Editing Abstract polytope (section)

=== Local topology ===

The amalgamation problem has, historically, been pursued according to ''local topology''. That is, rather than restricting ''K'' and ''L'' to be particular polytopes, they are allowed to be any polytope with a given [[topology]], that is, any polytope [[tessellation|tessellating]] a given [[manifold]]. If ''K'' and ''L'' are ''spherical'' (that is, tessellations of a topological [[sphere]]), then ''P'' is called ''locally spherical'' and corresponds itself to a tessellation of some manifold. For example, if ''K'' and ''L'' are both squares (and so are topologically the same as circles), ''P'' will be a tessellation of the plane, [[torus]] or [[Klein bottle]] by squares. A tessellation of an ''n''-dimensional manifold is actually a rank ''n''&nbsp;+&nbsp;1 polytope. This is in keeping with the common intuition that the [[Platonic solid]]s are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.

In general, an abstract polytope is called ''locally X'' if its facets and vertex figures are, topologically, either spheres or ''X'', but not both spheres. The [[11-cell]] and [[57-cell]] are examples of rank 4 (that is, four-dimensional) ''locally projective'' polytopes, since their facets and vertex figures are tessellations of [[real projective plane]]s. There is a weakness in this terminology however. It does not allow an easy way to describe a polytope whose facets are [[torus|tori]] and whose vertex figures are projective planes, for example. Worse still if different facets have different topologies, or no well-defined topology at all. However, much progress has been made on the complete classification of the locally toroidal regular polytopes {{sfn|McMullen|Schulte|2002}}