Abstract polytope
algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc
In mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying purely geometric properties such as the position of vertices.
A geometric polytope is said to be a realization of an abstract polytope in some real n-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
Principles
Traditional versus abstract polytopes
In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example, a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. They are said to be isomorphic or “structure preserving”.
This common structure may be represented in an underlying abstract polytope, a purely algebraic partially ordered set which captures the pattern of connections (or incidences) between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.
What is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa. For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope.
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This entry incorporates text from “Abstract polytope” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.