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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}.

Contents

History of discoveryEdit

  • 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 polypeta 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 groupsEdit

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
1 A6 [3,3,3,3,3]            
2 B6 [3,3,3,3,4]            
3 D6 [3,3,3,31,1]          
4 E6 [32,2,1]          
[3,32,2]        
 
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 familiesEdit

Uniform prism

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

# Coxeter group Notes
1 A5A1 [3,3,3,3,2]             Prism family based on 5-simplex
2 B5A1 [4,3,3,3,2]             Prism family based on 5-cube
3a D5A1 [32,1,1,2]           Prism family based on 5-demicube
# Coxeter group Notes
4 A3I2(p)A1 [3,3,2,p,2]             Prism family based on tetrahedral-p-gonal duoprisms
5 B3I2(p)A1 [4,3,2,p,2]             Prism family based on cubic-p-gonal duoprisms
6 H3I2(p)A1 [5,3,2,p,2]             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]             Family based on 5-cell-p-gonal duoprisms.
2 B4I2(p) [4,3,3,2,p]             Family based on tesseract-p-gonal duoprisms.
3 F4I2(p) [3,4,3,2,p]             Family based on 24-cell-p-gonal duoprisms.
4 H4I2(p) [5,3,3,2,p]             Family based on 120-cell-p-gonal duoprisms.
5 D4I2(p) [31,1,1,2,p]           Family based on demitesseract-p-gonal duoprisms.
# Coxeter group Notes
6 A32 [3,3,2,3,3]             Family based on tetrahedral duoprisms.
7 A3B3 [3,3,2,4,3]             Family based on tetrahedral-cubic duoprisms.
8 A3H3 [3,3,2,5,3]             Family based on tetrahedral-dodecahedral duoprisms.
9 B32 [4,3,2,4,3]             Family based on cubic duoprisms.
10 B3H3 [4,3,2,5,3]             Family based on cubic-dodecahedral duoprisms.
11 H32 [5,3,2,5,3]             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]             Family based on p,q,r-gonal triprisms

Enumerating the convex uniform 6-polytopesEdit

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

These fundamental families generate 153 nonprismatic convex uniform polypeta.

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

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

The B6 familyEdit

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

The D6 familyEdit

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

The E6 familyEdit

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