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).

2D
All regular polygons, including star polygons, are defined as uniform. This forms the basis for the recursive definition of "uniform polytope" in higher dimensions.

3D
The uniform polyhedra, made of uniform-polygonal faces, include the 9 regular polyhedra (5 Platonic solids and 4 Kepler–Poinsot polyhedra), infinite families of prisms and antiprisms (including those of star polygons), the 13 Archimedean solids (convex, regular-faced, vertex-transitive polyhedra), and approximately 50 uniform star polyhedra.

It is known that the set of uniform polyhedra is complete.

4D
The uniform polychora, made of uniform-polyhedral cells, include the 16 regular polychora (6 convex and 10 nonconvex), infinite families of duoprisms and duoantiprisms, prisms of each of the uniform polyhedra, convex modifications of regular polychora, thousands of star polychora, and oddities such as the grand antiprism.

It is not yet known if the set of uniform polychora is complete, although all convex ones are known.