Abstract polytope

An abstract polytope is a combinatorial structure (as opposed to an inherently geometric structure) that encodes the incidences between element s in a polytope. It circumvents many of the issues that often arise when trying to define polytopes, such as the definition of their interior, or whether any of 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, "ordinary" geometrical polytopes can be treated as 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 regularity, uniformity, or any others.

The study of abstract polytopes is useful because it allows for much simpler formal treatment of a wide variety of shapes. Many constructions, like vertex figures, pyramid products, or antiprisms that are tricky to define in geometric 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.

Hasse diagrams
The Hasse diagram of a partial order is a directed graph that serves as a useful graphical notation. Elements of the poset are drawn as nodes in the graph, and elements that are related in the partial order are connected by an edge, so that the smallest element goes below the largest. To reduce clutter, only direct relations are drawn, so that if a ≤ b ≤ c, the edge between a and c will be omitted.

Since abstract polytopes are partial orders, they also have Hasse diagrams. In their specific case, the nodes are usually grouped by rank. A node will only be connected to elements of the next and the previous rank. For example, a polygonal ("face") element will be linked to several dyad elements (its edges) as well as whatever cells it is a face of, and a dyad ("edge") element will be linked to two point elements as well as whatever faces it is an edge of. Connections only go from d-elements down to (d&minus;1)-elements, where d is any integer between 0 and the rank of P.

The Hasse diagram of a polytope is not to be confused with the abstract polytope itself. For instance, the Hasse diagram to the right has a tesseractic skeleton, but actually represents a tetrahedron.

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 duoprism s).

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 and some stephanoids, 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.