Abstract polytope

An abstract polytope is a combinatorial structure that encodes the incidences between elements in a polytope. It circumvents many of the issues that often arise when trying to define polytopes, like the definition of their interior, or whether any or the many degenerate cases are to be considered, by completely forgoing any consideration of space and treating polytopes in purely structural terms.

In the context of abstract polytopes, "normal" geometrical polytopes are often called concrete polytopes. Any concrete polytope with the same incidence relations as a given abstract polytope is called a realization of it. Abstract polytopes may have none or many representations with any desired properties, be it

The study of abstract polytopes is useful because it allows for much simpler formal treatment of a wide variety of concrete polytopes. Many constructions, like pyramid products or antiprisms that are very hard to define in concrete terms, can be very easily stated in terms of abstract polytopes. Any of these constructions can then be made concrete if we have a mapping from the vertices of the abstract polytope into points in a space. All other elements will be automatically defined, which allows us to consider conditions such as planarity as secondary.

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 ditrigonal icosahedron, 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.