Editing Abstract polytope (section)

== Realization ==
A set of points ''V'' in a Euclidean space equipped with a surjection from the vertex set of an abstract polytope ''P'' such that automorphisms of ''P'' induce [[isometry|isometric]] permutations of ''V'' is called a ''realization'' of an abstract polytope.<ref name="McMullen-Schulte">{{Harvnb|McMullen|Schulte|2002|p=121}}</ref>{{sfn|McMullen|1994|p=225}} Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces.{{sfn|McMullen |Schulte |2002|p=126}}{{sfn|McMullen|1994|p=229}}

If an abstract ''n''-polytope is realized in ''n''-dimensional space, such that the geometrical arrangement does not break any rules for traditional polytopes (such as curved faces, or ridges of zero size), then the realization is said to be ''faithful''. In general, only a restricted set of abstract polytopes of rank ''n'' may be realized faithfully in any given ''n''-space. The characterization of this effect is an outstanding problem.

For a regular abstract polytope, if the combinatorial automorphisms of the abstract polytope are realized by geometric symmetries then the geometric figure will be a regular polytope.

===Moduli space===
The group ''G'' of symmetries of a realization ''V'' of an abstract polytope ''P'' is generated by two reflections, the product of which translates each vertex of ''P'' to the next.{{sfn|McMullen |Schulte |2002|pp=140–141}}{{sfn|McMullen|1994|p=231}} The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection.{{sfn|McMullen |Schulte |2002|p=141}}{{sfn|McMullen|1994|p=231}}

Generally, the [[moduli space]] of realizations of an abstract polytope is a [[convex cone]] of infinite dimension.{{sfn|McMullen |Schulte |2002|p=127}}{{sfn|McMullen|1994|pp=229–230}} The realization cone of the abstract polytope has uncountably infinite [[algebraic dimension]] and cannot be [[Closed set|closed]] in the [[Euclidean topology]].{{sfn|McMullen |Schulte |2002|p=141}}{{sfn|McMullen|1994|p=232}}