# Uniform 6-polytope

In six-dimensional geometry, a uniform polypeton[1][2] (or uniform 6-polytope) is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

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

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

## History of discovery

• Regular polytopes: (convex faces)
• 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
• Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
• 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[3]
• Convex uniform polytopes:
• 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
• Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra)
• Ongoing: Thousands of nonconvex uniform polypeta are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be, although over 10000 convex and nonconvex uniform polytera are currently known, in particular 923 with 6-simplex symmetry. Participating researchers include Jonathan Bowers, Richard Klitzing and Norman Johnson.[4]

## Uniform 6-polytopes by fundamental Coxeter groups

Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.

There are four fundamental reflective symmety groups which generate 153 unique uniform 6-polytopes.

# Coxeter group Coxeter-Dynkin diagram
1A6 [3,3,3,3,3]
2B6[3,3,3,3,4]
3D6[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
1A5A1 [3,3,3,3,2]Prism family based on 5-simplex
2B5A1[4,3,3,3,2]Prism family based on 5-cube
3aD5A1 [32,1,1,2]Prism family based on 5-demicube
# Coxeter group Notes
4A3I2(p)A1 [3,3,2,p,2]Prism family based on tetrahedral-p-gonal duoprisms
5B3I2(p)A1 [4,3,2,p,2]Prism family based on cubic-p-gonal duoprisms
6H3I2(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
1A4I2(p) [3,3,3,2,p]Family based on 5-cell-p-gonal duoprisms.
2B4I2(p) [4,3,3,2,p]Family based on tesseract-p-gonal duoprisms.
3F4I2(p) [3,4,3,2,p]Family based on 24-cell-p-gonal duoprisms.
4H4I2(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
6A32 [3,3,2,3,3]Family based on tetrahedral duoprisms.
7A3B3 [3,3,2,4,3]Family based on tetrahedral-cubic duoprisms.
8A3H3 [3,3,2,5,3]Family based on tetrahedral-dodecahedral duoprisms.
9B32 [4,3,2,4,3]Family based on cubic duoprisms.
10B3H3 [4,3,2,5,3]Family based on cubic-dodecahedral duoprisms.
11H32 [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
1I2(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 -
• 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 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) 7213535217
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 prismated 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) 10570014701400630105
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)
641922401606012
37 t1{3,3,3,3,4}Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
765761200112048060
38 t2{3,3,3,3,4}Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
76636216028801440160
39 t2{4,3,3,3,3}Birectified 6-cube
Birectified hexeract (brox)
76636208032001920240
40 t1{4,3,3,3,3}Rectified 6-cube
Rectified hexeract (rax)
7644411201520960192
41 t0{4,3,3,3,3}6-cube
Hexeract (ax)
126016024019264
42 t0,1{3,3,3,3,4}Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
7657612001120540120
43 t0,2{3,3,3,3,4}Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
1361656504064003360480
44 t1,2{3,3,3,3,4}Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
1920480
45 t0,3{3,3,3,3,4}Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
7200960
46 t1,3{3,3,3,3,4}Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
86401440
47 t2,3{4,3,3,3,3}Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
3360960
48 t0,4{3,3,3,3,4}Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
5760960
49 t1,4{4,3,3,3,3}Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
115201920
50 t1,3{4,3,3,3,3}Bicantellated 6-cube
Small birhombated hexeract (saborx)
96001920
51 t1,2{4,3,3,3,3}Bitruncated 6-cube
Bitruncated hexeract (botox)
2880960
52 t0,5{4,3,3,3,3}Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
1920384
53 t0,4{4,3,3,3,3}Stericated 6-cube
Small cellated hexeract (scox)
5760960
54 t0,3{4,3,3,3,3}Runcinated 6-cube
Small prismated hexeract (spox)
76801280
55 t0,2{4,3,3,3,3}Cantellated 6-cube
Small rhombated hexeract (srox)
4800960
56 t0,1{4,3,3,3,3}Truncated 6-cube
Truncated hexeract (tox)
76444112015201152384
57 t0,1,2{3,3,3,3,4}Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
3840960
58 t0,1,3{3,3,3,3,4}Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
158402880
59 t0,2,3{3,3,3,3,4}Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
115202880
60 t1,2,3{3,3,3,3,4}Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
100802880
61 t0,1,4{3,3,3,3,4}Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
192003840
62 t0,2,4{3,3,3,3,4}Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
288005760
63 t1,2,4{3,3,3,3,4}Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
230405760
64 t0,3,4{3,3,3,3,4}Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
153603840
65 t1,2,4{4,3,3,3,3}Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
230405760
66 t1,2,3{4,3,3,3,3}Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
115203840
67 t0,1,5{3,3,3,3,4}Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
86401920
68 t0,2,5{3,3,3,3,4}Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
211203840
69 t0,3,4{4,3,3,3,3}Steriruncinated 6-cube
Celliprismated hexeract (copox)
153603840
70 t0,2,5{4,3,3,3,3}Penticantellated 6-cube
Terirhombated hexeract (topag)
211203840
71 t0,2,4{4,3,3,3,3}Stericantellated 6-cube
Cellirhombated hexeract (crax)
288005760
72 t0,2,3{4,3,3,3,3}Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
134403840
73 t0,1,5{4,3,3,3,3}Pentitruncated 6-cube
Teritruncated hexeract (tacog)
86401920
74 t0,1,4{4,3,3,3,3}Steritruncated 6-cube
Cellitruncated hexeract (catax)
192003840
75 t0,1,3{4,3,3,3,3}Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
172803840
76 t0,1,2{4,3,3,3,3}Cantitruncated 6-cube
Great rhombated hexeract (grox)
57601920
77 t0,1,2,3{3,3,3,3,4}Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
201605760
78 t0,1,2,4{3,3,3,3,4}Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
4608011520
79 t0,1,3,4{3,3,3,3,4}Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
4032011520
80 t0,2,3,4{3,3,3,3,4}Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
4032011520
81 t1,2,3,4{4,3,3,3,3}Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
3456011520
82 t0,1,2,5{3,3,3,3,4}Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
307207680
83 t0,1,3,5{3,3,3,3,4}Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
5184011520
84 t0,2,3,5{4,3,3,3,3}Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
4608011520
85 t0,2,3,4{4,3,3,3,3}Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
4032011520
86 t0,1,4,5{4,3,3,3,3}Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
307207680
87 t0,1,3,5{4,3,3,3,3}Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
5184011520
88 t0,1,3,4{4,3,3,3,3}Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
4032011520
89 t0,1,2,5{4,3,3,3,3}Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
307207680
90 t0,1,2,4{4,3,3,3,3}Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
4608011520
91 t0,1,2,3{4,3,3,3,3}Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
230407680
92 t0,1,2,3,4{3,3,3,3,4}Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
6912023040
93 t0,1,2,3,5{3,3,3,3,4}Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
8064023040
94 t0,1,2,4,5{3,3,3,3,4}Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
8064023040
95 t0,1,2,4,5{4,3,3,3,3}Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
8064023040
96 t0,1,2,3,5{4,3,3,3,3}Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
8064023040
97 t0,1,2,3,4{4,3,3,3,3}Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
6912023040
98 t0,1,2,3,4,5{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.

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

# Coxeter diagram Names Base point
(Alternately signed)
543210
99 = 6-demicube
Hemihexeract (hax)
(1,1,1,1,1,1)44252640640240320.8660254
100 = Cantic 6-cube
Truncated hemihexeract (thax)
(1,1,3,3,3,3)766362080320021604802.1794493
101 = Runcic 6-cube
Small rhombated hemihexeract (sirhax)
(1,1,1,3,3,3) 38406401.9364916
102 = Steric 6-cube
Small prismated hemihexeract (sophax)
(1,1,1,1,3,3) 33604801.6583123
103 = Pentic 6-cube
Small cellated demihexeract (sochax)
(1,1,1,1,1,3) 14401921.3228756
104 = Runcicantic 6-cube
Great rhombated hemihexeract (girhax)
(1,1,3,5,5,5) 576019203.2787192
105 = Stericantic 6-cube
Prismatotruncated hemihexeract (pithax)
(1,1,3,3,5,5) 1296028802.95804
106 = Steriruncic 6-cube
Prismatorhombated hemihexeract (prohax)
(1,1,1,3,5,5) 768019202.7838821
107 = Penticantic 6-cube
Cellitruncated hemihexeract (cathix)
(1,1,3,3,3,5) 960019202.5980761
108 = Pentiruncic 6-cube
Cellirhombated hemihexeract (crohax)
(1,1,1,3,3,5) 1056019202.3979158
109 = Pentisteric 6-cube
Celliprismated hemihexeract (cophix)
(1,1,1,1,3,5) 52809602.1794496
110 = Steriruncicantic 6-cube
Great prismated hemihexeract (gophax)
(1,1,3,5,7,7) 1728057604.0926762
111 = Pentiruncicantic 6-cube
Celligreatorhombated hemihexeract (cagrohax)
(1,1,3,5,5,7) 2016057603.7080991
112 = Pentistericantic 6-cube
Celliprismatotruncated hemihexeract (capthix)
(1,1,3,3,5,7) 2304057603.4278274
113 = Pentisteriruncic 6-cube
Celliprismatorhombated hemihexeract (caprohax)
(1,1,1,3,5,7) 1536038403.2787192
114 = 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 Names Element counts
5-faces 4-faces Cells Faces Edges Vertices
115221
Icosiheptaheptacontidipeton (jak)
99648108072021627
116Rectified 221
Rectified icosiheptaheptacontidipeton (rojak)
1261350432050402160216
117Truncated 221
Truncated icosiheptaheptacontidipeton (tojak)
1261350432050402376432
118Cantellated 221
Small rhombated icosiheptaheptacontidipeton (sirjak)
34239421512024480151202160
119Runcinated 221
Small demiprismated icosiheptaheptacontidipeton (shopjak)
3424662162001944086401080
120Demified icosiheptaheptacontidipeton (hejak)3422430720079203240432
121Bitruncated 221
Bitruncated icosiheptaheptacontidipeton (botajik)
2160
122Demirectified icosiheptaheptacontidipeton (harjak)1080
123Cantitruncated 221
Great rhombated icosiheptaheptacontidipeton (girjak)
4320
124Runcitruncated 221
Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)
4320
125Steritruncated 221
Cellitruncated icosiheptaheptacontidipeton (catjak)
2160
126Demitruncated icosiheptaheptacontidipeton (hotjak)2160
127Runcicantellated 221
Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)
6480
128Small demirhombated icosiheptaheptacontidipeton (shorjak)4320
129Small prismated icosiheptaheptacontidipeton (spojak)4320
130Tritruncated icosiheptaheptacontidipeton (titajak)4320
131Runcicantitruncated 221
Great demiprismated icosiheptaheptacontidipeton (ghopjak)
12960
132Stericantitruncated 221
Celligreatorhombated icosiheptaheptacontidipeton (cograjik)
12960
133Great demirhombated icosiheptaheptacontidipeton (ghorjak)8640
134Prismatotruncated icosiheptaheptacontidipeton (potjak)12960
135Demicellitruncated icosiheptaheptacontidipeton (hictijik)8640
136Prismatorhombated icosiheptaheptacontidipeton (projak)12960
137Great prismated icosiheptaheptacontidipeton (gapjak)25920
138Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)25920
# Coxeter diagram Names Element counts
5-faces 4-faces Cells Faces Edges Vertices
139 = 122
Pentacontatetrapeton (mo)
547022160216072072
140 = Rectified 122
Rectified pentacontatetrapeton (ram)
12615666480108006480720
141 = Birectified 122
Birectified pentacontatetrapeton (barm)
12622861080019440129602160
142 = Trirectified 122
Trirectified pentacontatetrapeton (trim)
5584608864064802160270
143 = Truncated 122
Truncated pentacontatetrapeton (tim)
136801440
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

### Non-Wythoffian 6-Polytopes

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

## Regular and uniform honeycombs

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[3[6]]12
2[4,33,4]35
3[4,3,31,1]
[4,33,4,1+]

47 (16 new)
4[31,1,3,31,1]
[1+,4,33,4,1+]

20 (3 new)

Regular and uniform honeycombs include:

Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1x[3[5],2,∞]
2x[4,3,31,1,2,∞]
3x[4,3,3,4,2,∞]
4x[31,1,1,1,2,∞]
5x[3,4,3,3,2,∞]
6xx[4,3,4,2,∞,2,∞]
7xx[4,31,1,2,∞,2,∞]
8xx[3[4],2,∞,2,∞]
9xxx[4,4,2,∞,2,∞,2,∞]
10xxx[6,3,2,∞,2,∞,2,∞]
11xxx[3[3],2,∞,2,∞,2,∞]
12xxxx[∞,2,∞,2,∞,2,∞,2,∞]
13xx[3[3],2,3[3],2,∞]
14xx[3[3],2,4,4,2,∞]
15xx[3[3],2,6,3,2,∞]
16xx[4,4,2,4,4,2,∞]
17xx[4,4,2,6,3,2,∞]
18xx[6,3,2,6,3,2,∞]
19x[3[4],2,3[3]]
20x[4,31,1,2,3[3]]
21x[4,3,4,2,3[3]]
22x[3[4],2,4,4]
23x[4,31,1,2,4,4]
24x[4,3,4,2,4,4]
25x[3[4],2,6,3]
26x[4,31,1,2,6,3]
27x[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 noncompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams.

 = [3,3[5]]: = [(3,3,3,3,3,4)]: = [(3,3,4,3,3,4)]: = [4,3,32,1]: = [3,4,31,1]: = [3,(3,4)1,1]: = [3,3,3,4,3]: = [3,3,4,3,3]: = [3,4,3,3,4]: = [32,1,1,1]: = [4,3,31,1,1]: = [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.

## Notes

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

## References

• 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
• Klitzing, Richard. "6D uniform polytopes (polypeta)".
• Klitzing, Richard. "Uniform polytopes truncation operators".

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family / /
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
Uniform 10-honeycomb {3[10]} δ10 hδ10 qδ10
Uniform n-honeycomb {3[n]} δn hδn qδn 1k22k1k21