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 April 2021, there are 2188 known uniform polychora (counting polyhedral prisms, but not the infinite duoprism and antiprism prism families). In 5D, there are 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

 * See also: List of uniform polyhedra

The uniform polyhedra, made of uniform-polygonal faces, include the 9 regular polyhedra (5 Platonic solids and 4 Kepler–Poinsot polyhedra), the 13 Archimedean solids (convex, derived from Platonic solids and retaining all symmetries of their predecessors), infinite families of prisms and antiprisms (including those of star polygons), and 53 uniform star polyhedra.

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

4D

 * See also: List of uniform polychora

The uniform polychora, made of uniform polyhedral cells, include the 16 regular polychora (6 convex and 10 nonconvex), the 40 analogues of the Archimedean solids, the prisms of each of the uniform polyhedra, thousands of uniform star polychora, infinite families of duoprisms and antiprism prisms, 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. For a long time the count stood at 1849, but in 2020 two new uniform polychora were found, the first to be confirmed in 14 years. Following these discoveries, two additional polychora were found, bringing the count to 1853 as of early October 2020. A while later another two with similar symmetry to the grand antiprism were found. Lastly in January 2021, a new snub regiment with 272 uniform members was found, bringing the count to 2127. As of April 2021, the regiment has 333 members, for a total of 2188 uniform polychora.