# 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.

## Definition

There are many definitions of polytopes depending on application. In the case of uniforms, there are some conventional rules to prevent degeneracy.

A uniform polytope is defined as a rank-n abstract polytope (implying dyadicity) along with a realization that maps its vertices to distinct points in n-dimensional Euclidean space such that:

• It is finite.
• All edges are of the same length.
• Every element is itself uniform. Therefore, every element of rank n ≥ 0 lives in an n-dimensional affine subspace, and no uniform polytope is skew.
• It is vertex-transitive, i.e. for every two vertices, there is a symmetry of the polytope that transforms one vertex and all its adjacent facets into the other vertex and all its adjacent facets.
• The polytope is not a compound and not fissary.
• No elements are doubled.

Some sources also add a requirement that the polytope lacks "hidden facets" that are not visible from the exterior. However, this is implied by the other conditions.

The exclusion of compounds is standard in the literature, but the exclusion of fissaries is specific to the enthusiast community, as fissaries are only a distinct concept from compounds in 4D and above.

## 0D-2D

As the nullitope has no elements and no edges, it is vacuously true that it is uniform, kickstarting the recursive definition of uniformity. It is also clear that the point and line segment are uniform.

The uniform polygons are precisely the regular polygons, including the regular star polygons.

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

Note that the cube is a square prism and the regular octahedron is a triangular antiprism. As both are Platonic solids, they appear twice on this list. Most resources on uniform polyhedra enumerate a total of 5 + 4 + 13 + 53 = 75 non-prismatic uniform polyhedra, conventionally labeling the cube and regular octahedron as "non-prismatic."

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 by professional mathematicians, enumerating the full set of uniform polychora is an open problem, and mostly the domain of the online enthusiast community.

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.

## Related concepts

Relaxing the definition to allow compounds (and more generally fissaries in 4D and above) has been examined. The uniform polyhedron compounds have been fully characterized. Some have degrees of freedom, where different parts of the compound can be rotated freely while maintaining uniformity.

If a polytope's facets are not just uniform but also regular, the polytope is called semi-regular. Semi-regularity and uniformity are identical up to and including three dimensions, but in 4D the semi-regular polychora are a proper subset of the uniform polychora. Semi-regularity was the original attempt to define uniform polytopes in 4D and above, but is now largely superseded by the current, broader definition of uniformity.

Dyadicity is baked into the conventional definition of abstract polytopes. If any even number of facets can meet at each ridge rather than just two, the resulting figures are more broadly polytopoids, and ones that are not dyadic (and therefore generally not considered valid polytopes) are called exotic. Uniformity can still be defined for exotic polytopoids. There is only one exotic uniform polyhedroid (3-polytopoid) that is not a compound: Skilling's figure.

The pseudo-uniform polytopes satisfy the definition of uniform polytopes, but the vertex figures are all identical under any isometry, and not all identical under the polytope's symmetry group. The pseudorhombicuboctahedron is a well-known example, and one of only two known pseudo-uniform polyhedra.

Some definitions of infinite uniform polytopes, especially apeirotopes, have been studied. Wachmann et al. studied a class of "uniform skew apeirohedra" in 1974, but it is not known whether the list is complete.

The semi-uniform polytopes relax the requirement of a single edge length, and all elements are merely isogonal rather than uniform. In two dimensions, examples include the rectangles, ditrigons, and tripods, which can be continuously varied. The semi-uniform polyhedra have not been fully characterized. So far, the definition and study of the semi-uniforms has only been within the amateur communities.

The concept of uniformity has also been applied to tilings of Euclidean and hyperbolic space. Uniformity also applies to polytwisters, although the exact definition is currently unclear.