Editing Abstract polytope (section)

== The amalgamation problem and universal polytopes ==

An important question in the theory of abstract polytopes is the ''amalgamation problem''. This is a series of questions such as

: For given abstract polytopes ''K'' and ''L'', are there any polytopes ''P'' whose facets are ''K'' and whose vertex figures are ''L'' ?
: If so, are they all finite ?
: What finite ones are there ?

For example, if ''K'' is the square, and ''L'' is the triangle, the answers to these questions are

: Yes, there are polytopes ''P'' with square faces, joined three per vertex (that is, there are polytopes of type {4,3}).
: Yes, they are all finite, specifically,
: There is the [[cube]], with six square faces, twelve edges and eight vertices, and the [[hemi-cube (geometry)|hemi-cube]], with three faces, six edges and four vertices.

It is known that if the answer to the first question is 'Yes' for some regular ''K'' and ''L'', then there is a unique polytope whose facets are ''K'' and whose vertex figures are ''L'', called the '''universal''' polytope with these facets and vertex figures, which '''covers''' all other such polytopes. That is, suppose ''P'' is the universal polytope with facets ''K'' and vertex figures ''L''. Then any other polytope ''Q'' with these facets and vertex figures can be written ''Q''=''P''/''N'', where
* ''N'' is a subgroup of the automorphism group of ''P'', and
* ''P''/''N'' is the collection of [[orbit (group theory)|orbits]] of elements of ''P'' under the action of ''N'', with the partial order induced by that of ''P''.
''Q''=''P''/''N'' is called a '''quotient''' of ''P'', and we say ''P'' '''covers''' ''Q''.

Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:
# Attempt to find the applicable universal polytope
# Attempt to classify its quotients.
These two problems are, in general, very difficult.

Returning to the example above, if ''K'' is the square, and ''L'' is the triangle, the universal polytope {''K'',''L''} is the cube (also written {4,3}). The hemicube is the quotient {4,3}/''N'', where ''N'' is a group of symmetries (automorphisms) of the cube with just two elements - the identity, and the symmetry that maps each corner (or edge or face) to its opposite.

If ''L'' is, instead, also a square, the universal polytope {''K'',''L''} (that is, {4,4}) is the tessellation of the Euclidean plane by squares. This tessellation has infinitely many quotients with square faces, four per vertex, some regular and some not. Except for the universal polytope itself, they all correspond to various ways to tessellate either a [[torus]] or an infinitely long [[cylinder (geometry)|cylinder]] with squares.

===The 11-cell and the 57-cell===

The [[11-cell]], discovered independently by [[H. S. M. Coxeter]] and [[Branko Grünbaum]], is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a ''locally'' projective polytope. It is self-dual and universal: it is the ''only'' polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.

The [[57-cell]] is also self-dual, with hemi-dodecahedral facets. It was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell. Like the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures. On the other hand, there are many other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}. The universal polytope with hemi-dodecahedral facets and icosahedral (not hemi-icosahedral) vertex figures is finite, but very large, with 10006920 facets and half as many vertices.

=== 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}}