Uniform 6-polytope

Last updated
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 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

Contents

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

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.

# Coxeter group Coxeter-Dynkin diagram
1A6[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
2B6[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
3D6[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

Uniform prism

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

# Coxeter group Notes
1A5A1[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.pngPrism family based on 5-simplex
2B5A1[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.pngPrism family based on 5-cube
3aD5A1[32,1,1,2]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngPrism family based on 5-demicube
# Coxeter group Notes
4A3I2(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.pngPrism family based on tetrahedral-p-gonal duoprisms
5B3I2(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.pngPrism family based on cubic-p-gonal duoprisms
6H3I2(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.pngPrism 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
1A4I2(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.pngFamily based on 5-cell-p-gonal duoprisms.
2B4I2(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.pngFamily based on tesseract-p-gonal duoprisms.
3F4I2(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.pngFamily based on 24-cell-p-gonal duoprisms.
4H4I2(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.pngFamily based on 120-cell-p-gonal duoprisms.
5D4I2(p)[31,1,1,2,p]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel p.pngCDel node.pngFamily based on demitesseract-p-gonal duoprisms.
# Coxeter group Notes
6A32[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.pngFamily based on tetrahedral duoprisms.
7A3B3[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.pngFamily based on tetrahedral-cubic duoprisms.
8A3H3[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.pngFamily based on tetrahedral-dodecahedral duoprisms.
9B32[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.pngFamily based on cubic duoprisms.
10B3H3[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.pngFamily based on cubic-dodecahedral duoprisms.
11H32[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.pngFamily 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
1I2(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.pngFamily based on p,q,r-gonal triprisms

Enumerating the convex uniform 6-polytopes

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:

  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

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).

# Coxeter-Dynkin Johnson naming system
Bowers name and (acronym)
Base pointElement counts
543210
1CDel 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)7213535217
2CDel 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)146314017510521
3CDel 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)146314017512642
4CDel 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)148424535021035
5CDel 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)35210560805525105
6CDel 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)1484245385315105
7CDel 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)35210560805630210
8CDel 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)7045513301610840140
9CDel 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 birhombated heptapeton (sabril)
(0,0,0,1,1,2,2)7045512951610840140
10CDel 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)70560182028001890420
11CDel 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)1484280490420140
12CDel 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)70455129519601470420
13CDel 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)4932998015401260420
14CDel 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)70560182030102520840
15CDel 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)10570014701400630105
16CDel 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)84714210025201260210
17CDel 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)105945294037802100420
18CDel 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)1051050346550403150630
19CDel 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)84714231035702520630
20CDel 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)10511554410714050401260
21CDel 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)105700199526601680420
22CDel 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)1059453360567044101260
23CDel 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)10510503675588044101260
24CDel 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)847142520441037801260
25CDel 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)10511554620861075602520
26CDel 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 (staff)
(0,1,1,1,1,1,2)12643463049021042
27CDel 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)12682617851820945210
28CDel 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)1261246357043402310420
29CDel 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)1261351409553903360840
30CDel 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)12614915565861056701260
31CDel 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)12615965250756050401260
32CDel 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)126170168251155088202520
33CDel 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)1261176378052503360840
34CDel 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)126159665101134088202520
35CDel 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)1261806840016800151205040

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.

# Coxeter-Dynkin diagram Schläfli symbol NamesElement counts
543210
36CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0{3,3,3,3,4} 6-orthoplex
Hexacontatetrapeton (gee)
641922401606012
37CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1{3,3,3,3,4} Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
765761200112048060
38CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt2{3,3,3,3,4} Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
76636216028801440160
39CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt2{4,3,3,3,3} Birectified 6-cube
Birectified hexeract (brox)
76636208032001920240
40CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt1{4,3,3,3,3} Rectified 6-cube
Rectified hexeract (rax)
7644411201520960192
41CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0{4,3,3,3,3} 6-cube
Hexeract (ax)
126016024019264
42CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngt0,1{3,3,3,3,4} Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
7657612001120540120
43CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,2{3,3,3,3,4} Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
1361656504064003360480
44CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,2{3,3,3,3,4} Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
1920480
45CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,3{3,3,3,3,4} Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
7200960
46CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,3{3,3,3,3,4} Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
86401440
47CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt2,3{4,3,3,3,3} Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
3360960
48CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,4{3,3,3,3,4} Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
5760960
49CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,4{4,3,3,3,3} Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
115201920
50CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt1,3{4,3,3,3,3} Bicantellated 6-cube
Small birhombated hexeract (saborx)
96001920
51CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt1,2{4,3,3,3,3} Bitruncated 6-cube
Bitruncated hexeract (botox)
2880960
52CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,5{4,3,3,3,3} Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
1920384
53CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt0,4{4,3,3,3,3} Stericated 6-cube
Small cellated hexeract (scox)
5760960
54CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,3{4,3,3,3,3} Runcinated 6-cube
Small prismated hexeract (spox)
76801280
55CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,2{4,3,3,3,3} Cantellated 6-cube
Small rhombated hexeract (srox)
4800960
56CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,1{4,3,3,3,3} Truncated 6-cube
Truncated hexeract (tox)
76444112015201152384
57CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngt0,1,2{3,3,3,3,4} Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
3840960
58CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngt0,1,3{3,3,3,3,4} Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
158402880
59CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,2,3{3,3,3,3,4} Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
115202880
60CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,2,3{3,3,3,3,4} Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
100802880
61CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngt0,1,4{3,3,3,3,4} Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
192003840
62CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,2,4{3,3,3,3,4} Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
288005760
63CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,2,4{3,3,3,3,4} Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
230405760
64CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,3,4{3,3,3,3,4} Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
153603840
65CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt1,2,4{4,3,3,3,3} Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
230405760
66CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt1,2,3{4,3,3,3,3} Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
115203840
67CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngt0,1,5{3,3,3,3,4} Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
86401920
68CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,2,5{3,3,3,3,4} Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
211203840
69CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt0,3,4{4,3,3,3,3} Steriruncinated 6-cube
Celliprismated hexeract (copox)
153603840
70CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,2,5{4,3,3,3,3} Penticantellated 6-cube
Terirhombated hexeract (topag)
211203840
71CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt0,2,4{4,3,3,3,3} Stericantellated 6-cube
Cellirhombated hexeract (crax)
288005760
72CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,2,3{4,3,3,3,3} Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
134403840
73CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngt0,1,5{4,3,3,3,3} Pentitruncated 6-cube
Teritruncated hexeract (tacog)
86401920
74CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngt0,1,4{4,3,3,3,3} Steritruncated 6-cube
Cellitruncated hexeract (catax)
192003840
75CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,1,3{4,3,3,3,3} Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
172803840
76CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngt0,1,2{4,3,3,3,3} Cantitruncated 6-cube
Great rhombated hexeract (grox)
57601920
77CDel 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.pngt0,1,2,3{3,3,3,3,4} Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
201605760
78CDel 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.pngt0,1,2,4{3,3,3,3,4} Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
4608011520
79CDel 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.pngt0,1,3,4{3,3,3,3,4} Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
4032011520
80CDel 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.pngt0,2,3,4{3,3,3,3,4} Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
4032011520
81CDel 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.pngt1,2,3,4{4,3,3,3,3} Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
3456011520
82CDel 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.pngt0,1,2,5{3,3,3,3,4} Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
307207680
83CDel 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.pngt0,1,3,5{3,3,3,3,4} Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
5184011520
84CDel 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.pngt0,2,3,5{4,3,3,3,3} Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
4608011520
85CDel 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.pngt0,2,3,4{4,3,3,3,3} Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
4032011520
86CDel 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.pngt0,1,4,5{4,3,3,3,3} Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
307207680
87CDel 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.pngt0,1,3,5{4,3,3,3,3} Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
5184011520
88CDel 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.pngt0,1,3,4{4,3,3,3,3} Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
4032011520
89CDel 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.pngt0,1,2,5{4,3,3,3,3} Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
307207680
90CDel 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.pngt0,1,2,4{4,3,3,3,3} Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
4608011520
91CDel 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.pngt0,1,2,3{4,3,3,3,3} Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
230407680
92CDel 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.pngt0,1,2,3,4{3,3,3,3,4} Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
6912023040
93CDel 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.pngt0,1,2,3,5{3,3,3,3,4} Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
8064023040
94CDel 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.pngt0,1,2,4,5{3,3,3,3,4} Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
8064023040
95CDel 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.pngt0,1,2,4,5{4,3,3,3,3} Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
8064023040
96CDel 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.pngt0,1,2,3,5{4,3,3,3,3} Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
8064023040
97CDel 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.pngt0,1,2,3,4{4,3,3,3,3} Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
6912023040
98CDel 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.pngt0,1,2,3,4,5{4,3,3,3,3} Omnitruncated 6-cube
Great teri-hexeractihexacontitetrapeton (gotaxog)
13824046080

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.

# Coxeter diagram NamesBase point
(Alternately signed)
Element countsCircumrad
543210
99CDel 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)44252640640240320.8660254
100CDel 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)766362080320021604802.1794493
101CDel 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)38406401.9364916
102CDel 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)33604801.6583123
103CDel 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)14401921.3228756
104CDel 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)576019203.2787192
105CDel 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)1296028802.95804
106CDel 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)768019202.7838821
107CDel 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)960019202.5980761
108CDel 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)1056019202.3979158
109CDel 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)52809602.1794496
110CDel 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)1728057604.0926762
111CDel 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)2016057603.7080991
112CDel 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)2304057603.4278274
113CDel 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)1536038403.2787192
114CDel 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)34560115204.5552168

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.

# Coxeter diagram NamesElement counts
5-faces4-facesCellsFacesEdgesVertices
115CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 221
Icosiheptaheptacontidipeton (jak)
99648108072021627
116CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png Rectified 221
Rectified icosiheptaheptacontidipeton (rojak)
1261350432050402160216
117CDel 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)
1261350432050402376432
118CDel 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)
34239421512024480151202160
119CDel 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)
3424662162001944086401080
120CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngDemified icosiheptaheptacontidipeton (hejak)3422430720079203240432
121CDel 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
122CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngDemirectified icosiheptaheptacontidipeton (harjak)1080
123CDel 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
124CDel 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
125CDel 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
126CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngDemitruncated icosiheptaheptacontidipeton (hotjak)2160
127CDel 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
128CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngSmall demirhombated icosiheptaheptacontidipeton (shorjak)4320
129CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngSmall prismated icosiheptaheptacontidipeton (spojak)4320
130CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngTritruncated icosiheptaheptacontidipeton (titajak)4320
131CDel 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
132CDel 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
133CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngGreat demirhombated icosiheptaheptacontidipeton (ghorjak)8640
134CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngPrismatotruncated icosiheptaheptacontidipeton (potjak)12960
135CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngDemicellitruncated icosiheptaheptacontidipeton (hictijik)8640
136CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngPrismatorhombated icosiheptaheptacontidipeton (projak)12960
137CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngGreat prismated icosiheptaheptacontidipeton (gapjak)25920
138CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngDemicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)25920
# Coxeter diagram NamesElement counts
5-faces4-facesCellsFacesEdgesVertices
139CDel 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)
547022160216072072
140CDel 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)
12615666480108006480720
141CDel 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)
12622861080019440129602160
142CDel 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)
5584608864064802160270
143CDel 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)
136801440
144CDel 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
145CDel 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
146CDel 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
147CDel 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
148CDel 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
149CDel 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
150CDel 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<