Polytope

A **polytope** is a generalization of 2-dimensional polygons and 3-dimensional polyhedra to any amount of dimensions. N-dimensional polytopes are bound by (n-1)-dimensional **facets** that meet each other at points known as vertices. The constituent parts of a polytope are known as its **elements** or its subfacets (latter term can be synonymous with ridges).

0-dimensional subfacets of a polytope are known as vertices. They make up the boundary of a 1-dimensional polytope (such as a line segment or a comtelon).

1-dimensional subfacets of a polytope are known as edges. They make up the boundary of a polygon. The total length of all the edges of a polytope is known as the edge length, or perimeter (for polygons).

2-dimensional subfacets of a polytope are known as faces. They make up the boundary of a polyhedron, and the total area of the faces of a polytope is known as the surface area, or just area (for polygons).

3-dimensional subfacets of a polytope are known as cells. They make up the boundary of a 4-dimensional polychoron, and the total volume of the cells is known as the surcell volume, or just volume (for polyhedra).

Higher dimensional subfacets that make up polytopes are named after metric prefixes after "tera-" (past yotta- uses an extended system). 4-dimensional subfacets are known as tera, 5-dimensional subfacets are known as peta, and so on. Polytera are 5D shapes bound by tera, polypeta are 6D shapes bound by peta, et cetera.

The n-D subfacet of a polytope is itself.

The (n-1)-D subfacets of a polytope are known as facets. The facets of a polygon are its edges, the facets of a polyhedron are its faces, and the facets of a polychoron are its cells.

The (n-2)-D subfacets of a polytope are known as **ridges**. The ridges of a polytope should always be the contact region of two facets, otherwise it is an exotic polytope. The ridges of a polygon are its vertices; of a polyhedron, its edges; of a polychoron, its faces; and of a polyteron are its cells.

The (n-3)-D subfacets of a polytope are known as **peaks**. The peaks of a polyhedron are its vertices; of a polychoron, its edges; and of a polyteron, its faces.

Polytopes that are degenerate in Euclidean space, but can exist as a tiling of a hypersphere are known as **degenerate polytopes**. The simplest possible non-degenerate polytopes are known as simplices.

Polytopes created from repeated Cartesian products of a line segment are known as hypercubes.

Polytopes created from repeatedly taking the bipyramid of a line segment are known as cross polytopes. They are the duals of hypercubes.

Polytopes that are vertex-transitive and whose faces are regular polygons are known as uniform polytopes, uniform polytopes whose subfacets are all congruent are known as regular polytopes. All simplices, hypercubes, and crosses are regular polytopes.

Polytopes embedded in a complex space are known as complex polytopes or **polycomtopes**.

Polytopes that have more than two facets meeting at a ridge are known as exotic polytopes.

The polytope that corresponds to the empty set is known as a null polytope. It is a proper subset of every shape except itself.

Uniform polytopes can be represented symbolically with Schläfli symbols, the number of entries is always one less than the dimensionality of the polytope. Polytopes with palindromic Schläfli symbols are self-dual.