It is defined (required) to have flat faces which intersect at straight edges. It must have only straight edges, which intersect at vertices. Furthermore, polytopes must encircle (encompass) an inner region such that no portion of the inner space may be traced to any portion of the outer space without crossing the boundary of the polytope. Any locale on the surface of the polytope must be accessible to any other locale on the surface by traveling along surface, never having to leave it. The edges of the inner-space must exist at the surface of the polytope, which would be a flat face, straight edge, or a vertex.
Polytopes are just the generic term of a geometric shape that has no curves.
- A polytope existing in two-dimensional Euclidean space is called a polygon.
- A polytope existing in three-dimensional Euclidean space is called a polyhedron.
- A polytope existing in four-dimensional Euclidean space is called a polychoron or polyhedroid.
- A polytope existing in five-dimensional Euclidean space is called a polyteron.
- A polytope existing in six-dimensional Euclidean space is called a polypeton.
In space greater than three-dimensions, the flat faces, straight edges, and vertices that form the boundaries of the polytope have their hyper-equivalences. These form multi-dimensional boundaries, such as:
All of which are regarded as a non-curved element of space that bounds the shape.