space or as regular skew polyhedra in 4-space (where they are made of the square faces of duoprisms).

The uniform polyhedra the dodecadodecahedron and ditrigonary dodecadodecahedron as well as their duals the medial rhombic triacontahedron and medial triambic icosahedron are regular when seen as abstract polytopes. Along with the excavated dodecahedron, they are called the regular polyhedra of index 2 (where the fully regular polyhedra are of index 1). There are higher indexes that include more polyhedra.