ABSTRACT

A polytope is a generalization of a polygon and of a polyhedra to higher dimensions. An abstract polytope is a structure which considers only the combinatorial properties of a traditional polytope, ignoring for instance angles and lengths. We will define an abstract regular poly- tope as a poset. Examples of non-traditional abstract polyhedra are the projective polytopes and the toroidal maps. There are four projective polyhedra: the hemicube, hemi-octahedron, hemi-dodecahedron, and the hemi-icosahedron. The 11-cell and the 57-cell are examples of polytopes of dimension 4. Toroidal maps are tessellations of the 2-dimensional surface of the torus. A tessellation of an n-dimensional manifold is actually a rank (n + 1)-polytope (a poly- tope of dimension n).

References

[1] P. McMullen and E. Schulte, Abstract regular polytopes, volume 92 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2002.