# Uniform polytope

A uniform polytope is an isogonal polytope that has only one edge length and whose elements are also uniform. Regular polytopes are also uniform polytopes. Many uniform polytopes can be derived from Wythoffian construction. 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 1297 non-prismatic uniform polytera, plus 4 known infinite sets, one of which was discovered in February 2022. In 6D, there are 41348 known uniform polypeta plus 10 infinite sets, excluding the atypical howar, chowar, and dittinta regiments. The number in 7D and higher has not been established, as many of the regiments of Wythoffian polytopes are not counted yet.

## 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.

Some uniform polygons
Pentagon Pentagram Octagram
convex nonconvex nonconvex

## 3D

The set of uniform polyhedra is known to be complete, and is classified like so:

• 5 convex regular Platonic solids
• 4 nonconvex regular Kepler–Poinsot polyhedra
• 13 convex nonregular Archimedean solids
• 53 nonconvex nonregular uniform star polyhedra
• the infinite families of prisms and antiprisms, including those based on star polygons

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 four faces meeting at some 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.

Some uniform polyhedra
Dodecahedron Small rhombicuboctahedron Great dodecahedron Enneagonal antiprism Small icosicosidodecahedron
Platonic solid Archimedean solid Kepler-Poinsot polyhedron antiprism uniform star polyhedron

## 4D

The known uniform polychora, made of uniform polyhedral cells, are classified like so:

• 6 convex regular polychora (4D analogues of the Platonic solids), proven complete
• 10 nonconvex regular polychora (4D analogues of the Kepler–Poinsot solids), proven complete
• 40 convex nonregular polychora (4D analogues of the Archimedean solids), proven complete
• 2000+ known nonconvex nonregular uniform polychora
• infinite families of duoprisms and antiprism prisms

While the convex uniform polychora and regular polychora have been proven complete, enumerating the full set of uniform polychora is an open problem.

For a long time the complete list of uniform polychora not including the infinite families 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 305 fissary uniform polychora excluded from the main count.

Some uniform polychora
Tesseract Grand hexacosichoron Truncated pentachoron Truncated icosahedral prism Triangular-pentagonal duoprism
regular, convex regular, nonconvex convex convex infinite family, convex

## 5D

There are only three regular polytera: the hexateron, the penteract, and the triacontaditeron. In general, for 5D and above, there are only three regular polytopes: the simplex (An symmetry), and the hypercube and the orthoplex (both Bn). There are no nonconvex regular polytopes in 5D and above.

The full set of convex nonregular uniform polytera (the 5D analogues of the Archimedean solids together with the convex prisms) is not known. The known set comprises the 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 1297 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, it is likely there are fewer uniform polytera than uniform polychora.

## 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. 41348 non-prismatic uniform polypeta are known, but many regiments are uncounted in 7D and higher.