# Uniform 6-polytope

 6-simplex Truncated 6-simplex Rectified 6-simplex Cantellated 6-simplex Runcinated 6-simplex Stericated 6-simplex Pentellated 6-simplex 6-orthoplex Truncated 6-orthoplex Rectified 6-orthoplex Cantellated 6-orthoplex Runcinated 6-orthoplex Stericated 6-orthoplex Cantellated 6-cube Runcinated 6-cube Stericated 6-cube Pentellated 6-cube 6-cube Truncated 6-cube Rectified 6-cube 6-demicube Truncated 6-demicube Cantellated 6-demicube Runcinated 6-demicube Stericated 6-demicube 221 122 Truncated 221 Truncated 122

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

The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope.

The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}.

## History of discovery

• Regular polytopes: (convex faces)
• 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
• Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
• 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1]
• Convex uniform polytopes:
• 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
• Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra)
• Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete.[2][3]

## 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
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 families

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-polytopes

• 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 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 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 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
(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
(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
(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
(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
(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
(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
(0,1,2,3,4,5,6) 126 1806 8400 16800 15120 5040

### 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 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 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 Names Base point
(Alternately signed)
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 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 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 = Omnitruncated 122
Great prismated pentacontatetrapeton (gopam)
51840

### Triaprisms

Uniform triaprisms, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2. {4}×{4}×{4} makes a lower symmetry form of the 6-cube.

The extended f-vector is (p,p,1)*(q,q,1)*(r,r,1)=(pqr,3pqr,3pqr+pq+pr+qr,3p(p+1),3p,1).

Coxeter diagram Names Element counts
5-faces 4-faces Cells Faces Edges Vertices
{p}×{q}×{r} [4] p+q+r pq+pr+qr+p+q+r pqr+2(pq+pr+qr) 3pqr+pq+pr+qr 3pqr pqr
{p}×{p}×{p} 3p 3p(p+1) p2(p+6) 3p2(p+1) 3p3 p3
{3}×{3}×{3} (trittip) 9 36 81 99 81 27
{4}×{4}×{4} = 6-cube 12 60 160 240 192 64

### Non-Wythoffian 6-polytopes

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

## Regular and uniform honeycombs

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 $\displaystyle{ {\tilde{A}}_5 }$ [3[6]] 12
2 $\displaystyle{ {\tilde{C}}_5 }$ [4,33,4] 35
3 $\displaystyle{ {\tilde{B}}_5 }$ [4,3,31,1]
[4,33,4,1+]

47 (16 new)
4 $\displaystyle{ {\tilde{D}}_5 }$ [31,1,3,31,1]
[1+,4,33,4,1+]

20 (3 new)

Regular and uniform honeycombs include:

• $\displaystyle{ {\tilde{A}}_5 }$ There are 12 unique uniform honeycombs, including:
• $\displaystyle{ {\tilde{C}}_5 }$ There are 35 uniform honeycombs, including:
• Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,33,4}, =
• $\displaystyle{ {\tilde{B}}_5 }$ There are 47 uniform honeycombs, 16 new, including:
• $\displaystyle{ {\tilde{D}}_5 }$, [31,1,3,31,1]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,33,4}, = . The other two new ones are = , = .
Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1 $\displaystyle{ {\tilde{A}}_4 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[5],2,∞]
2 $\displaystyle{ {\tilde{B}}_4 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,3,31,1,2,∞]
3 $\displaystyle{ {\tilde{C}}_4 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,3,3,4,2,∞]
4 $\displaystyle{ {\tilde{D}}_4 }$x$\displaystyle{ {\tilde{I}}_1 }$ [31,1,1,1,2,∞]
5 $\displaystyle{ {\tilde{F}}_4 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3,4,3,3,2,∞]
6 $\displaystyle{ {\tilde{C}}_3 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,3,4,2,∞,2,∞]
7 $\displaystyle{ {\tilde{B}}_3 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,31,1,2,∞,2,∞]
8 $\displaystyle{ {\tilde{A}}_3 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[4],2,∞,2,∞]
9 $\displaystyle{ {\tilde{C}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,4,2,∞,2,∞,2,∞]
10 $\displaystyle{ {\tilde{H}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [6,3,2,∞,2,∞,2,∞]
11 $\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[3],2,∞,2,∞,2,∞]
12 $\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$x$\displaystyle{ {\tilde{I}}_1 }$ [∞,2,∞,2,∞,2,∞,2,∞]
13 $\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[3],2,3[3],2,∞]
14 $\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{B}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[3],2,4,4,2,∞]
15 $\displaystyle{ {\tilde{A}}_2 }$x$\displaystyle{ {\tilde{G}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [3[3],2,6,3,2,∞]
16 $\displaystyle{ {\tilde{B}}_2 }$x$\displaystyle{ {\tilde{B}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,4,2,4,4,2,∞]
17 $\displaystyle{ {\tilde{B}}_2 }$x$\displaystyle{ {\tilde{G}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [4,4,2,6,3,2,∞]
18 $\displaystyle{ {\tilde{G}}_2 }$x$\displaystyle{ {\tilde{G}}_2 }$x$\displaystyle{ {\tilde{I}}_1 }$ [6,3,2,6,3,2,∞]
19 $\displaystyle{ {\tilde{A}}_3 }$x$\displaystyle{ {\tilde{A}}_2 }$ [3[4],2,3[3]]
20 $\displaystyle{ {\tilde{B}}_3 }$x$\displaystyle{ {\tilde{A}}_2 }$ [4,31,1,2,3[3]]
21 $\displaystyle{ {\tilde{C}}_3 }$x$\displaystyle{ {\tilde{A}}_2 }$ [4,3,4,2,3[3]]
22 $\displaystyle{ {\tilde{A}}_3 }$x$\displaystyle{ {\tilde{B}}_2 }$ [3[4],2,4,4]
23 $\displaystyle{ {\tilde{B}}_3 }$x$\displaystyle{ {\tilde{B}}_2 }$ [4,31,1,2,4,4]
24 $\displaystyle{ {\tilde{C}}_3 }$x$\displaystyle{ {\tilde{B}}_2 }$ [4,3,4,2,4,4]
25 $\displaystyle{ {\tilde{A}}_3 }$x$\displaystyle{ {\tilde{G}}_2 }$ [3[4],2,6,3]
26 $\displaystyle{ {\tilde{B}}_3 }$x$\displaystyle{ {\tilde{G}}_2 }$ [4,31,1,2,6,3]
27 $\displaystyle{ {\tilde{C}}_3 }$x$\displaystyle{ {\tilde{G}}_2 }$ [4,3,4,2,6,3]

### Regular and uniform hyperbolic honeycombs

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

 $\displaystyle{ {\bar{P}}_5 }$ = [3,3[5]]: $\displaystyle{ {\widehat{AU}}_5 }$ = [(3,3,3,3,3,4)]: $\displaystyle{ {\widehat{AR}}_5 }$ = [(3,3,4,3,3,4)]: $\displaystyle{ {\bar{S}}_5 }$ = [4,3,32,1]: $\displaystyle{ {\bar{O}}_5 }$ = [3,4,31,1]: $\displaystyle{ {\bar{N}}_5 }$ = [3,(3,4)1,1]: $\displaystyle{ {\bar{U}}_5 }$ = [3,3,3,4,3]: $\displaystyle{ {\bar{X}}_5 }$ = [3,3,4,3,3]: $\displaystyle{ {\bar{R}}_5 }$ = [3,4,3,3,4]: $\displaystyle{ {\bar{Q}}_5 }$ = [32,1,1,1]: $\displaystyle{ {\bar{M}}_5 }$ = [4,3,31,1,1]: $\displaystyle{ {\bar{L}}_5 }$ = [31,1,1,1,1]:

## Notes on the Wythoff construction for the uniform 6-polytopes

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} Any regular 6-polytope
Rectified t1{p,q,r,s,t} 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} Birectification reduces cells to their duals.
Truncated t0,1{p,q,r,s,t} 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.
Bitruncated t1,2{p,q,r,s,t} Bitrunction transforms cells to their dual truncation.
Tritruncated t2,3{p,q,r,s,t} Tritruncation transforms 4-faces to their dual truncation.
Cantellated t0,2{p,q,r,s,t} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Bicantellated t1,3{p,q,r,s,t} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Runcinated t0,3{p,q,r,s,t} Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated t1,4{p,q,r,s,t} Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s,t} 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} 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} All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied.

• List of regular polytopes#Higher dimensions

## Notes

1. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
2. Uniform Polypeta, Jonathan Bowers
3. Uniform polytope

## References

• 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
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