Uniform 6-polytope

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Graphs of three regular and related uniform polytopes
6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t01.svg
Truncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t1.svg
Rectified 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t02.svg
Cantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t03.svg
Runcinated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t04.svg
Stericated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t05.svg
Pentellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-cube t5.svg
6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t45.svg
Truncated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t4.svg
Rectified 6-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t35.svg
Cantellated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t25.svg
Runcinated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6-cube t15.svg
Stericated 6-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
6-cube t02.svg
Cantellated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t03.svg
Runcinated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t04.svg
Stericated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t05.svg
Pentellated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-cube t0.svg
6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t01.svg
Truncated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t1.svg
Rectified 6-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-demicube t0 D6.svg
6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-demicube t01 D6.svg
Truncated 6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-demicube t02 D6.svg
Cantellated 6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-demicube t03 D6.svg
Runcinated 6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-demicube t04 D6.svg
Stericated 6-demicube
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t0 E6.svg
122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t1 E6.svg
Truncated 221
CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t2 E6.svg
Truncated 122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

In six-dimensional geometry, a uniform polypeton[1][2] (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[edit]

  • 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.[3]
  • 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.[4]

Uniform 6-polytopes by fundamental Coxeter groups[edit]

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.

# Coxeter group Coxeter-Dynkin diagram
1 A6 [3,3,3,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
2 B6 [3,3,3,3,4] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
3 D6 [3,3,3,31,1] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
4 E6 [32,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
[3,32,2] CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
Coxeter diagram finite rank6 correspondence.png
Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

Uniform prismatic families[edit]

Uniform prism

There are 6 categorical uniform prisms based on the uniform 5-polytopes.

# Coxeter group Notes
1 A5A1 [3,3,3,3,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on 5-simplex
2 B5A1 [4,3,3,3,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on 5-cube
3a D5A1 [32,1,1,2] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on 5-demicube
# Coxeter group Notes
4 A3I2(p)A1 [3,3,2,p,2] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on tetrahedral-p-gonal duoprisms
5 B3I2(p)A1 [4,3,2,p,2] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on cubic-p-gonal duoprisms
6 H3I2(p)A1 [5,3,2,p,2] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.png Prism family based on dodecahedral-p-gonal duoprisms

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:

# Coxeter group Notes
1 A4I2(p) [3,3,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png Family based on 5-cell-p-gonal duoprisms.
2 B4I2(p) [4,3,3,2,p] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png Family based on tesseract-p-gonal duoprisms.
3 F4I2(p) [3,4,3,2,p] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png Family based on 24-cell-p-gonal duoprisms.
4 H4I2(p) [5,3,3,2,p] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png Family based on 120-cell-p-gonal duoprisms.
5 D4I2(p) [31,1,1,2,p] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.png Family based on demitesseract-p-gonal duoprisms.
# Coxeter group Notes
6 A32 [3,3,2,3,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Family based on tetrahedral duoprisms.
7 A3B3 [3,3,2,4,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Family based on tetrahedral-cubic duoprisms.
8 A3H3 [3,3,2,5,3] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Family based on tetrahedral-dodecahedral duoprisms.
9 B32 [4,3,2,4,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png Family based on cubic duoprisms.
10 B3H3 [4,3,2,5,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Family based on cubic-dodecahedral duoprisms.
11 H32 [5,3,2,5,3] CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png Family based on dodecahedral duoprisms.

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.

# Coxeter group Notes
1 I2(p)I2(q)I2(r) [p,2,q,2,r] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel q.pngCDel node.pngCDel 2.pngCDel node.pngCDel r.pngCDel node.png Family based on p,q,r-gonal triprisms

Enumerating the convex uniform 6-polytopes[edit]

  • Simplex family: A6 [34] - CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    • 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular:
      1. {34} - 6-simplex - CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
  • Hypercube/orthoplex family: B6 [4,34] - CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    • 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms:
      1. {4,33} — 6-cube (hexeract) - CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
      2. {33,4} — 6-orthoplex, (hexacross) - CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
  • Demihypercube D6 family: [33,1,1] - CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
    • 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
      1. {3,32,1}, 121 6-demicube (demihexeract) - CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png; also as h{4,33}, CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
      2. {3,3,31,1}, 211 6-orthoplex - CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
  • E6 family: [33,1,1] - CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
    • 39 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including:
      1. {3,3,32,1}, 221 - CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
      2. {3,32,2}, 122 - CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

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], [32,1,1,2].

In addition, there are infinitely many uniform 6-polytope based on:

  1. Duoprism prism families: [3,3,2,p,2], [4,3,2,p,2], [5,3,2,p,2].
  2. Duoprism families: [3,3,3,2,p], [4,3,3,2,p], [5,3,3,2,p].
  3. Triaprism family: [p,2,q,2,r].

The A6 family[edit]

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.

# Coxeter-Dynkin Johnson naming system
Bowers name and (acronym)
Base point Element counts
5 4 3 2 1 0
1 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 6-simplex
heptapeton (hop)
(0,0,0,0,0,0,1) 7 21 35 35 21 7
2 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Rectified 6-simplex
rectified heptapeton (ril)
(0,0,0,0,0,1,1) 14 63 140 175 105 21
3 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Truncated 6-simplex
truncated heptapeton (til)
(0,0,0,0,0,1,2) 14 63 140 175 126 42
4 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Birectified 6-simplex
birectified heptapeton (bril)
(0,0,0,0,1,1,1) 14 84 245 350 210 35
5 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Cantellated 6-simplex
small rhombated heptapeton (sril)
(0,0,0,0,1,1,2) 35 210 560 805 525 105
6 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Bitruncated 6-simplex
bitruncated heptapeton (batal)
(0,0,0,0,1,2,2) 14 84 245 385 315 105
7 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Cantitruncated 6-simplex
great rhombated heptapeton (gril)
(0,0,0,0,1,2,3) 35 210 560 805 630 210
8 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Runcinated 6-simplex
small prismated heptapeton (spil)
(0,0,0,1,1,1,2) 70 455 1330 1610 840 140
9 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Bicantellated 6-simplex
small prismated heptapeton (sabril)
(0,0,0,1,1,2,2) 70 455 1295 1610 840 140
10 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Runcitruncated 6-simplex
prismatotruncated heptapeton (patal)
(0,0,0,1,1,2,3) 70 560 1820 2800 1890 420
11 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Tritruncated 6-simplex
tetradecapeton (fe)
(0,0,0,1,2,2,2) 14 84 280 490 420 140
12 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Runcicantellated 6-simplex
prismatorhombated heptapeton (pril)
(0,0,0,1,2,2,3) 70 455 1295 1960 1470 420
13 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Bicantitruncated 6-simplex
great birhombated heptapeton (gabril)
(0,0,0,1,2,3,3) 49 329 980 1540 1260 420
14 CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Runcicantitruncated 6-simplex
great prismated heptapeton (gapil)
(0,0,0,1,2,3,4) 70 560 1820 3010 2520 840
15 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Stericated 6-simplex
small cellated heptapeton (scal)
(0,0,1,1,1,1,2) 105 700 1470 1400 630 105
16 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Biruncinated 6-simplex
small biprismato-tetradecapeton (sibpof)
(0,0,1,1,1,2,2) 84 714 2100 2520 1260 210
17 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Steritruncated 6-simplex
cellitruncated heptapeton (catal)
(0,0,1,1,1,2,3) 105 945 2940 3780 2100 420
18 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Stericantellated 6-simplex
cellirhombated heptapeton (cral)
(0,0,1,1,2,2,3) 105 1050 3465 5040 3150 630
19 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Biruncitruncated 6-simplex
biprismatorhombated heptapeton (bapril)
(0,0,1,1,2,3,3) 84 714 2310 3570 2520 630
20 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Stericantitruncated 6-simplex
celligreatorhombated heptapeton (cagral)
(0,0,1,1,2,3,4) 105 1155 4410 7140 5040 1260
21 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Steriruncinated 6-simplex
celliprismated heptapeton (copal)
(0,0,1,2,2,2,3) 105 700 1995 2660 1680 420
22 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Steriruncitruncated 6-simplex
celliprismatotruncated heptapeton (captal)
(0,0,1,2,2,3,4) 105 945 3360 5670 4410 1260
23 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Steriruncicantellated 6-simplex
celliprismatorhombated heptapeton (copril)
(0,0,1,2,3,3,4) 105 1050 3675 5880 4410 1260
24 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Biruncicantitruncated 6-simplex
great biprismato-tetradecapeton (gibpof)
(0,0,1,2,3,4,4) 84 714 2520 4410 3780 1260
25 CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Steriruncicantitruncated 6-simplex
great cellated heptapeton (gacal)
(0,0,1,2,3,4,5) 105 1155 4620 8610 7560 2520
26 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Pentellated 6-simplex
small teri-tetradecapeton (staf)
(0,1,1,1,1,1,2) 126 434 630 490 210 42
27 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentitruncated 6-simplex
teracellated heptapeton (tocal)
(0,1,1,1,1,2,3) 126 826 1785 1820 945 210
28 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Penticantellated 6-simplex
teriprismated heptapeton (topal)
(0,1,1,1,2,2,3) 126 1246 3570 4340 2310 420
29 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Penticantitruncated 6-simplex
terigreatorhombated heptapeton (togral)
(0,1,1,1,2,3,4) 126 1351 4095 5390 3360 840
30 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentiruncitruncated 6-simplex
tericellirhombated heptapeton (tocral)
(0,1,1,2,2,3,4) 126 1491 5565 8610 5670 1260
31 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Pentiruncicantellated 6-simplex
teriprismatorhombi-tetradecapeton (taporf)
(0,1,1,2,3,3,4) 126 1596 5250 7560 5040 1260
32 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentiruncicantitruncated 6-simplex
terigreatoprismated heptapeton (tagopal)
(0,1,1,2,3,4,5) 126 1701 6825 11550 8820 2520
33 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentisteritruncated 6-simplex
tericellitrunki-tetradecapeton (tactaf)
(0,1,2,2,2,3,4) 126 1176 3780 5250 3360 840
34 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentistericantitruncated 6-simplex
tericelligreatorhombated heptapeton (tacogral)
(0,1,2,2,3,4,5) 126 1596 6510 11340 8820 2520
35 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Omnitruncated 6-simplex
great teri-tetradecapeton (gotaf)
(0,1,2,3,4,5,6) 126 1806 8400 16800 15120 5040

The B6 family[edit]

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.

# Coxeter-Dynkin diagram Schläfli symbol Names Element counts
5 4 3 2 1 0
36 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0{3,3,3,3,4} 6-orthoplex
Hexacontatetrapeton (gee)
64 192 240 160 60 12
37 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1{3,3,3,3,4} Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
76 576 1200 1120 480 60
38 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t2{3,3,3,3,4} Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
76 636 2160 2880 1440 160
39 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t2{4,3,3,3,3} Birectified 6-cube
Birectified hexeract (brox)
76 636 2080 3200 1920 240
40 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t1{4,3,3,3,3} Rectified 6-cube
Rectified hexeract (rax)
76 444 1120 1520 960 192
41 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0{4,3,3,3,3} 6-cube
Hexeract (ax)
12 60 160 240 192 64
42 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1{3,3,3,3,4} Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
76 576 1200 1120 540 120
43 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2{3,3,3,3,4} Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
136 1656 5040 6400 3360 480
44 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,2{3,3,3,3,4} Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
1920 480
45 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,3{3,3,3,3,4} Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
7200 960
46 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,3{3,3,3,3,4} Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
8640 1440
47 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t2,3{4,3,3,3,3} Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
3360 960
48 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,4{3,3,3,3,4} Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
5760 960
49 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,4{4,3,3,3,3} Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
11520 1920
50 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t1,3{4,3,3,3,3} Bicantellated 6-cube
Small birhombated hexeract (saborx)
9600 1920
51 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t1,2{4,3,3,3,3} Bitruncated 6-cube
Bitruncated hexeract (botox)
2880 960
52 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,5{4,3,3,3,3} Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
1920 384
53 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,4{4,3,3,3,3} Stericated 6-cube
Small cellated hexeract (scox)
5760 960
54 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,3{4,3,3,3,3} Runcinated 6-cube
Small prismated hexeract (spox)
7680 1280
55 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,2{4,3,3,3,3} Cantellated 6-cube
Small rhombated hexeract (srox)
4800 960
56 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,1{4,3,3,3,3} Truncated 6-cube
Truncated hexeract (tox)
76 444 1120 1520 1152 384
57 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2{3,3,3,3,4} Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
3840 960
58 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,3{3,3,3,3,4} Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
15840 2880
59 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,3{3,3,3,3,4} Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
11520 2880
60 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,2,3{3,3,3,3,4} Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
10080 2880
61 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,4{3,3,3,3,4} Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
19200 3840
62 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,4{3,3,3,3,4} Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
28800 5760
63 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,2,4{3,3,3,3,4} Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
23040 5760
64 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,3,4{3,3,3,3,4} Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
15360 3840
65 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,2,4{4,3,3,3,3} Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
23040 5760
66 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t1,2,3{4,3,3,3,3} Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
11520 3840
67 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,5{3,3,3,3,4} Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
8640 1920
68 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,5{3,3,3,3,4} Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
21120 3840
69 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,3,4{4,3,3,3,3} Steriruncinated 6-cube
Celliprismated hexeract (copox)
15360 3840
70 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,5{4,3,3,3,3} Penticantellated 6-cube
Terirhombated hexeract (topag)
21120 3840
71 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,2,4{4,3,3,3,3} Stericantellated 6-cube
Cellirhombated hexeract (crax)
28800 5760
72 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,2,3{4,3,3,3,3} Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
13440 3840
73 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,1,5{4,3,3,3,3} Pentitruncated 6-cube
Teritruncated hexeract (tacog)
8640 1920
74 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,1,4{4,3,3,3,3} Steritruncated 6-cube
Cellitruncated hexeract (catax)
19200 3840
75 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,1,3{4,3,3,3,3} Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
17280 3840
76 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,1,2{4,3,3,3,3} Cantitruncated 6-cube
Great rhombated hexeract (grox)
5760 1920
77 CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,3{3,3,3,3,4} Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
20160 5760
78 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,4{3,3,3,3,4} Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
46080 11520
79 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,3,4{3,3,3,3,4} Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
40320 11520
80 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,3,4{3,3,3,3,4} Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
40320 11520
81 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t1,2,3,4{4,3,3,3,3} Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
34560 11520
82 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,5{3,3,3,3,4} Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
30720 7680
83 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,3,5{3,3,3,3,4} Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
51840 11520
84 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,2,3,5{4,3,3,3,3} Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
46080 11520
85 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,2,3,4{4,3,3,3,3} Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
40320 11520
86 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,4,5{4,3,3,3,3} Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
30720 7680
87 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,1,3,5{4,3,3,3,3} Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
51840 11520
88 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,1,3,4{4,3,3,3,3} Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
40320 11520
89 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,1,2,5{4,3,3,3,3} Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
30720 7680
90 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,1,2,4{4,3,3,3,3} Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
46080 11520
91 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png t0,1,2,3{4,3,3,3,3} Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
23040 7680
92 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,3,4{3,3,3,3,4} Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
69120 23040
93 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,3,5{3,3,3,3,4} Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
80640 23040
94 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,4,5{3,3,3,3,4} Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
80640 23040
95 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,4,5{4,3,3,3,3} Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
80640 23040
96 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png t0,1,2,3,5{4,3,3,3,3} Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
80640 23040
97 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png t0,1,2,3,4{4,3,3,3,3} Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
69120 23040
98 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png t0,1,2,3,4,5{4,3,3,3,3} Omnitruncated 6-cube
Great teri-hexeractihexacontitetrapeton (gotaxog)
138240 46080

The D6 family[edit]

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.

# Coxeter diagram Names Base point
(Alternately signed)
Element counts Circumrad
5 4 3 2 1 0
99 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 6-demicube
Hemihexeract (hax)
(1,1,1,1,1,1) 44 252 640 640 240 32 0.8660254
100 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Cantic 6-cube
Truncated hemihexeract (thax)
(1,1,3,3,3,3) 76 636 2080 3200 2160 480 2.1794493
101 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Runcic 6-cube
Small rhombated hemihexeract (sirhax)
(1,1,1,3,3,3) 3840 640 1.9364916
102 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Steric 6-cube
Small prismated hemihexeract (sophax)
(1,1,1,1,3,3) 3360 480 1.6583123
103 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Pentic 6-cube
Small cellated demihexeract (sochax)
(1,1,1,1,1,3) 1440 192 1.3228756
104 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png Runcicantic 6-cube
Great rhombated hemihexeract (girhax)
(1,1,3,5,5,5) 5760 1920 3.2787192
105 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Stericantic 6-cube
Prismatotruncated hemihexeract (pithax)
(1,1,3,3,5,5) 12960 2880 2.95804
106 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png Steriruncic 6-cube
Prismatorhombated hemihexeract (prohax)
(1,1,1,3,5,5) 7680 1920 2.7838821
107 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Penticantic 6-cube
Cellitruncated hemihexeract (cathix)
(1,1,3,3,3,5) 9600 1920 2.5980761
108 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Pentiruncic 6-cube
Cellirhombated hemihexeract (crohax)
(1,1,1,3,3,5) 10560 1920 2.3979158
109 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentisteric 6-cube
Celliprismated hemihexeract (cophix)
(1,1,1,1,3,5) 5280 960 2.1794496
110 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Steriruncicantic 6-cube
Great prismated hemihexeract (gophax)
(1,1,3,5,7,7) 17280 5760 4.0926762
111 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png Pentiruncicantic 6-cube
Celligreatorhombated hemihexeract (cagrohax)
(1,1,3,5,5,7) 20160 5760 3.7080991
112 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentistericantic 6-cube
Celliprismatotruncated hemihexeract (capthix)
(1,1,3,3,5,7) 23040 5760 3.4278274
113 CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentisteriruncic 6-cube
Celliprismatorhombated hemihexeract (caprohax)
(1,1,1,3,5,7) 15360 3840 3.2787192
114 CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png Pentisteriruncicantic 6-cube
Great cellated hemihexeract (gochax)
(1,1,3,5,7,9) 34560 11520 4.5552168

The E6 family[edit]

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.

# Coxeter diagram Names Element counts
5-faces 4-faces Cells Faces Edges Vertices
115 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 221
Icosiheptaheptacontidipeton (jak)
99 648 1080 720 216 27
116 CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Rectified 221
Rectified icosiheptaheptacontidipeton (rojak)
126 1350 4320 5040 2160 216
117 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Truncated 221
Truncated icosiheptaheptacontidipeton (tojak)
126 1350 4320 5040 2376 432
118 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Cantellated 221
Small rhombated icosiheptaheptacontidipeton (sirjak)
342 3942 15120 24480 15120 2160
119 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Runcinated 221
Small demiprismated icosiheptaheptacontidipeton (shopjak)
342 4662 16200 19440 8640 1080
120 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Demified icosiheptaheptacontidipeton (hejak) 342 2430 7200 7920 3240 432
121 CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Bitruncated 221
Bitruncated icosiheptaheptacontidipeton (botajik)
2160
122 CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Demirectified icosiheptaheptacontidipeton (harjak) 1080
123 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Cantitruncated 221
Great rhombated icosiheptaheptacontidipeton (girjak)
4320
124 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Runcitruncated 221
Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)
4320
125 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Steritruncated 221
Cellitruncated icosiheptaheptacontidipeton (catjak)
2160
126 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Demitruncated icosiheptaheptacontidipeton (hotjak) 2160
127 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Runcicantellated 221
Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)
6480
128 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Small demirhombated icosiheptaheptacontidipeton (shorjak) 4320
129 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Small prismated icosiheptaheptacontidipeton (spojak) 4320
130 CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Tritruncated icosiheptaheptacontidipeton (titajak) 4320
131 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Runcicantitruncated 221
Great demiprismated icosiheptaheptacontidipeton (ghopjak)
12960
132 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Stericantitruncated 221
Celligreatorhombated icosiheptaheptacontidipeton (cograjik)
12960
133 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Great demirhombated icosiheptaheptacontidipeton (ghorjak) 8640
134 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Prismatotruncated icosiheptaheptacontidipeton (potjak) 12960
135 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Demicellitruncated icosiheptaheptacontidipeton (hictijik) 8640
136 CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Prismatorhombated icosiheptaheptacontidipeton (projak) 12960
137 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Great prismated icosiheptaheptacontidipeton (gapjak) 25920
138 CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik) 25920
# Coxeter diagram Names Element counts
5-faces 4-faces Cells Faces Edges Vertices
139 CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 122
Pentacontatetrapeton (mo)
54 702 2160 2160 720 72
140 CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Rectified 122
Rectified pentacontatetrapeton (ram)
126 1566 6480 10800 6480 720
141 CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Birectified 122
Birectified pentacontatetrapeton (barm)
126 2286 10800 19440 12960 2160
142 CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Trirectified 122
Trirectified pentacontatetrapeton (trim)
558 4608 8640 6480 2160 270
143 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Truncated 122
Truncated pentacontatetrapeton (tim)
13680 1440
144 CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Bitruncated 122
Bitruncated pentacontatetrapeton (bitem)
6480
145 CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png Tritruncated 122
Tritruncated pentacontatetrapeton (titam)
8640
146 CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Cantellated 122
Small rhombated pentacontatetrapeton (sram)
6480
147 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.png = CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png Cantitruncated 122
Great rhombated pentacontatetrapeton (gram)
12960
148 CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Runcinated 122
Small prismated pentacontatetrapeton (spam)
2160
149 CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Bicantellated 122
Small birhombated pentacontatetrapeton (sabrim)
6480
150 CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png Bicantitruncated 122
Great birhombated pentacontatetrapeton (gabrim)
12960
151 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png Runcitruncated 122
Prismatotruncated pentacontatetrapeton (patom)
12960
152 CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png Runcicantellated 122
Prismatorhombated pentacontatetrapeton (prom)
25920
153 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes 11.png = CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea 1.png Omnitruncated 122
Great prismated pentacontatetrapeton (gopam)
51840

Non-Wythoffian 6-Polytopes[edit]

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[edit]

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

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

# Coxeter group Coxeter diagram Forms
1 {\tilde{A}}_5 [3[6]] CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png 12
2 {\tilde{C}}_5 [4,33,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 35
3 {\tilde{B}}_5 [4,3,31,1]
[4,33,4,1+]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
47 (16 new)
4 {\tilde{D}}_5 [31,1,3,31,1]
[1+,4,33,4,1+]
CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
20 (3 new)

Regular and uniform honeycombs include:

Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1 {\tilde{A}}_4x{\tilde{I}}_1 [3[5],2,∞] CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
2 {\tilde{B}}_4x{\tilde{I}}_1 [4,3,31,1,2,∞] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
3 {\tilde{C}}_4x{\tilde{I}}_1 [4,3,3,4,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
4 {\tilde{D}}_4x{\tilde{I}}_1 [31,1,1,1,2,∞] CDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
5 {\tilde{F}}_4x{\tilde{I}}_1 [3,4,3,3,2,∞] CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
6 {\tilde{C}}_3x{\tilde{I}}_1x{\tilde{I}}_1 [4,3,4,2,∞,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
7 {\tilde{B}}_3x{\tilde{I}}_1x{\tilde{I}}_1 [4,31,1,2,∞,2,∞] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
8 {\tilde{A}}_3x{\tilde{I}}_1x{\tilde{I}}_1 [3[4],2,∞,2,∞] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
9 {\tilde{C}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1 [4,4,2,∞,2,∞,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
10 {\tilde{H}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1 [6,3,2,∞,2,∞,2,∞] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
11 {\tilde{A}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1 [3[3],2,∞,2,∞,2,∞] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
12 {\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1 [∞,2,∞,2,∞,2,∞,2,∞] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
13 {\tilde{A}}_2x{\tilde{A}}_2x{\tilde{I}}_1 [3[3],2,3[3],2,∞] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
14 {\tilde{A}}_2x{\tilde{B}}_2x{\tilde{I}}_1 [3[3],2,4,4,2,∞] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
15 {\tilde{A}}_2x{\tilde{G}}_2x{\tilde{I}}_1 [3[3],2,6,3,2,∞] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
16 {\tilde{B}}_2x{\tilde{B}}_2x{\tilde{I}}_1 [4,4,2,4,4,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
17 {\tilde{B}}_2x{\tilde{G}}_2x{\tilde{I}}_1 [4,4,2,6,3,2,∞] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
18 {\tilde{G}}_2x{\tilde{G}}_2x{\tilde{I}}_1 [6,3,2,6,3,2,∞] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png
19 {\tilde{A}}_3x{\tilde{A}}_2 [3[4],2,3[3]] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png
20 {\tilde{B}}_3x{\tilde{A}}_2 [4,31,1,2,3[3]] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png
21 {\tilde{C}}_3x{\tilde{A}}_2 [4,3,4,2,3[3]] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel split1.pngCDel branch.png
22 {\tilde{A}}_3x{\tilde{B}}_2 [3[4],2,4,4] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
23 {\tilde{B}}_3x{\tilde{B}}_2 [4,31,1,2,4,4] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
24 {\tilde{C}}_3x{\tilde{B}}_2 [4,3,4,2,4,4] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
25 {\tilde{A}}_3x{\tilde{G}}_2 [3[4],2,6,3] CDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
26 {\tilde{B}}_3x{\tilde{G}}_2 [4,31,1,2,6,3] CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
27 {\tilde{C}}_3x{\tilde{G}}_2 [4,3,4,2,6,3] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png

Regular and uniform hyperbolic honeycombs[edit]

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.

Hyperbolic noncompact groups

{\bar{P}}_5 = [3,3[5]]: CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
{\widehat{AU}}_5 = [(3,3,3,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.png

{\widehat{AR}}_5 = [(3,3,4,3,3,4)]: CDel label4.pngCDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel branch.pngCDel label4.png

{\bar{S}}_5 = [4,3,32,1]: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
{\bar{O}}_5 = [3,4,31,1]: CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{N}}_5 = [3,(3,4)1,1]: CDel nodea.pngCDel 4a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png

{\bar{U}}_5 = [3,3,3,4,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{X}}_5 = [3,3,4,3,3]: CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{\bar{R}}_5 = [3,4,3,3,4]: CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

{\bar{Q}}_5 = [32,1,1,1]: CDel nodea.pngCDel 3a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png

{\bar{M}}_5 = [4,3,31,1,1]: CDel nodea.pngCDel 4a.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
{\bar{L}}_5 = [31,1,1,1,1]: CDel node.pngCDel branch3.pngCDel splitsplit2.pngCDel node.pngCDel split1.pngCDel nodes.png

Notes on the Wythoff construction for the uniform 6-polytopes[edit]

Construction of the reflective 6-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter-Dynkin diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 6-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.

Here's the primary operators available for constructing and naming the uniform 6-polytopes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

Operation Extended
Schläfli symbol
Coxeter-
Dynkin
diagram
Description
Parent t0{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Any regular 6-polytope
Rectified t1{p,q,r,s,t} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png The edges are fully truncated into single points. The 6-polytope now has the combined faces of the parent and dual.
Birectified t2{p,q,r,s,t} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Birectification reduces cells to their duals.
Truncated t0,1{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 6-polytope. The 6-polytope has its original faces doubled in sides, and contains the faces of the dual.
Cube truncation sequence.svg
Bitruncated t1,2{p,q,r,s,t} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Bitrunction transforms cells to their dual truncation.
Tritruncated t2,3{p,q,r,s,t} CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Tritruncation transforms 4-faces to their dual truncation.
Cantellated t0,2{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Cube cantellation sequence.svg
Bicantellated t1,3{p,q,r,s,t} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Runcinated t0,3{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node.pngCDel t.pngCDel node.png Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated t1,4{p,q,r,s,t} CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.pngCDel t.pngCDel node.png Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node 1.pngCDel t.pngCDel node.png Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated t0,5{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.pngCDel node.pngCDel s.pngCDel node.pngCDel t.pngCDel node 1.png Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. (expansion operation for polypeta)
Omnitruncated t0,1,2,3,4,5{p,q,r,s,t} CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.pngCDel r.pngCDel node 1.pngCDel s.pngCDel node 1.pngCDel t.pngCDel node 1.png All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied.

See also[edit]

Notes[edit]

  1. ^ A proposed name polypeton (plural: polypeta) has been advocated, from the Greek root poly- meaning "many", a shortened penta- meaning "five", and suffix -on. "Five" refers to the dimension of the 5-polytope facets.
  2. ^ Ditela, polytopes and dyads
  3. ^ T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  4. ^ Uniform Polypeta and Other Six Dimensional Shapes

References[edit]

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Richard Klitzing, 6D, uniform polytopes (polypeta)
  • Richard Klitzing, Uniform polytopes, truncation operators

External links[edit]