Abstract polytope

In mathematics, an abstract polytope is a combinatorial structure with properties similar to those shared by a more classical polytope. Abstract polytopes correspond to the structures of polygons, polyhedra, tessellations of the plane and higher-dimensional spaces, tessellations or decompositions of other manifolds such as the torus or projective plane, and many other objects (such as the 11-cell and the 57-cell) that don't fit well into "normal" Euclidean spaces.

Definition

An abstract polytope P in n dimensions is a partially-ordered set (poset) with a rank function (having range {−1, 0, ..., n} that satisfies the following four properties:

1. It has a unique minimal face (F₋₁) and a unique maximal face (F_n).

2. Every flag (i.e every maximal totally ordered subset) has exactly n + 2 elements.

Given faces F, G of P with F < G, the section G/F = {H | F < H < G} has rank G/F = rank G − rank F − 1.

3. It is strongly connected, that is, any flag can be changed into any other by changing just one element at a time.

4. If G/F = {H | F < H < G} has rank 1, then it contains just four elements: F, G and two others.

That is, an abstract polytope is an incidence geometry defined on different types of objects, satisfying certain axioms, supposed to represent the vertices, edges and so on — the faces — of the polytope.

The definition is inspired by noting certain properties of the traditional polytopes in Euclidean space, such that the set of j-faces (−1 ≤ j ≤ n) of such a polytope forms an abstract polytope.

Example

The faces of a polytope form a lattice called its face lattice, where the partial ordering is by set containment of faces. To ensure that every pair of faces has a join and a meet in the face lattice, the polytope itself and the empty set are considered 'faces'. The whole polytope is the unique maximum element of the lattice. The empty set, considered to be a −1-dimensional face of every polytope, is the unique minimum element of the lattice.

The poset may be represented in a Hasse diagram, as shown. In the diagram, each rank or row corresponds to the geometric elements of appropriate dimensionality. The illustration shows a 3-polytope with elements of rank −1, 0, 1, 2 and 3 corresponding to the null polytope, vertices, edges, faces and whole body of the polytope; the minimal rank is always −1. The cardinality of each rank is the number of elements; the cardinality of the maximal and minimal ranks is always 1. The above example may be tabulated:

Element	Rank	Cardinality
Body	3	1
Face	2	5
Edge	1	8
Vertex	0	5
Null	−1	1

Properties

The maximal face is the whole polytope, the minimal face is the empty set.

For an n-dimensional polytope, a flag is a set comprising one element from each rank {empty set, vertex, edge, ..., facet, whole polytope}, such for any element G and element of higher rank H, G < H. So for a three-dimensional polytope such as the cube, the flags have five elements : empty set, vertex, edge, face, cube; moreover the edge is a side of the face, and the vertex is an end of the edge.

Requiring abstract polytopes to be strongly connected eliminates, for example, objects such as pairs of cubes (either disjoint, or connected at a single vertex)

For a cube, let {0,V,E,F,C} be a flag. Then the section E/0 contains four elements: the edge E, the empty set 0, and the two vertices at the ends of the edge. The section F/V contains the face F, the vertex V, and the two edges joined to both F and V.

Realizations

The hemicube is constructed from the cube by treating opposite edges (likewise faces and corners) as really the *same* edge. It has 3 faces, 6 edges, and 4 corners.

Any ordinary geometrical polytope is said to be a realization of some abstract polytope: The geometric pyramid to the right of the Hasse diagram above is a realization of the poset represented. So also are tessellations or tilings of the plane, or other piecewise manifolds in two and higher dimensions. The latter include, for example, the projective polytopes. These can be obtained from a polytope with a central symmetry by identifying opposite vertices, edges, faces and so forth. In three dimensions, this gives the hemi-cube and the hemi-dodecahedron, and their duals, the hemi-octahedron and the hemi-icosahedron.

More generally, a realisation of a regular abstract polytope is a collection of points in space (corresponding to the vertices of the polytope) which is at least as symmetrical as the original abstract polytope. For example, the set of points {(0,0), (0,1), (1,0), (1,1)} is a realisation of the abstract 4-gon (the square). It is not the only realisation, however - one could choose, instead, the set of vertices of a tetrahedron. For every symmetry of the square, there exists a corresponding symmetry of the tetrahedron.

In fact, every abstract polytope with v vertices has at least one realisation, as the vertices of a (v-1)-dimensional simplex. It is usually desirable to seek lower-dimensional realisations.

If an abstract polytope of rank n is realised in n-dimensional space, such that the geometrical arrangement does not break any rules for geometric 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 realised faithfully in any given n-space. The characterisation of this effect is an outstanding problem.

Duality

Every abstract polytope of rank n has a dual twin, where the n and −1 elements are exchanged, likewise the n − 1 and 0 elements, n − 2 and 1 elements, and so on.

The Hasse diagram of the dual may be obtained simply by inverting the ranking of the original.

From this, we can readily see that the diagram of a self-dual polytope must be symmetrical about a horizontal axis corresponding to rank ${\frac {(n-1)}{2}}$ . The square pyramid example above is seen to be self-dual.

The amalgamation problem

The term "abstract polytope" and the definition given above were introduced in Egon Schulte's doctoral dissertation, although earlier work by Branko Grünbaum, H. S. M. Coxeter and Jacques Tits laid the groundwork. Since then, research in the theory of abstract polytopes has focused almost exclusively on regular abstract polytopes, that is, those whose automorphism groups act transitively on the set of flags of the polytope.

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 edge (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, with three faces, six edges and four 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 tesselating a given manifold. If K and L are spherical (that is, tesselations of a topological sphere), then P is called locally spherical and corresponds itself to a tesselation of some manifold. For example, if K and L are both squares, P will be a tesselation of the plane, torus or Klein bottle by squares. Note that a tesselation of an n-dimensional manifold is actually a rank n + 1 polytope. This is in keeping with the common intuition that the platonic solids are three dimensional, even though they can be regarded as tesselations 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 tesselations of projective planes. There is a weakness in this terminology however. It does not allow an easy way to describe a polytope whose facets are 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.

Notable Abstract Polytopes

The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grünbaum, is a rank 4 (that is, four dimensional) abstract polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tesselation (in the usual sense) of any manifold. Instead, the 11-cell is a locally projective polytope. The 11-cell is not only beautiful in the mathematical sense, it is also historically important as one of the first abstract polytopes discovered. It is self-dual and universal - in fact, 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 universal, indeed, it is 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 Schlãfli 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.

References

Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0
Jaron's World: Shapes in Other Dimensions, Discover mag., Apr 2007