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In mathematics, an abstract polytope, informally speaking, is a structure which considers only the ''combinatorial'' properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc. No space, such as Euclidean space, is required to contain it. The abstract formulation embodies the combinatorial properties as a partially ordered set or poset. The abstract definition allows some more general combinatorial structures than the traditional concept of a polytope, and allows many new objects that have no counterpart in traditional theory. The term ''polytope'' is a generalisation of polygons and polyhedra into any number of dimensions. == Traditional versus abstract polytopes == In Euclidean geometry, the six quadrilaterals above are all different. Yet they have something in common that is not shared by a triangle or a cube, for example. The elegant, but geographically inaccurate, London Tube map provides all the ''relevant'' information to go from ''A'' to ''B''. An even better example is an electrical circuit diagram or schematic; the final layout of wires and parts is often unrecognisable at first glance. In each of these examples, the ''connections'' between elements are the same, regardless of the ''physical layout''. The objects are said to be ''combinatorially equivalent''. This equivalence is what is encapsulated in the concept of an abstract polytope. So, combinatorially, our six quadrilaterals are all the “same”. More rigorously, they are said to be isomorphic or “structure preserving”. Properties, particularly measurable ones, of traditional polytopes such as angles, edge-lengths, skewness, and convexity ''have no meaning for an abstract polytope''. Other traditional concepts may carry over, ''but not always identically''. Care must be exercised, for what is true for traditional 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 so for abstract polytopes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Abstract polytope」の詳細全文を読む スポンサード リンク
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