Uniform polytope

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A uniform polytope is a planar isogonal (vertex-transitive) polytope that has only one edge length and whose elements are also uniform. Self-intersection is allowed, but certain degeneracies such as doubled elements are not (see below). They encompass regular polytopes and all must have regular faces. The study of uniform polytopes originated as a generalization of the 3-dimensional Archimedean solids to permit non-convex polyhedra with regular faces, and later to arbitrary dimensions.

A major historical project in the study of polytopes was the classification of all uniform polyhedra, which was completed in 1975 (see list of uniform polyhedra). In four dimensions, the convex uniform polychora were enumerated in the 20th century, while the non-convex uniform polychora have mostly been studied in the enthusiast community starting in the 90's. These efforts, led by Jonathan Bowers and the hi.gher.space community, have not yet been proven complete. In five dimensions, despite professional research, it is not even known whether the set of convex uniform 5-polytopes is complete.

Wythoffian construction is a common method for discovering and describing uniform polytopes, although it is not capable of constructing all uniform polytopes. Some uniform polytopes fall into (countably) infinite families, such as the 2D regular polygons, the 3D uniform prisms and antiprisms, and other higher-dimensional polytopes formed by prism products.

Definition[edit | edit source]

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 (although uniforms are not necessarily aploic).

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.

By dimension[edit | edit source]

0D-2D[edit | edit source]

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[edit | edit source]

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

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

4D[edit | edit source]

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
  • 2191 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. The current count is 2191, with the last two found in April 2023. 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[edit | edit source]

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+[edit | edit source]

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.

Notable subfamilies of uniforms[edit | edit source]

The regular polytopes are a particularly important infinite family of uniforms, containing the simplices, hypercubes, orthoplexes, and special cases for dimensions less than 5.

Wythoffian construction is a common method for discovering and describing uniform polytopes, although it is not capable of constructing all uniform polytopes. This divides uniform polytopes into Wythoffian and non-Wythoffian classes.

The demihypercubes are an infinite family of convex uniforms, one in each dimension greater than 2, formed by alternation of the hypercubes.

The demicrosses are an infinite family of self-intersecting uniforms, one in each dimension greater than 2.

The prism product of a uniform n-polytope and a uniform m-polytope of the same edge length is a uniform (n + m)-polytope. This generates infinite families of convex uniform polytopes in every dimension; in general these are not considered interesting, and uniform polytope finding is largely concentrated on polytopes not formed by prism products.

The contact polytopes of the E6, E7, and E8 lattices, respectively in 6, 7, and 8 dimensions, are convex uniform polytopes. Further uniform polytopes may be generated through operations including truncation and rectification.

Related concepts[edit | edit source]

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. During a computer search by J. C. Skilling that proved the uniform polyhedra complete, it was demonstrated that there exists a 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.

External links[edit | edit source]