In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.
The simplest uniform polypeta are regular polytopes: the 6-simplex { 3,3,3,3,3}, the 6-cube (hexeract) { 4,3,3,3,3}, and the 6-orthoplex (hexacross) { 3,3,3,3,4}.
History of discovery
Regular polytopes: (convex faces) 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. Convex uniform polytopes: 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra) Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete.
Uniform 6-polytopes by fundamental Coxeter groups
Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.
There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes.
Uniform prismatic families
Uniform prism
There are 6 categorical uniform prisms based on the uniform 5-polytopes.
Uniform duoprism
There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Five are formed as the product of a uniform 4-polytope with a regular polygon, and six are formed by the product of two uniform polyhedra:
Uniform triaprism
There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.
Enumerating the convex uniform 6-polytopes
Simplex family: A6 [34] -
35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular: { 34} - 6-simplex - Hypercube/orthoplex family: B6 [4,34] - 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms: { 4,33} — 6-cube (hexeract) - { 33,4} — 6-orthoplex, (hexacross) - Demihypercube D6 family: [33,1,1] - 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including: { 3,32,1}, 121 6-demicube (demihexeract) - ; also as h{ 4,33}, { 3,3,31,1}, 211 6-orthoplex - , a half symmetry form of . E6 family: [33,1,1] - 39 uniform 6-polytopes as permutations of rings in the group diagram, including: { 3,3,32,1}, 221 - { 3,32,2}, 122 -
These fundamental families generate 153 nonprismatic convex uniform polypeta.
In addition, there are 57 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes: [3,3,3,3,2], [4,3,3,3,2], [32,1,1,2], excluding the penteract prism as a duplicate of the hexeract.
In addition, there are infinitely many uniform 6-polytope based on: Duoprism prism families: [3,3,2,p,2], [4,3,2,p,2], [5,3,2,p,2]. Duoprism families: [3,3,3,2,p], [4,3,3,2,p], [5,3,3,2,p]. Triaprism family: [p,2,q,2,r].
The A6 family
There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram. All 35 are enumerated below. They are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex (heptapeton). Bowers-style acronym names are given in parentheses for cross-referencing.
The A6 family has symmetry of order 5040 (7 factorial).
The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1).