Uniform polytope

A uniform polytope is an isogonal polytope that can be represented with only one edge length and whose elements are also uniform geometrically (this includes regular polygons). Regular polytopes are also uniform polytopes. Most uniform polytopes can be derived from a Wythoffian construction, but there are some uniform polytopes, such as the grand antiprism, that are not Wythoffian-constructible. Infinite sets of uniform polytopes can be created from the Cartesian product of two uniform polytopes, with one being a regular polygon or a 3D antiprism.

Besides the infinite sets mentioned above, there are an infinite number of uniform polytopes in 2D (the regular polygons). The list of 75 uniform polyhedra, plus infinite families of prisms and antiprisms, was proven to be complete by John Skilling in 1975. Jonathan Bowers has been searching for uniform polytopes in 4D and higher since 1990. As of 2020, there are 1849 known uniform polychora (counting polyhedral prisms, but not the infinite duoprism and antiprism prism families), and at least 1293 non-prismatic uniform polytera. The number in 6D and higher has not been established, as many of the regiments of Wythoffian polytopes are not counted yet (the number of uniform polypeta in counted regiments currently stands at 10107, with 15 regiments still uncounted as of July 2020).