Uniform 6 polytope - Alchetron, The Free Social Encyclopedia

In six-dimensional geometry, a uniform polypeton (or 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 polypeta 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Ongoing: Thousands of nonconvex uniform polypeta are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although over 10000 convex and nonconvex uniform polytera are currently known, in particular 923 with 6-simplex symmetry. Participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson.

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 symmety 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: [3^3,1,1] -

47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:

{3,3^2,1}, 1₂₁ 6-demicube (demihexeract) - ; also as h{4,3³},
{3,3,3^1,1}, 2₁₁ 6-orthoplex -

E₆ family: [3^3,1,1] -

39 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:

{3,3,3^2,1}, 2₂₁ -
{3,3^2,2}, 1₂₂ -

These fundamental families generate 153 nonprismatic convex uniform polypeta.

In addition, there are 105 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes: [3,3,3,3,2], [4,3,3,3,2], [5,3,3,3,2], [3^2,1,1,2].

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).

See also list of A6 polytopes for graphs of these polytopes.

The B6 family

There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

The B₆ family has symmetry of order 46080 (6 factorial x 2⁶).

They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing.

See also list of B6 polytopes for graphs of these polytopes.

The D6 family

The D₆ family has symmetry of order 23040 (6 factorial x 2⁵).

This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D₆ Coxeter-Dynkin diagram. Of these, 31 (2×16−1) are repeated from the B₆ family and 16 are unique to this family. The 16 unique forms are enumerated below. Bowers-style acronym names are given for cross-referencing.

See list of D6 polytopes for Coxeter plane graphs of these polytopes.

The E6 family

There are 39 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Bowers-style acronym names are given for cross-referencing. The E₆ family has symmetry of order 51,840.

See also list of E6 polytopes for graphs of these polytopes.

Non-Wythoffian 6-Polytopes

In 6 dimensions and above, there are infinite amount of non-Wythoffian convex uniform polytopes as the Cartesian product of the Grand antiprism in 4 dimensions and a regular polygon in 2 dimensions. It is not yet proven whether or not there are more.

Regular and uniform honeycombs

There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space:

Regular and uniform honeycombs include:

A ~ 5 There are 12 unique uniform honeycombs, including:

5-simplex honeycomb

Truncated 5-simplex honeycomb

Omnitruncated 5-simplex honeycomb

C ~ 5 There are 35 uniform honeycombs, including:

Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,3³,4}, =

B ~ 5 There are 47 uniform honeycombs, 16 new, including:

The uniform alternated hypercube honeycomb, 5-demicubic honeycomb, with symbols h{4,3³,4}, = =

D ~ 5 , [3^1,1,3,3^1,1]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,3³,4}, = . The other two new ones are = , = .

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 12 noncompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.

References

Uniform 6-polytope Wikipedia

(Text) CC BY-SA

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