Abstract Polytope - Realizations

Realizations

Any traditional polytope is an example of a realization of its underlying abstract polytope: The traditional pyramid to the left of the Hasse diagram above is a realization of the poset represented. So also are tessellations or tilings of the plane, or other piecewise linear manifolds in two and higher dimensions. The latter include, for example, the projective polytopes. These can be obtained from a polytope with 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 realization of a regular abstract polytope is a collection of points in space (corresponding to the vertices of the polytope), together with the face structure induced on it by the polytope, which is at least as symmetrical as the original abstract polytope; that is, all combinatorial automorphisms of the abstract polytopes have been realized by geometric symmetries. 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 regular tetrahedron. For every symmetry of the square, there exists a corresponding symmetry of the regular tetrahedron. (There are, however, more symmetries of the regular tetrahedron than there are of the abstract 4-gon.)

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

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 characterisation of this effect is an outstanding problem.