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This is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group. For example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. do not exist for abstract polytopes.
An abstract polytope is a partially ordered set P (whose elements are called faces) with properties modeling those of the inclusions of faces of convex polytopes.The rank (or dimension) of an abstract polytope is determined by the length of the maximal ordered chains of its faces, and an abstract polytope of rank n is called an abstract n-polytope.
An abstract polygon is an algebraic partially ordered set representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a realization of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.
An abstract polytope is said to be regular if its combinatorial symmetries are transitive on its flags – that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.
The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({n/m}#{} where n is odd) or an antiprism ({n/m}#{} where n is even). All polygons in 3 space have an even number of vertices and edges.
A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions (such as a 4-polytope in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.