Polytope

A polytope is a geometric object in any dimension that contains elements which all lie on a particular hyperplane. This means that all of its elements must be flat, and are therefore representable with a finite number of vertices. Polytopes can either be convex or nonconvex, and have varying symmetries that are defined by vertices and a symmetry axis.

If a polytope is isogonal and has (geometrically) one type of edge, and all of its elements are realizable as such, then it is called a uniform polytope. If the requirement of uniform elements is removed, allowing for Johnson solids as cells, then it is called a scaliform polytope, and if the requirement of vertex-transitivity is removed, then it is called a convex regular-faced polytope.