Uniform 4-polytope


In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.

History of discovery

  • Convex Regular polytopes:
  • * 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star 4-polytopes
  • * 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures small stellated dodecahedron| and great dodecahedron|.
  • * 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder .
  • Convex semiregular polytopes:
  • * 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell.
  • * 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and grand antiprism were missing from her list.
  • * 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
  • * 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.
  • Convex uniform polytopes:
  • *1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
  • * Convex uniform 4-polytopes:
  • **1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
  • ** 1966 Norman Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.
  • ** 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism.
  • ** 1998-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration. Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly and choros. The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi, tri-prefixes added when the first ring was on the second or third nodes.
  • ** 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.
  • ** 2008: The Symmetries of Things was published by John H. Conway and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, and all of Johnson's names were included in the book index.
  • Nonregular uniform star 4-polytopes:
  • *1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.
  • *1990-2006: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes had been identified by Jonathan Bowers and George Olshevsky, with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 non-prismatic uniform polyhedra, but not the infinite categories of duoprisms or prisms of antiprisms.
  • *2020-2023: 342 new polychora were found, bringing up the total number of known uniform 4-polytopes to 2191. The list has not been proven complete.

    Regular 4-polytopes

Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. Regular 4-polytopes can be expressed with Schläfli symbol have cells of type, faces of type, edge figures, and vertex figures.
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which becomes cells, and which becomes the vertex figure.
Existence as a finite 4-polytope is dependent upon an inequality:
The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent:

Symmetry of uniform 4-polytopes in four dimensions

The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups:
  1. =
  2. =
The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3:
  • =
  • =
    • There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 =, B4 =, D4 =, F4 =, H4 =. There are also 3 prismatic groups A3A1 =, B3A1 =, H3A1 =, and duoprismatic groups: I2×I2 =. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes.
    Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form , have an extended symmetry, a,b,a, doubling the symmetry order. This includes , , and . Uniform polytopes in these group with symmetric rings contain this extended symmetry.
    If all mirrors of a given color are unringed in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed, an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is [|not generally adjustable to create uniform solutions].

    Enumeration

    There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms.
    • 5 are polyhedral prisms based on the Platonic solids
    • 13 are polyhedral prisms based on the Archimedean solids
    • 9 are in the self-dual regular A4 group family.
    • 9 are in the self-dual regular F4 group family.
    • 15 are in the regular B4 group family
    • 15 are in the regular H4 group family.
    • 1 special snub form in the group family.
    • 1 special non-Wythoffian 4-polytope, the grand antiprism.
    • TOTAL: 68 − 4 = 64
    These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets.
    In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms:
    • Set of uniform antiprismatic prisms - sr× - Polyhedral prisms of two antiprisms.
    • Set of uniform duoprisms - × - A Cartesian product of two polygons.

      The A4 family

    The 5-cell has Pentachoric symmetry|diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way.
    Facets are given, grouped in their Coxeter diagram locations by removing specified nodes.
    The three uniform 4-polytopes forms marked with an asterisk, *, have the higher extended pentachoric symmetry, of order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup +, order 60, or its doubling +, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform.