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 October 2021, there are 2189 known uniform polychora (counting polyhedral prisms, but not the infinite duoprism and antiprism prism families). In 5D, there are at least 1294 non-prismatic uniform polytera, plus 4 known infinite sets, one of which was discovered in February 2022. 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), 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.

The platonic solids were known in ancient Greece, although Plato was probably not their discoverer (he may have known about the cuboctahedron). The Kepler–Poinsot polyhedra were known as geometrical designs before Kepler, but Kepler first discovered the stellated dodecahedra as regular polyhedra, and Poinsot rediscovered them and discovered their duals. The Archimedean solids were discovered by Archimedes, although his book on them is lost. The uniform star polyhedra were discovered by faceting by Edmund Hess, Albert Badoureau, Johann Pitsch, H. S. M. Coxeter, and J. C. P. Miller, the latter of whom created the complete list. S. P. Sopov proved the list complete.

There are also six degenerate cases with tetradic edges. Five of them can be seen as compounds, but the sixth, the great disnub dirhombidodecahedron, cannot. It was discovered by J. Skilling in 1975.

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, but it is known that the set of convex ones is complete. 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. 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 at that time. In October 2021, one additional uniform polychoron was found, bringing the count up to 2189. There are also at least 255 fissary uniform polychora excluded from the main count.

5D
The convex uniform polytera include 19 A5 polytera, 31 B5 polytera (15 also uniform under D5 symmetry), 8 D5 polytera that are not also B5, 46 polychoric prisms, and infinitely many polygonal duoprismatic prisms and polygon-polyhedron duoprisms. Including nonconvex ones, there are currently 1294 known uniform polytera excluding prismatics (but including the penteract). Due to the lack of analogs for F4 and H4 symmetries which result in the most complex polychora, there are fewer uniform polytera than uniform polychora. From 5D onwards, there are only three regular convex polytopes and no nonconvex ones: the simplex with An symmetry and the hypercube and orthoplex both with Bn, but the demihypercubic Dn symmetries begin to produce unique polytopes.

6D
In the sixth through eighth dimensions, the gosset symmetries En appear, but like pentagonal Hn symmetries they are sporadic, not continuing to 9D or above. At least 10369(?) uniform polypeta are known, but some regiments are not counted.