Editing Polytope (section)

==Generalisations of a polytope==

===Infinite polytopes===
{{Main|Apeirotope}}
Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold,  this idea may be extended to infinite manifolds. [[Tessellation|plane tilings]], space-filling ([[Honeycomb (geometry)|honeycombs]]) and [[hyperbolic tiling]]s are in this sense polytopes, and are sometimes called [[apeirotope]]s because they have infinitely many cells.

Among these, there are regular forms including the [[regular skew polyhedron|regular skew polyhedra]] and the infinite series of tilings represented by the regular [[apeirogon]], square tiling, cubic honeycomb, and so on.

===Abstract polytopes===
{{Main|Abstract polytope}}

The theory of [[abstract polytope]]s attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define an intuitive underlying space, such as the [[11-cell]].

An abstract polytope is a [[partially ordered set]] of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a realization in some real space of the associated abstract polytope.<ref>{{citation | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | title = Abstract Regular Polytopes | edition = 1st | publisher = [[Cambridge University Press]] | isbn = 0-521-81496-0 | date = December 2002 | url-access = registration | url = https://archive.org/details/abstractregularp0000mcmu }}</ref>

===Complex polytopes===
{{Main|Complex polytope}}

Structures analogous to polytopes exist in complex [[Hilbert space]]s <math> \Complex^n</math> where ''n'' real dimensions are accompanied by ''n'' [[imaginary number|imaginary]] ones. [[Regular complex polytope]]s are more appropriately treated as [[configuration (polytope)|configurations]].<ref>Coxeter, H.S.M.; ''Regular Complex Polytopes'', 1974</ref>