Different Approaches To Definition
The term polytope is a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called polytopes. They represent different approaches of generalizing the convex polytopes to include other objects with similar properties and aesthetic beauty.
The original approach broadly followed by Schläfli, Gossett and others begins with the 0-dimensional point as a 0-polytope (vertex). A 1-dimensional 1-polytope (edge) is constructed by bounding a line segment with two 0-polytopes. Then 2-polytopes (polygons) are defined as plane objects whose bounding facets (edges) are 1-polytopes, 3-polytopes (polyhedra) are defined as solids whose facets (faces) are 2-polytopes, and so forth.
A polytope may also be regarded as a tessellation of some given manifold. Convex polytopes are equivalent to tilings of the sphere, while others may be tilings of other elliptic, flat or toroidal surfaces – see elliptic tiling and toroidal polyhedron. Under this definition, plane tilings and space tilings (honeycombs) are considered to be polytopes, and are sometimes classed as apeirotopes because they have infinitely many cells; tilings of hyperbolic spaces are also included under this definition.
An alternative approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two. However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics.
The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define clearly a natural underlying space, such as the 11-cell.
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