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.

Graphs of three regular and related uniform polytopes


Truncated 6-simplex

Rectified 6-simplex

Cantellated 6-simplex

Runcinated 6-simplex

Stericated 6-simplex

Pentellated 6-simplex


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


Truncated 6-cube

Rectified 6-cube


Truncated 6-demicube

Cantellated 6-demicube

Runcinated 6-demicube

Stericated 6-demicube



Truncated 221

Truncated 122

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    • Ongoing: Thousands of nonconvex uniform polypeta are known, but mostly unpublished.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 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 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
4 E6 [32,2,1]

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.
5D4I2(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 - , 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 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
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) 146314017510521
3 Truncated 6-simplex
truncated heptapeton (til)
(0,0,0,0,0,1,2) 146314017512642
4 Birectified 6-simplex
birectified heptapeton (bril)
(0,0,0,0,1,1,1) 148424535021035
5 Cantellated 6-simplex
small rhombated heptapeton (sril)
(0,0,0,0,1,1,2) 35210560805525105
6 Bitruncated 6-simplex
bitruncated heptapeton (batal)
(0,0,0,0,1,2,2) 1484245385315105
7 Cantitruncated 6-simplex
great rhombated heptapeton (gril)
(0,0,0,0,1,2,3) 35210560805630210
8 Runcinated 6-simplex
small prismated heptapeton (spil)
(0,0,0,1,1,1,2) 7045513301610840140
9 Bicantellated 6-simplex
small birhombated heptapeton (sabril)
(0,0,0,1,1,2,2) 7045512951610840140
10 Runcitruncated 6-simplex
prismatotruncated heptapeton (patal)
(0,0,0,1,1,2,3) 70560182028001890420
11 Tritruncated 6-simplex
tetradecapeton (fe)
(0,0,0,1,2,2,2) 1484280490420140
12 Runcicantellated 6-simplex
prismatorhombated heptapeton (pril)
(0,0,0,1,2,2,3) 70455129519601470420
13 Bicantitruncated 6-simplex
great birhombated heptapeton (gabril)
(0,0,0,1,2,3,3) 4932998015401260420
14 Runcicantitruncated 6-simplex
great prismated heptapeton (gapil)
(0,0,0,1,2,3,4) 70560182030102520840
15 Stericated 6-simplex
small cellated heptapeton (scal)
(0,0,1,1,1,1,2) 10570014701400630105
16 Biruncinated 6-simplex
small biprismato-tetradecapeton (sibpof)
(0,0,1,1,1,2,2) 84714210025201260210
17 Steritruncated 6-simplex
cellitruncated heptapeton (catal)
(0,0,1,1,1,2,3) 105945294037802100420
18 Stericantellated 6-simplex
cellirhombated heptapeton (cral)
(0,0,1,1,2,2,3) 1051050346550403150630
19 Biruncitruncated 6-simplex
biprismatorhombated heptapeton (bapril)
(0,0,1,1,2,3,3) 84714231035702520630
20 Stericantitruncated 6-simplex
celligreatorhombated heptapeton (cagral)
(0,0,1,1,2,3,4) 10511554410714050401260
21 Steriruncinated 6-simplex
celliprismated heptapeton (copal)
(0,0,1,2,2,2,3) 105700199526601680420
22 Steriruncitruncated 6-simplex
celliprismatotruncated heptapeton (captal)
(0,0,1,2,2,3,4) 1059453360567044101260
23 Steriruncicantellated 6-simplex
celliprismatorhombated heptapeton (copril)
(0,0,1,2,3,3,4) 10510503675588044101260
24 Biruncicantitruncated 6-simplex
great biprismato-tetradecapeton (gibpof)
(0,0,1,2,3,4,4) 847142520441037801260
25 Steriruncicantitruncated 6-simplex
great cellated heptapeton (gacal)
(0,0,1,2,3,4,5) 10511554620861075602520
26 Pentellated 6-simplex
small teri-tetradecapeton (staff)
(0,1,1,1,1,1,2) 12643463049021042
27 Pentitruncated 6-simplex
teracellated heptapeton (tocal)
(0,1,1,1,1,2,3) 12682617851820945210
28 Penticantellated 6-simplex
teriprismated heptapeton (topal)
(0,1,1,1,2,2,3) 1261246357043402310420
29 Penticantitruncated 6-simplex
terigreatorhombated heptapeton (togral)
(0,1,1,1,2,3,4) 1261351409553903360840
30 Pentiruncitruncated 6-simplex
tericellirhombated heptapeton (tocral)
(0,1,1,2,2,3,4) 12614915565861056701260
31 Pentiruncicantellated 6-simplex
teriprismatorhombi-tetradecapeton (taporf)
(0,1,1,2,3,3,4) 12615965250756050401260
32 Pentiruncicantitruncated 6-simplex
terigreatoprismated heptapeton (tagopal)
(0,1,1,2,3,4,5) 126170168251155088202520
33 Pentisteritruncated 6-simplex
tericellitrunki-tetradecapeton (tactaf)
(0,1,2,2,2,3,4) 1261176378052503360840
34 Pentistericantitruncated 6-simplex
tericelligreatorhombated heptapeton (tacogral)
(0,1,2,2,3,4,5) 126159665101134088202520
35 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 Names Element counts
36 t0{3,3,3,3,4}6-orthoplex
Hexacontatetrapeton (gee)
37 t1{3,3,3,3,4}Rectified 6-orthoplex
Rectified hexacontatetrapeton (rag)
38 t2{3,3,3,3,4}Birectified 6-orthoplex
Birectified hexacontatetrapeton (brag)
39 t2{4,3,3,3,3}Birectified 6-cube
Birectified hexeract (brox)
40 t1{4,3,3,3,3}Rectified 6-cube
Rectified hexeract (rax)
41 t0{4,3,3,3,3}6-cube
Hexeract (ax)
42 t0,1{3,3,3,3,4}Truncated 6-orthoplex
Truncated hexacontatetrapeton (tag)
43 t0,2{3,3,3,3,4}Cantellated 6-orthoplex
Small rhombated hexacontatetrapeton (srog)
44 t1,2{3,3,3,3,4}Bitruncated 6-orthoplex
Bitruncated hexacontatetrapeton (botag)
45 t0,3{3,3,3,3,4}Runcinated 6-orthoplex
Small prismated hexacontatetrapeton (spog)
46 t1,3{3,3,3,3,4}Bicantellated 6-orthoplex
Small birhombated hexacontatetrapeton (siborg)
47 t2,3{4,3,3,3,3}Tritruncated 6-cube
Hexeractihexacontitetrapeton (xog)
48 t0,4{3,3,3,3,4}Stericated 6-orthoplex
Small cellated hexacontatetrapeton (scag)
49 t1,4{4,3,3,3,3}Biruncinated 6-cube
Small biprismato-hexeractihexacontitetrapeton (sobpoxog)
50 t1,3{4,3,3,3,3}Bicantellated 6-cube
Small birhombated hexeract (saborx)
51 t1,2{4,3,3,3,3}Bitruncated 6-cube
Bitruncated hexeract (botox)
52 t0,5{4,3,3,3,3}Pentellated 6-cube
Small teri-hexeractihexacontitetrapeton (stoxog)
53 t0,4{4,3,3,3,3}Stericated 6-cube
Small cellated hexeract (scox)
54 t0,3{4,3,3,3,3}Runcinated 6-cube
Small prismated hexeract (spox)
55 t0,2{4,3,3,3,3}Cantellated 6-cube
Small rhombated hexeract (srox)
56 t0,1{4,3,3,3,3}Truncated 6-cube
Truncated hexeract (tox)
57 t0,1,2{3,3,3,3,4}Cantitruncated 6-orthoplex
Great rhombated hexacontatetrapeton (grog)
58 t0,1,3{3,3,3,3,4}Runcitruncated 6-orthoplex
Prismatotruncated hexacontatetrapeton (potag)
59 t0,2,3{3,3,3,3,4}Runcicantellated 6-orthoplex
Prismatorhombated hexacontatetrapeton (prog)
60 t1,2,3{3,3,3,3,4}Bicantitruncated 6-orthoplex
Great birhombated hexacontatetrapeton (gaborg)
61 t0,1,4{3,3,3,3,4}Steritruncated 6-orthoplex
Cellitruncated hexacontatetrapeton (catog)
62 t0,2,4{3,3,3,3,4}Stericantellated 6-orthoplex
Cellirhombated hexacontatetrapeton (crag)
63 t1,2,4{3,3,3,3,4}Biruncitruncated 6-orthoplex
Biprismatotruncated hexacontatetrapeton (boprax)
64 t0,3,4{3,3,3,3,4}Steriruncinated 6-orthoplex
Celliprismated hexacontatetrapeton (copog)
65 t1,2,4{4,3,3,3,3}Biruncitruncated 6-cube
Biprismatotruncated hexeract (boprag)
66 t1,2,3{4,3,3,3,3}Bicantitruncated 6-cube
Great birhombated hexeract (gaborx)
67 t0,1,5{3,3,3,3,4}Pentitruncated 6-orthoplex
Teritruncated hexacontatetrapeton (tacox)
68 t0,2,5{3,3,3,3,4}Penticantellated 6-orthoplex
Terirhombated hexacontatetrapeton (tapox)
69 t0,3,4{4,3,3,3,3}Steriruncinated 6-cube
Celliprismated hexeract (copox)
70 t0,2,5{4,3,3,3,3}Penticantellated 6-cube
Terirhombated hexeract (topag)
71 t0,2,4{4,3,3,3,3}Stericantellated 6-cube
Cellirhombated hexeract (crax)
72 t0,2,3{4,3,3,3,3}Runcicantellated 6-cube
Prismatorhombated hexeract (prox)
73 t0,1,5{4,3,3,3,3}Pentitruncated 6-cube
Teritruncated hexeract (tacog)
74 t0,1,4{4,3,3,3,3}Steritruncated 6-cube
Cellitruncated hexeract (catax)
75 t0,1,3{4,3,3,3,3}Runcitruncated 6-cube
Prismatotruncated hexeract (potax)
76 t0,1,2{4,3,3,3,3}Cantitruncated 6-cube
Great rhombated hexeract (grox)
77 t0,1,2,3{3,3,3,3,4}Runcicantitruncated 6-orthoplex
Great prismated hexacontatetrapeton (gopog)
78 t0,1,2,4{3,3,3,3,4}Stericantitruncated 6-orthoplex
Celligreatorhombated hexacontatetrapeton (cagorg)
79 t0,1,3,4{3,3,3,3,4}Steriruncitruncated 6-orthoplex
Celliprismatotruncated hexacontatetrapeton (captog)
80 t0,2,3,4{3,3,3,3,4}Steriruncicantellated 6-orthoplex
Celliprismatorhombated hexacontatetrapeton (coprag)
81 t1,2,3,4{4,3,3,3,3}Biruncicantitruncated 6-cube
Great biprismato-hexeractihexacontitetrapeton (gobpoxog)
82 t0,1,2,5{3,3,3,3,4}Penticantitruncated 6-orthoplex
Terigreatorhombated hexacontatetrapeton (togrig)
83 t0,1,3,5{3,3,3,3,4}Pentiruncitruncated 6-orthoplex
Teriprismatotruncated hexacontatetrapeton (tocrax)
84 t0,2,3,5{4,3,3,3,3}Pentiruncicantellated 6-cube
Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog)
85 t0,2,3,4{4,3,3,3,3}Steriruncicantellated 6-cube
Celliprismatorhombated hexeract (coprix)
86 t0,1,4,5{4,3,3,3,3}Pentisteritruncated 6-cube
Tericelli-hexeractihexacontitetrapeton (tactaxog)
87 t0,1,3,5{4,3,3,3,3}Pentiruncitruncated 6-cube
Teriprismatotruncated hexeract (tocrag)
88 t0,1,3,4{4,3,3,3,3}Steriruncitruncated 6-cube
Celliprismatotruncated hexeract (captix)
89 t0,1,2,5{4,3,3,3,3}Penticantitruncated 6-cube
Terigreatorhombated hexeract (togrix)
90 t0,1,2,4{4,3,3,3,3}Stericantitruncated 6-cube
Celligreatorhombated hexeract (cagorx)
91 t0,1,2,3{4,3,3,3,3}Runcicantitruncated 6-cube
Great prismated hexeract (gippox)
92 t0,1,2,3,4{3,3,3,3,4}Steriruncicantitruncated 6-orthoplex
Great cellated hexacontatetrapeton (gocog)
93 t0,1,2,3,5{3,3,3,3,4}Pentiruncicantitruncated 6-orthoplex
Terigreatoprismated hexacontatetrapeton (tagpog)
94 t0,1,2,4,5{3,3,3,3,4}Pentistericantitruncated 6-orthoplex
Tericelligreatorhombated hexacontatetrapeton (tecagorg)
95 t0,1,2,4,5{4,3,3,3,3}Pentistericantitruncated 6-cube
Tericelligreatorhombated hexeract (tocagrax)
96 t0,1,2,3,5{4,3,3,3,3}Pentiruncicantitruncated 6-cube
Terigreatoprismated hexeract (tagpox)
97 t0,1,2,3,4{4,3,3,3,3}Steriruncicantitruncated 6-cube
Great cellated hexeract (gocax)
98 t0,1,2,3,4,5{4,3,3,3,3}Omnitruncated 6-cube
Great teri-hexeractihexacontitetrapeton (gotaxog)

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)
Element counts Circumrad
99 = 6-demicube
Hemihexeract (hax)
100 = Cantic 6-cube
Truncated hemihexeract (thax)
101 = Runcic 6-cube
Small rhombated hemihexeract (sirhax)
102 = Steric 6-cube
Small prismated hemihexeract (sophax)
103 = Pentic 6-cube
Small cellated demihexeract (sochax)
104 = Runcicantic 6-cube
Great rhombated hemihexeract (girhax)
105 = Stericantic 6-cube
Prismatotruncated hemihexeract (pithax)
106 = Steriruncic 6-cube
Prismatorhombated hemihexeract (prohax)
107 = Penticantic 6-cube
Cellitruncated hemihexeract (cathix)
108 = Pentiruncic 6-cube
Cellirhombated hemihexeract (crohax)
109 = Pentisteric 6-cube
Celliprismated hemihexeract (cophix)
110 = Steriruncicantic 6-cube
Great prismated hemihexeract (gophax)
111 = Pentiruncicantic 6-cube
Celligreatorhombated hemihexeract (cagrohax)
112 = Pentistericantic 6-cube
Celliprismatotruncated hemihexeract (capthix)
113 = Pentisteriruncic 6-cube
Celliprismatorhombated hemihexeract (caprohax)
114 = Pentisteriruncicantic 6-cube
Great cellated hemihexeract (gochax)

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
Icosiheptaheptacontidipeton (jak)
116Rectified 221
Rectified icosiheptaheptacontidipeton (rojak)
117Truncated 221
Truncated icosiheptaheptacontidipeton (tojak)
118Cantellated 221
Small rhombated icosiheptaheptacontidipeton (sirjak)
119Runcinated 221
Small demiprismated icosiheptaheptacontidipeton (shopjak)
120Demified icosiheptaheptacontidipeton (hejak)3422430720079203240432
121Bitruncated 221
Bitruncated icosiheptaheptacontidipeton (botajik)
122Demirectified icosiheptaheptacontidipeton (harjak)1080
123Cantitruncated 221
Great rhombated icosiheptaheptacontidipeton (girjak)
124Runcitruncated 221
Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak)
125Steritruncated 221
Cellitruncated icosiheptaheptacontidipeton (catjak)
126Demitruncated icosiheptaheptacontidipeton (hotjak)2160
127Runcicantellated 221
Demiprismatorhombated icosiheptaheptacontidipeton (haprojak)
128Small demirhombated icosiheptaheptacontidipeton (shorjak)4320
129Small prismated icosiheptaheptacontidipeton (spojak)4320
130Tritruncated icosiheptaheptacontidipeton (titajak)4320
131Runcicantitruncated 221
Great demiprismated icosiheptaheptacontidipeton (ghopjak)
132Stericantitruncated 221
Celligreatorhombated icosiheptaheptacontidipeton (cograjik)
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)
140 = Rectified 122
Rectified pentacontatetrapeton (ram)
141 = Birectified 122
Birectified pentacontatetrapeton (barm)
142 = Trirectified 122
Trirectified pentacontatetrapeton (trim)
143 = Truncated 122
Truncated pentacontatetrapeton (tim)
144 = Bitruncated 122
Bitruncated pentacontatetrapeton (bitem)
145 = Tritruncated 122
Tritruncated pentacontatetrapeton (titam)
146 = Cantellated 122
Small rhombated pentacontatetrapeton (sram)
147 = Cantitruncated 122
Great rhombated pentacontatetrapeton (gram)
148 = Runcinated 122
Small prismated pentacontatetrapeton (spam)
149 = Bicantellated 122
Small birhombated pentacontatetrapeton (sabrim)
150 = Bicantitruncated 122
Great birhombated pentacontatetrapeton (gabrim)
151 = Runcitruncated 122
Prismatotruncated pentacontatetrapeton (patom)
152 = Runcicantellated 122
Prismatorhombated pentacontatetrapeton (prom)
153 = Omnitruncated 122
Great prismated pentacontatetrapeton (gopam)

Non-Wythoffian 6-Polytopes

In 6 dimensions and above, there are an 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

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

47 (16 new)

20 (3 new)

Regular and uniform honeycombs include:

Prismatic groups
# Coxeter group Coxeter-Dynkin diagram

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.

Hyperbolic noncompact groups

= [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
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.

See also

  • List of regular polytopes#Higher dimensions


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


  • 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 / 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 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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