abstract polytopes

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• As he and others refined these ideas, such sets came to be called abstract polytopes.
• In general, only a restricted set of abstract polytopes of rank n may be realized faithfully in any given n-space.
• Note, however, that this definition does not work for abstract polytopes.
• An important question in the theory of abstract polytopes is the amalgamation problem.
• These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements.
• Regularity has a related, though different meaning for abstract polytopes, since angles and lengths of edges have no meaning.
• The term traditional will be used, somewhat loosely, to refer to what is generally understood by polytope, excluding our abstract polytopes.
• The 11-cell is not only beautiful in the mathematical sense, it is also historically important as one of the first non-traditional abstract polytopes discovered.
• The abstract polytopes arose out of an attempt to study polytopes apart from the geometrical space they are embedded in.
• For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not so for abstract polytopes.
• With the digon, and many other abstract polytopes, vertex notation cannot be used.
• But non-regular classical polytopes can have regular abstract equivalents, since abstract polytopes don't care about angles and edge lengths, for example.
• The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties.
• Some notable examples of abstract polytopes that do not appear elsewhere in this list are the 11-cell and the 57-cell.
• Thus, any geometric polytope may be described by the appropriate abstract poset, though not all abstract polytopes have proper geometric realizations.
• Since then, research in the theory of abstract polytopes has focused mostly on regular polytopes, that is, those whose automorphism groups act transitively on the set of flags of the polytope.
• In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, the requirement for convexity is relaxed.
• Indeed, much of the study of symmetric but chiral polytopes has been carried out in the framework of abstract polytopes, because of the paucity of geometric examples.
• Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.
• The definition of regularity in terms of the transitivity of flags as given in the introduction applies to abstract polytopes.