Editing Abstract polytope (section)

=== Traditional versus abstract polytopes ===
[[Image:Isomorphic Tetragons.svg|thumb|275px|Isomorphic quadrilaterals.]]

In Euclidean geometry, two shapes that are not [[Similar (geometry)|similar]] can nonetheless share a common structure. For example, a [[square]] and a [[trapezoid]] both comprise an alternating chain of four [[vertex (geometry)|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.<ref>{{Harvnb |McMullen |Schulte |2002 |loc=p. 31}}</ref>

====Realizations====
A traditional polytope is said to be a ''realization'' of the associated abstract polytope. A realization is a mapping or injection of the abstract object into a real space, typically [[Euclidean space|Euclidean]], to construct a traditional polytope as a real geometric figure.

The six quadrilaterals shown are all distinct realizations of the abstract quadrilateral, each with different geometric properties. Some of them do not conform to traditional definitions of a quadrilateral and are said to be ''unfaithful'' realizations. A conventional polytope is a faithful realization.