Paracompact uniform honeycombs

In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.

Example paracompact regular honeycombs

{3,3,6}

{6,3,3}

{4,3,6}

{6,3,4}

{5,3,6}

{6,3,5}

{6,3,6}

{3,6,3}

{4,4,3}

{3,4,4}

{4,4,4}

Regular paracompact honeycombs

Of the uniform paracompact H3 honeycombs, 11 are regular, meaning that their group of symmetries acts transitively on their flags. These have Schläfli symbol {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}, and are shown below. Four have finite Ideal polyhedral cells: {3,3,6}, {4,3,6}, {3,4,4}, and {5,3,6}.

11 paracompact regular honeycombs

{6,3,3}

{6,3,4}

{6,3,5}

{6,3,6}

{4,4,3}

{4,4,4}

{3,3,6}

{4,3,6}

{5,3,6}

{3,6,3}

{3,4,4}
Name Schläfli
Symbol
{p,q,r}
Coxeter
Cell
type
{p,q}
Face
type
{p}
Edge
figure
{r}
Vertex
figure

{q,r}
Dual Coxeter
group
Order-6 tetrahedral honeycomb{3,3,6}{3,3}{3}{6}{3,6}{6,3,3}[6,3,3]
Hexagonal tiling honeycomb{6,3,3}{6,3}{6}{3}{3,3}{3,3,6}
Order-4 octahedral honeycomb{3,4,4}{3,4}{3}{4}{4,4}{4,4,3}[4,4,3]
Square tiling honeycomb{4,4,3}{4,4}{4}{3}{4,3}{3,4,4}
Triangular tiling honeycomb{3,6,3}{3,6}{3}{3}{6,3}Self-dual[3,6,3]
Order-6 cubic honeycomb{4,3,6}{4,3}{4}{4}{3,6}{6,3,4}[6,3,4]
Order-4 hexagonal tiling honeycomb{6,3,4}{6,3}{6}{4}{3,4}{4,3,6}
Order-4 square tiling honeycomb{4,4,4}{4,4}{4}{4}{4,4}Self-dual[4,4,4]
Order-6 dodecahedral honeycomb{5,3,6}{5,3}{5}{5}{3,6}{6,3,5}[6,3,5]
Order-5 hexagonal tiling honeycomb{6,3,5}{6,3}{6}{5}{3,5}{5,3,6}
Order-6 hexagonal tiling honeycomb{6,3,6}{6,3}{6}{6}{3,6}Self-dual[6,3,6]

Coxeter groups of paracompact uniform honeycombs

These graphs show subgroup relations of paracompact hyperbolic Coxeter groups. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry.

This is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains (rank 4 paracompact coxeter groups). The honeycombs are indexed here for cross-referencing duplicate forms, with brackets around the nonprimary constructions.

The alternations are listed, but are either repeats or don't generate uniform solutions. Single-hole alternations represent a mirror removal operation. If an end-node is removed, another simplex (tetrahedral) family is generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored. These nonsimplectic (pyramidal) Coxeter groups are not enumerated on this page, except as special cases of half groups of the tetrahedral ones. Six uniform honeycombs that arise here as alternations have been numbered 152 to 157, after the 151 Wythoffian forms not requiring alternation for their construction.

Tetrahedral hyperbolic paracompact group summary
Coxeter group Simplex
volume
Commutator subgroup Unique honeycomb count
[6,3,3]0.0422892336[1+,6,(3,3)+] = [3,3[3]]+15
[4,4,3]0.0763304662[1+,4,1+,4,3+]15
[3,3[3]]0.0845784672[3,3[3]]+4
[6,3,4]0.1057230840[1+,6,3+,4,1+] = [3[]x[]]+15
[3,41,1]0.1526609324[3+,41+,1+]4
[3,6,3]0.1691569344[3+,6,3+]8
[6,3,5]0.1715016613[1+,6,(3,5)+] = [5,3[3]]+15
[6,31,1]0.2114461680[1+,6,(31,1)+] = [3[]x[]]+4
[4,3[3]]0.2114461680[1+,4,3[3]]+ = [3[]x[]]+4
[4,4,4]0.2289913985[4+,4+,4+]+6
[6,3,6]0.2537354016[1+,6,3+,6,1+] = [3[3,3]]+8
[(4,4,3,3)]0.3053218647[(4,1+,4,(3,3)+)]4
[5,3[3]]0.3430033226[5,3[3]]+4
[(6,3,3,3)]0.3641071004[(6,3,3,3)]+9
[3[]x[]]0.4228923360[3[]x[]]+1
[41,1,1]0.4579827971[1+,41+,1+,1+]0
[6,3[3]]0.5074708032[1+,6,3[3]] = [3[3,3]]+2
[(6,3,4,3)]0.5258402692[(6,3+,4,3+)]9
[(4,4,4,3)]0.5562821156[(4,1+,4,1+,4,3+)]9
[(6,3,5,3)]0.6729858045[(6,3,5,3)]+9
[(6,3,6,3)]0.8457846720[(6,3+,6,3+)]5
[(4,4,4,4)]0.9159655942[(4+,4+,4+,4+)]1
[3[3,3]]1.014916064[3[3,3]]+0

The complete list of nonsimplectic (non-tetrahedral) paracompact Coxeter groups was published by P. Tumarkin in 2003.[1] The smallest paracompact form in H3 can be represented by or , or [,3,3,] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4] : = . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is or , constructed as [4,4,1+,4] = [,4,4,] : = .

Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1+,4)] = [((3,,3)),((3,,3))] or , [(3,4,4,1+,4)] = [((4,,3)),((3,,4))] or , [(4,4,4,1+,4)] = [((4,,4)),((4,,4))] or . = , = , = .

Another nonsimplectic half groups is .

A radical nonsimplectic subgroup is , which can be doubled into a triangular prism domain as .

Pyramidal hyperbolic paracompact group summary
Dimension Rank Graphs
H3 5

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Linear graphs

[6,3,3] family

# Honeycomb name
Coxeter diagram:
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
1
2
3
4
1 hexagonal

{6,3,3}
- - - (4)

(6.6.6)

Tetrahedron
2 rectified hexagonal

t1{6,3,3} or r{6,3,3}
(2)

(3.3.3)
- - (3)

(3.6.3.6)

Triangular prism
3 rectified order-6 tetrahedral

t1{3,3,6} or r{3,3,6}
(6)

(3.3.3.3)
- - (2)

(3.3.3.3.3.3)

Hexagonal prism
4 order-6 tetrahedral

{3,3,6}
(∞)

(3.3.3)
- - -
Triangular tiling
5 truncated hexagonal

t0,1{6,3,3} or t{6,3,3}
(1)

(3.3.3)
- - (3)

(3.12.12)

Triangular pyramid
6 cantellated hexagonal

t0,2{6,3,3} or rr{6,3,3}
(1)

3.3.3.3
(2)

(4.4.3)
- (2)

(3.4.6.4)
7 runcinated hexagonal

t0,3{6,3,3}
(1)

(3.3.3)
(3)

(4.4.3)
(3)

(4.4.6)
(1)

(6.6.6)
8 cantellated order-6 tetrahedral

t0,2{3,3,6} or rr{3,3,6}
(1)

(3.4.3.4)
- (2)

(4.4.6)
(2)

(3.6.3.6)
9 bitruncated hexagonal

t1,2{6,3,3} or 2t{6,3,3}
(2)

(3.6.6)
- - (2)

(6.6.6)
10 truncated order-6 tetrahedral

t0,1{3,3,6} or t{3,3,6}
(6)

(3.6.6)
- - (1)

(3.3.3.3.3.3)
11 cantitruncated hexagonal

t0,1,2{6,3,3} or tr{6,3,3}
(1)

(3.6.6)
(1)

(4.4.3)
- (2)

(4.6.12)
12 runcitruncated hexagonal

t0,1,3{6,3,3}
(1)

(3.4.3.4)
(2)

(4.4.3)
(1)

(4.4.12)
(1)

(3.12.12)
13 runcitruncated order-6 tetrahedral

t0,1,3{3,3,6}
(1)

(3.6.6)
(1)

(4.4.6)
(2)

(4.4.6)
(1)

(3.4.6.4)
14 cantitruncated order-6 tetrahedral

t0,1,2{3,3,6} or tr{3,3,6}
(2)

(4.6.6)
- (1)

(4.4.6)
(1)

(6.6.6)
15 omnitruncated hexagonal

t0,1,2,3{6,3,3}
(1)

(4.6.6)
(1)

(4.4.6)
(1)

(4.4.12)
(1)

(4.6.12)
Alternated forms
# Honeycomb name
Coxeter diagram:
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
1
2
3
4
Alt
[137] alternated hexagonal
() =
- - (4)

(3.3.3.3.3.3)
(4)

(3.3.3)

(3.6.6)
[138] cantic hexagonal
(1)

(3.3.3.3)
- (2)

(3.6.3.6)
(2)

(3.6.6)
[139] runcic hexagonal
(1)

(4.4.4)
(1)

(4.4.3)
(1)

(3.3.3.3.3.3)
(3)

(3.4.3.4)
[140] runcicantic hexagonal
(1)

(3.6.6)
(1)

(4.4.3)
(1)

(3.6.3.6)
(2)

(4.6.6)
Nonuniform snub rectified order-6 tetrahedral

sr{3,3,6}

Irr. (3.3.3)
Nonuniform cantic snub order-6 tetrahedral

sr3{3,3,6}
Nonuniform omnisnub order-6 tetrahedral

ht0,1,2,3{6,3,3}

Irr. (3.3.3)

[6,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or

# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
1
2
3
16 (Regular) order-4 hexagonal

{6,3,4}
- - - (8)


(6.6.6)

(3.3.3.3)
17 rectified order-4 hexagonal

t1{6,3,4} or r{6,3,4}
(2)


(3.3.3.3)
- - (4)


(3.6.3.6)

(4.4.4)
18 rectified order-6 cubic

t1{4,3,6} or r{4,3,6}
(6)


(3.4.3.4)
- - (2)


(3.3.3.3.3.3)

(6.4.4)
19 order-6 cubic

{4,3,6}
(20)


(4.4.4)
- - -
(3.3.3.3.3.3)
20 truncated order-4 hexagonal

t0,1{6,3,4} or t{6,3,4}
(1)


(3.3.3.3)
- - (4)


(3.12.12)
21 bitruncated order-6 cubic

t1,2{6,3,4} or 2t{6,3,4}
(2)


(4.6.6)
- - (2)


(6.6.6)
22 truncated order-6 cubic

t0,1{4,3,6} or t{4,3,6}
(6)


(3.8.8)
- - (1)


(3.3.3.3.3.3)
23 cantellated order-4 hexagonal

t0,2{6,3,4} or rr{6,3,4}
(1)


(3.4.3.4)
(2)


(4.4.4)
- (2)


(3.4.6.4)
24 cantellated order-6 cubic

t0,2{4,3,6} or rr{4,3,6}
(2)


(3.4.4.4)
- (2)


(4.4.6)
(1)


(3.6.3.6)
25 runcinated order-6 cubic

t0,3{6,3,4}
(1)


(4.4.4)
(3)


(4.4.4)
(3)


(4.4.6)
(1)


(6.6.6)
26 cantitruncated order-4 hexagonal

t0,1,2{6,3,4} or tr{6,3,4}
(1)


(4.6.6)
(1)


(4.4.4)
- (2)


(4.6.12)
27 cantitruncated order-6 cubic

t0,1,2{4,3,6} or tr{4,3,6}
(2)


(4.6.8)
- (1)


(4.4.6)
(1)


(6.6.6)
28 runcitruncated order-4 hexagonal

t0,1,3{6,3,4}
(1)


(3.4.4.4)
(1)


(4.4.4)
(2)


(4.4.12)
(1)


(3.12.12)
29 runcitruncated order-6 cubic

t0,1,3{4,3,6}
(1)


(3.8.8)
(2)


(4.4.8)
(1)


(4.4.6)
(1)


(3.4.6.4)
30 omnitruncated order-6 cubic

t0,1,2,3{6,3,4}
(1)


(4.6.8)
(1)


(4.4.8)
(1)


(4.4.12)
(1)


(4.6.12)
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
1
2
3
Alt
[87] alternated order-6 cubic

h{4,3,6}

(3.3.3)
   
(3.3.3.3.3.3)


(3.6.3.6)
[88] cantic order-6 cubic

h2{4,3,6}
(2)

(3.6.6)
- - (1)

(3.6.3.6)
(2)

(6.6.6)
[89] runcic order-6 cubic

h3{4,3,6}
(1)

(3.3.3)
- - (1)

(6.6.6)
(3)

(3.4.6.4)
[90] runcicantic order-6 cubic

h2,3{4,3,6}
(1)

(3.6.6)
- - (1)

(3.12.12)
(2)

(4.6.12)
[141] alternated order-4 hexagonal

h{6,3,4}
- -
(3.3.3.3.3.3)

(3.3.3.3)

(4.6.6)
[142] cantic order-4 hexagonal

h1{6,3,4}
(1)

(3.4.3.4)
- (2)

(3.6.3.6)
(2)

(4.6.6)
[143] runcic order-4 hexagonal

h3{6,3,4}
(1)

(4.4.4)
(1)

(4.4.3)
(1)

(3.3.3.3.3.3)
(3)

(3.4.4.4)
[144] runcicantic order-4 hexagonal

h2,3{6,3,4}
(1)

(3.8.8)
(1)

(4.4.3)
(1)

(3.6.3.6)
(2)

(4.6.8)
[151] quarter order-4 hexagonal

q{6,3,4}
(3)
(1)
- (1)
(3)
Nonuniform bisnub order-6 cubic

2s{4,3,6}


(3.3.3.3.3.3)
- -

(3.3.3.3.6)

+(3.3.3)
Nonuniform runcic bisnub order-6 cubic
Nonuniform snub rectified order-6 cubic

sr{4,3,6}


(3.3.3.3.3)


(3.3.3)


(3.3.3.3)


(3.3.3.3.6)

+(3.3.3)
Nonuniform runcic snub rectified order-6 cubic

sr3{4,3,6}
Nonuniform snub rectified order-4 hexagonal

sr{6,3,4}


(3.3.3.3.3.3)


(3.3.3)
-

(3.3.3.3.6)

+(3.3.3)
Nonuniform runcisnub rectified order-4 hexagonal

sr3{6,3,4}
Nonuniform omnisnub rectified order-6 cubic

ht0,1,2,3{6,3,4}


(3.3.3.3.4)


(3.3.3.4)


(3.3.3.6)


(3.3.3.3.6)

+(3.3.3)

[6,3,5] family

# Honeycomb name
Coxeter diagram
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
31 order-5 hexagonal

{6,3,5}
- - - (20)

(6)3

Icosahedron
32 rectified order-5 hexagonal

t1{6,3,5} or r{6,3,5}
(2)

(3.3.3.3.3)
- - (5)

(3.6)2

(5.4.4)
33 rectified order-6 dodecahedral

t1{5,3,6} or r{5,3,6}
(5)

(3.5.3.5)
- - (2)

(3)6

(6.4.4)
34 order-6 dodecahedral

{5,3,6}

(5.5.5)
- - - ()

(3)6
35 truncated order-5 hexagonal

t0,1{6,3,5} or t{6,3,5}
(1)

(3.3.3.3.3)
- - (5)

3.12.12
36 cantellated order-5 hexagonal

t0,2{6,3,5} or rr{6,3,5}
(1)

(3.5.3.5)
(2)

(5.4.4)
- (2)

3.4.6.4
37 runcinated order-6 dodecahedral

t0,3{6,3,5}
(1)

(5.5.5)
- (6)

(6.4.4)
(1)

(6)3
38 cantellated order-6 dodecahedral

t0,2{5,3,6} or rr{5,3,6}
(2)

(4.3.4.5)
- (2)

(6.4.4)
(1)

(3.6)2
39 bitruncated order-6 dodecahedral

t1,2{6,3,5} or 2t{6,3,5}
(2)

(5.6.6)
- - (2)

(6)3
40 truncated order-6 dodecahedral

t0,1{5,3,6} or t{5,3,6}
(6)

(3.10.10)
- - (1)

(3)6
41 cantitruncated order-5 hexagonal

t0,1,2{6,3,5} or tr{6,3,5}
(1)

(5.6.6)
(1)

(5.4.4)
- (2)

4.6.10
42 runcitruncated order-5 hexagonal

t0,1,3{6,3,5}
(1)

(4.3.4.5)
(1)

(5.4.4)
(2)

(12.4.4)
(1)

3.12.12
43 runcitruncated order-6 dodecahedral

t0,1,3{5,3,6}
(1)

(3.10.10)
(1)

(10.4.4)
(2)

(6.4.4)
(1)

3.4.6.4
44 cantitruncated order-6 dodecahedral

t0,1,2{5,3,6} or tr{5,3,6}
(1)

(4.6.10)
- (2)

(6.4.4)
(1)

(6)3
45 omnitruncated order-6 dodecahedral

t0,1,2,3{6,3,5}
(1)

(4.6.10)
(1)

(10.4.4)
(1)

(12.4.4)
(1)

4.6.12
Alternated forms
# Honeycomb name
Coxeter diagram
Schläfli symbol
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
Alt
[145] alternated order-5 hexagonal

h{6,3,5}
- - - (20)

(3)6
(12)

(3)5

(5.6.6)
[146] cantic order-5 hexagonal

h2{6,3,5}
(1)

(3.5.3.5)
- (2)

(3.6.3.6)
(2)

(5.6.6)
[147] runcic order-5 hexagonal

h3{6,3,5}
(1)

(5.5.5)
(1)

(4.4.3)
(1)

(3.3.3.3.3.3)
(3)

(3.4.5.4)
[148] runcicantic order-5 hexagonal

h2,3{6,3,5}
(1)

(3.10.10)
(1)

(4.4.3)
(1)

(3.6.3.6)
(2)

(4.6.10)
Nonuniform snub rectified order-6 dodecahedral

sr{5,3,6}

(3.3.5.3.5)
-
(3.3.3.3)

(3.3.3.3.3.3)

irr. tet
Nonuniform omnisnub order-5 hexagonal

ht0,1,2,3{6,3,5}

(3.3.5.3.5)

(3.3.3.5)

(3.3.3.6)

(3.3.6.3.6)

irr. tet

[6,3,6] family

There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or

# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
1
2
3
46 order-6 hexagonal

{6,3,6}
- - - (20)

(6.6.6)

(3.3.3.3.3.3)
47 rectified order-6 hexagonal

t1{6,3,6} or r{6,3,6}
(2)

(3.3.3.3.3.3)
- - (6)

(3.6.3.6)

(6.4.4)
48 truncated order-6 hexagonal

t0,1{6,3,6} or t{6,3,6}
(1)

(3.3.3.3.3.3)
- - (6)

(3.12.12)
49 cantellated order-6 hexagonal

t0,2{6,3,6} or rr{6,3,6}
(1)

(3.6.3.6)
(2)

(4.4.6)
- (2)

(3.6.4.6)
50 Runcinated order-6 hexagonal

t0,3{6,3,6}
(1)

(6.6.6)
(3)

(4.4.6)
(3)

(4.4.6)
(1)

(6.6.6)
51 cantitruncated order-6 hexagonal

t0,1,2{6,3,6} or tr{6,3,6}
(1)

(6.6.6)
(1)

(4.4.6)
- (2)

(4.6.12)
52 runcitruncated order-6 hexagonal

t0,1,3{6,3,6}
(1)

(3.6.4.6)
(1)

(4.4.6)
(2)

(4.4.12)
(1)

(3.12.12)
53 omnitruncated order-6 hexagonal

t0,1,2,3{6,3,6}
(1)

(4.6.12)
(1)

(4.4.12)
(1)

(4.4.12)
(1)

(4.6.12)
[1] bitruncated order-6 hexagonal

t1,2{6,3,6} or 2t{6,3,6}
(2)

(6.6.6)
- - (2)

(6.6.6)
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol
Cells by location and count per vertex Vertex figure Picture
0
1
2
3
Alt
[47] rectified order-6 hexagonal

q{6,3,6} = r{6,3,6}
(2)

(3.3.3.3.3.3)
- - (6)

(3.6.3.6)

(6.4.4)
[54] triangular
() =
h{6,3,6} = {3,6,3}
- - -

(3.3.3.3.3.3)


(3.3.3.3.3.3)

{6,3}
[55] cantic order-6 hexagonal
( ) =
h2{6,3,6} = r{3,6,3}
(1)

(3.6.3.6)
- (2)

(6.6.6)
(2)

(3.6.3.6)
[149] runcic order-6 hexagonal

h3{6,3,6}
(1)

(6.6.6)
(1)

(4.4.3)
(3)

(3.4.6.4)
(1)

(3.3.3.3.3.3)
[150] runcicantic order-6 hexagonal

h2,3{6,3,6}
(1)

(3.12.12)
(1)

(4.4.3)
(2)

(4.6.12)
(1)

(3.6.3.6)
[137] alternated hexagonal
() =
2s{6,3,6} = h{6,3,3}


(3.3.3.3.6)
- -

(3.3.3.3.6)

+(3.3.3)

(3.6.6)
Nonuniform snub rectified order-6 hexagonal

sr{6,3,6}


(3.3.3.3.3.3)


(3.3.3.3)
-

(3.3.3.3.6)

+(3.3.3)
Nonuniform alternated runcinated order-6 hexagonal

ht0,3{6,3,6}


(3.3.3.3.3.3)


(3.3.3.3)


(3.3.3.3)


(3.3.3.3.3.3)

+(3.3.3)
Nonuniform omnisnub order-6 hexagonal

ht0,1,2,3{6,3,6}


(3.3.3.3.6)


(3.3.3.6)


(3.3.3.6)


(3.3.3.3.6)

+(3.3.3)

[3,6,3] family

There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
1
2
3
54 triangular

{3,6,3}
- - - ()

{3,6}

{6,3}
55 rectified triangular

t1{3,6,3} or r{3,6,3}
(2)

(6)3
- - (3)

(3.6)2

(3.4.4)
56 cantellated triangular

t0,2{3,6,3} or rr{3,6,3}
(1)

(3.6)2
(2)

(4.4.3)
- (2)

(3.6.4.6)
57 runcinated triangular

t0,3{3,6,3}
(1)

(3)6
(6)

(4.4.3)
(6)

(4.4.3)
(1)

(3)6
58 bitruncated triangular

t1,2{3,6,3} or 2t{3,6,3}
(2)

(3.12.12)
- - (2)

(3.12.12)
59 cantitruncated triangular

t0,1,2{3,6,3} or tr{3,6,3}
(1)

(3.12.12)
(1)

(4.4.3)
- (2)

(4.6.12)
60 runcitruncated triangular

t0,1,3{3,6,3}
(1)

(3.6.4.6)
(1)

(4.4.3)
(2)

(4.4.6)
(1)

(6)3
61 omnitruncated triangular

t0,1,2,3{3,6,3}
(1)

(4.6.12)
(1)

(4.4.6)
(1)

(4.4.6)
(1)

(4.6.12)
[1] truncated triangular

t0,1{3,6,3} or t{3,6,3} = {6,3,3}
(1)

(6)3
- - (3)

(6)3

{3,3}
Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
1
2
3
Alt
[56] cantellated triangular
=
s2{3,6,3}
(1)

(3.6)2
- - (2)

(3.6.4.6)

(3.4.4)
[60] runcitruncated triangular
=
s2,3{3,6,3}
(1)

(6)3
- (1)

(4.4.3)
(1)

(3.6.4.6)
(2)

(4.4.6)
[137] alternated hexagonal
( ) = ()
s{3,6,3}

(3)6
- -
(3)6

+(3)3

(3.6.6)
Scaliform runcisnub triangular

s3{3,6,3}

r{6,3}
-
(3.4.4)

(3)6

tricup
Nonuniform omnisnub triangular tiling honeycomb

ht0,1,2,3{3,6,3}

(3.3.3.3.6)

(3)4

(3)4

(3.3.3.3.6)

+(3)3

[4,4,3] family

There are 15 forms, generated by ring permutations of the Coxeter group: [4,4,3] or

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
1
2
3
62 square
=
{4,4,3}
- - - (6)


Cube
63 rectified square
=
t1{4,4,3} or r{4,4,3}
(2)

- - (3)



Triangular prism
64 rectified order-4 octahedral

t1{3,4,4} or r{3,4,4}
(4)

- - (2)

65 order-4 octahedral

{3,4,4}
()

- - -
66 truncated square
=
t0,1{4,4,3} or t{4,4,3}
(1)

- - (3)

67 truncated order-4 octahedral

t0,1{3,4,4} or t{3,4,4}
(4)

- - (1)

68 bitruncated square

t1,2{4,4,3} or 2t{4,4,3}
(2)

- - (2)

69 cantellated square

t0,2{4,4,3} or rr{4,4,3}
(1)

(2)

- (2)

70 cantellated order-4 octahedral

t0,2{3,4,4} or rr{3,4,4}
(2)

- (2)

(1)

71 runcinated square

t0,3{4,4,3}
(1)

(3)

(3)

(1)

72 cantitruncated square

t0,1,2{4,4,3} or tr{4,4,3}
(1)

(1)

- (2)

73 cantitruncated order-4 octahedral

t0,1,2{3,4,4} or tr{3,4,4}
(2)

- (1)

(1)

74 runcitruncated square

t0,1,3{4,4,3}
(1)

(1)

(2)

(1)

75 runcitruncated order-4 octahedral

t0,1,3{3,4,4}
(1)

(2)

(1)

(1)

76 omnitruncated square

t0,1,2,3{4,4,3}
(1)

(1)

(1)

(1)

Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
1
2
3
Alt
[83]alternated square

h{4,4,3}
---{4,3}
(4.3.4.3)
[84]cantic square

h2{4,4,3}

(3.4.3.4)
-
(3.8.8)

(4.8.8)
[85]runcic square

h3{4,4,3}

(3.3.3.3)
-
(3.4.4.4)

(4.4.4)
[86] runcicantic square

(4.6.6)
-
(3.4.4.4)

(4.8.8)
[153]alternated rectified square

hr{4,4,3}
--{}x{3}
157--{}x{6}
Scaliformsnub order-4 octahedral
= =
s{3,4,4}
--{}v{4}
Scaliformruncisnub order-4 octahedral

s3{3,4,4}
cup-4
152snub square
=
s{4,4,3}

--
{3,3}
Nonuniformsnub rectified order-4 octahedral

sr{3,4,4}
-irr. {3,3}
Nonuniformalternated runcitruncated square

ht0,1,3{3,4,4}
irr. {}v{4}
Nonuniformomnisnub square

ht0,1,2,3{4,4,3}




irr. {3,3}

[4,4,4] family

There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or .

# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Symmetry Vertex figure Picture
0
1
2
3
77 order-4 square

{4,4,4}
---
[4,4,4]

Cube
78 truncated order-4 square

t0,1{4,4,4} or t{4,4,4}

--
[4,4,4]
79 bitruncated order-4 square

t1,2{4,4,4} or 2t{4,4,4}

--
[[4,4,4]]
80 runcinated order-4 square

t0,3{4,4,4}




[[4,4,4]]
81 runcitruncated order-4 square

t0,1,3{4,4,4}




[4,4,4]
82 omnitruncated order-4 square

t0,1,2,3{4,4,4}




[[4,4,4]]
[62]square

t1{4,4,4} or r{4,4,4}

--
[4,4,4]
Square tiling
[63]rectified square

t0,2{4,4,4} or rr{4,4,4}


-
[4,4,4]
[66]truncated order-4 square

t0,1,2{4,4,4} or tr{4,4,4}


-
[4,4,4]
Alternated constructions
# Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in honeycomb
Symmetry Vertex figure Picture
0
1
2
3
Alt
[62] Square
( ) =

(4.4.4.4)
- -
(4.4.4.4)
[1+,4,4,4]
=[4,4,4]
[63]rectified square
=
s2{4,4,4}


-
[4+,4,4]
[77]order-4 square
---

[1+,4,4,4]
=[4,4,4]


Cube
[78] truncated order-4 square

(4.8.8)
-
(4.8.8)
-
(4.4.4.4)
[1+,4,4,4]
=[4,4,4]
[79] bitruncated order-4 square

(4.8.8)
- -
(4.8.8)

(4.8.8)
[1+,4,4,4]
=[4,4,4]
[81]runcitruncated order-4 square tiling
=
s2,3{4,4,4}




[4,4,4]
[83]alternated square
( ) ↔
hr{4,4,4}

--
[4,1+,4,4]
(4.3.4.3)
[104]quarter order-4 square

q{4,4,4}
[[1+,4,4,4,1+]]
=[[4[4]]]
153alternated rectified square tiling


hrr{4,4,4}


-
[((2+,4,4)),4]
154alternated runcinated order-4 square tiling

ht0,3{4,4,4}




[[(4,4,4,2+)]]
Scaliformsnub order-4 square tiling

s{4,4,4}

--
[4+,4,4]
Nonuniformruncic snub order-4 square tiling

s3{4,4,4}
[4+,4,4]
Nonuniformbisnub order-4 square tiling

2s{4,4,4}

--
[[4,4+,4]]
[152]snub square tiling

sr{4,4,4}


-
[(4,4)+,4]
Nonuniformalternated runcitruncated order-4 square tiling

ht0,1,3{4,4,4}




[((2,4)+,4,4)]
Nonuniformomnisnub order-4 square tiling

ht0,1,2,3{4,4,4}




[[4,4,4]]+

Tridental graphs

[3,41,1] family

There are 11 forms (of which only 4 are not shared with the [4,4,3] family), generated by ring permutations of the Coxeter group:

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
0'
3
83 alternated square
- -
(4.4.4)

(4.4.4.4)

(4.3.4.3)
84 cantic square

(3.4.3.4)
-
(3.8.8)

(4.8.8)
85 runcic square

(4.4.4.4)
-
(3.4.4.4)

(4.4.4.4)
86 runcicantic square

(4.6.6)
-
(3.4.4.4)

(4.8.8)
[63] rectified square

(4.4.4)
-
(4.4.4)

(4.4.4.4)
[64] rectified order-4 octahedral

(3.4.3.4)
-
(3.4.3.4)

(4.4.4.4)
[65] order-4 octahedral

(4.4.4.4)
-
(4.4.4.4)
-
[67] truncated order-4 octahedral

(4.6.6)
-
(4.6.6)

(4.4.4.4)
[68] bitruncated square

(3.8.8)
-
(3.8.8)

(4.8.8)
[70] cantellated order-4 octahedral

(3.4.4.4)

(4.4.4)

(3.4.4.4)


(4.4.4.4)
[73] cantitruncated order-4 octahedral

(4.6.8)

(4.4.4)

(4.6.8)

(4.8.8)
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
0'
3
Alt
Scaliformsnub order-4 octahedral
= =
s{3,41,1}
--irr. {}v{4}
Nonuniform snub rectified order-4 octahedral

sr{3,41,1}

(3.3.3.3.4)

(3.3.3)

(3.3.3.3.4)

(3.3.4.3.4)

+(3.3.3)

[4,41,1] family

There are 7 forms, (all shared with [4,4,4] family), generated by ring permutations of the Coxeter group:

# Honeycomb name
Coxeter diagram
Cells by location Vertex figure Picture
0
1
0'
3
[62] Square
( ) =

(4.4.4.4)
-
(4.4.4.4)

(4.4.4.4)
[62] Square
( ) =

(4.4.4.4)
-
(4.4.4.4)

(4.4.4.4)
[63] rectified square
( ) =

(4.4.4.4)

(4.4.4)

(4.4.4.4)

(4.4.4.4)
[66] truncated square
( ) =

(4.8.8)

(4.4.4)

(4.8.8)

(4.8.8)
[77] order-4 square

(4.4.4.4)
-
(4.4.4.4)
-
[78] truncated order-4 square

(4.8.8)
-
(4.8.8)

(4.4.4.4)
[79] bitruncated order-4 square

(4.8.8)
-
(4.8.8)

(4.8.8)
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
0'
3
Alt
[77] order-4 square
( ) =
- -

Cube
[78] truncated order-4 square
( ) = ( )
[83] Alternated square
-

Scaliform Snub order-4 square
-
Nonuniform -
Nonuniform -
[153] ( )
= ( )
Nonuniform Snub square


(3.3.4.3.4)


(3.3.3)


(3.3.4.3.4)


(3.3.4.3.4)

+(3.3.3)

[6,31,1] family

There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group: [6,31,1] or .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
0'
3
87 alternated order-6 cubic
- - ()

(3.3.3.3.3)
()

(3.3.3)


(3.6.3.6)
88 cantic order-6 cubic
(1)

(3.6.3.6)
- (2)

(6.6.6)
(2)

(3.6.6)
89 runcic order-6 cubic
(1)

(6.6.6)
- (3)

(3.4.6.4)
(1)

(3.3.3)
90 runcicantic order-6 cubic
(1)

(3.12.12)
- (2)

(4.6.12)
(1)

(3.6.6)
[16] order-4 hexagonal
(4)

(6.6.6)
- (4)

(6.6.6)
-
(3.3.3.3)
[17] rectified order-4 hexagonal
(2)

(3.6.3.6)
- (2)

(3.6.3.6)
(2)

(3.3.3.3)
[18] rectified order-6 cubic
(1)

(3.3.3.3.3)
- (1)

(3.3.3.3.3)
(6)

(3.4.3.4)
[20] truncated order-4 hexagonal
(2)

(3.12.12)
- (2)

(3.12.12)
(1)

(3.3.3.3)
[21] bitruncated order-6 cubic
(1)

(6.6.6)
- (1)

(6.6.6)
(2)

(4.6.6)
[24] cantellated order-6 cubic
(1)

(3.4.6.4)
(2)

(4.4.4)
(1)

(3.4.6.4)
(1)

(3.4.3.4)
[27] cantitruncated order-6 cubic
(1)

(4.6.12)
(1)

(4.4.4)
(1)

(4.6.12)
(1)

(4.6.6)
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
0'
3
Alt
[141] alternated order-4 hexagonal

(4.6.6)
Nonuniform bisnub order-4 hexagonal
Nonuniform snub rectified order-4 hexagonal

(3.3.3.3.6)

(3.3.3)

(3.3.3.3.6)

(3.3.3.3.3)

+(3.3.3)

Cyclic graphs

[(4,4,3,3)] family

There are 11 forms, 4 unique to this family, generated by ring permutations of the Coxeter group: , with .

# Honeycomb name
Coxeter diagram
Cells by location Vertex figure Picture
0
1
2
3
91 tetrahedral-square
- (6)


(444)
(8)


(333)
(12)


(3434)


(3444)
92 cyclotruncated square-tetrahedral


(444)


(488)


(333)


(388)
93 cyclotruncated tetrahedral-square
(1)


(3333)
(1)


(444)
(4)


(366)
(4)


(466)
94 truncated tetrahedral-square
(1)


(3444)
(1)


(488)
(1)


(366)
(2)


(468)
[64]( ) =
rectified order-4 octahedral


(3434)


(4444)


(3434)


(3434)
[65]( ) =
order-4 octahedral


(3333)
-

(3333)


(3333)
[67]( ) =
truncated order-4 octahedral


(466)


(4444)


(3434)


(466)
[83] alternated square
() =


(444)


(4444)
-

(444)

(4.3.4.3)
[84] cantic square
() =


(388)


(488)


(3434)


(388)
[85] runcic square
() =


(3444)


(3434)


(3333)


(3444)
[86] runcicantic square
() =


(468)


(488)


(466)


(468)
# Honeycomb name
Coxeter diagram
Cells by location Vertex figure Picture
0
1
2
3
Alt
Scaliformsnub order-4 octahedral
= =
--irr. {}v{4}
Nonuniform
155alternated tetrahedral-square
r{4,3}

[(4,4,4,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
95 cubic-square
(8)

(4.4.4)
- (6)

(4.4.4.4)
(12)

(4.4.4.4)

(3.4.4.4)
96 octahedral-square

(3.4.3.4)

(3.3.3.3)
-
(4.4.4.4)

(4.4.4.4)
97 cyclotruncated cubic-square
(4)

(3.8.8)
(1)

(3.3.3.3)
(1)

(4.4.4.4)
(4)

(4.8.8)
98 cyclotruncated square-cubic
(1)

(4.4.4)
(1)

(4.4.4)
(3)

(4.8.8)
(3)

(4.8.8)
99 cyclotruncated octahedral-square
(4)

(4.6.6)
(4)

(4.6.6)
(1)

(4.4.4.4)
(1)

(4.4.4.4)
100 rectified cubic-square
(1)

(3.4.3.4)
(2)

(3.4.4.4)
(1)

(4.4.4.4)
(2)

(4.4.4.4)
101 truncated cubic-square
(1)

(4.8.8)
(1)

(3.4.4.4)
(2)

(4.8.8)
(1)

(4.8.8)
102 truncated octahedral-square
(2)

(4.6.8
(1)

(4.6.6)
(1)

(4.4.4.4)
(1)

(4.8.8)
103 omnitruncated octahedral-square
(1)

(4.6.8)
(1)

(4.6.8)
(1)

(4.8.8)
(1)

(4.8.8)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
Alt
156 alternated cubic-square
-



(3.4.4.4)
Nonuniform snub octahedral-square




Nonuniform cyclosnub square-cubic




Nonuniform cyclosnub octahedral-square




Nonuniform omnisnub cubic-square

(3.3.3.3.4)

(3.3.3.3.4)

(3.3.4.3.4)

(3.3.4.3.4)

+(3.3.3)

[(4,4,4,4)] family

There are 5 forms, 1 unique, generated by ring permutations of the Coxeter group: . Repeat constructions are related as: , , and .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
104 quarter order-4 square

(4.8.8)

(4.4.4.4)

(4.4.4.4)

(4.8.8)
[62] square

(4.4.4.4)

(4.4.4.4)

(4.4.4.4)

(4.4.4.4)
[77] order-4 square
( ) =

(4.4.4.4)
-
(4.4.4.4)

(4.4.4.4)

(4.4.4.4)
[78] truncated order-4 square
( ) =

(4.8.8)

(4.4.4.4)

(4.8.8)

(4.8.8)
[79] bitruncated order-4 square

(4.8.8)

(4.8.8)

(4.8.8)

(4.8.8)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
Alt
[83] alternated square
() =
(6)

(4.4.4.4)
(6)

(4.4.4.4)
(6)

(4.4.4.4)
(6)

(4.4.4.4)
(8)

(4.4.4)

(4.3.4.3)
[77] alternated order-4 square

-

Nonsimplectic cantic order-4 square




Nonuniform cyclosnub square




Nonuniform snub order-4 square




Nonuniform bisnub order-4 square

(3.3.4.3.4)

(3.3.4.3.4)

(3.3.4.3.4)

(3.3.4.3.4)

+(3.3.3)

[(6,3,3,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
105 tetrahedral-hexagonal
(4)

(3.3.3)
- (4)

(6.6.6)
(6)

(3.6.3.6)

(3.4.3.4)
106 tetrahedral-triangular


(3.3.3.3)


(3.3.3)
-

(3.3.3.3.3.3)

(3.4.6.4)
107 cyclotruncated tetrahedral-hexagonal
(3)

(3.6.6)
(1)

(3.3.3)
(1)

(6.6.6)
(3)

(6.6.6)
108 cyclotruncated hexagonal-tetrahedral
(1)

(3.3.3)
(1)

(3.3.3)
(4)

(3.12.12)
(4)

(3.12.12)
109 cyclotruncated tetrahedral-triangular
(6)

(3.6.6)
(6)

(3.6.6)
(1)

(3.3.3.3.3.3)
(1)

(3.3.3.3.3.3)
110 rectified tetrahedral-hexagonal
(1)

(3.3.3.3)
(2)

(3.4.3.4)
(1)

(3.6.3.6)
(2)

(3.4.6.4)
111 truncated tetrahedral-hexagonal
(1)

(3.6.6)
(1)

(3.4.3.4)
(1)

(3.12.12)
(2)

(4.6.12)
112 truncated tetrahedral-triangular
(2)

(4.6.6)
(1)

(3.6.6)
(1)

(3.4.6.4)
(1)

(6.6.6)
113 omnitruncated tetrahedral-hexagonal
(1)

(4.6.6)
(1)

(4.6.6)
(1)

(4.6.12)
(1)

(4.6.12)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
Alt
Nonuniform omnisnub tetrahedral-hexagonal

(3.3.3.3.3)

(3.3.3.3.3)

(3.3.3.3.6)

(3.3.3.3.6)

+(3.3.3)

[(6,3,4,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group:

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
114 octahedral-hexagonal
(6)

(3.3.3.3)
- (8)

(6.6.6)
(12)

(3.6.3.6)
115 cubic-triangular
()

(3.4.3.4)
()

(4.4.4)
- ()

(3.3.3.3.3.3)

(3.4.6.4)
116 cyclotruncated octahedral-hexagonal
(3)

(4.6.6)
(1)

(4.4.4)
(1)

(6.6.6)
(3)

(6.6.6)
117 cyclotruncated hexagonal-octahedral
(1)

(3.3.3.3)
(1)

(3.3.3.3)
(4)

(3.12.12)
(4)

(3.12.12)
118 cyclotruncated cubic-triangular
(6)

(3.8.8)
(6)

(3.8.8)
(1)

(3.3.3.3.3.3)
(1)

(3.3.3.3.3.3)
119 rectified octahedral-hexagonal
(1)

(3.4.3.4)
(2)

(3.4.4.4)
(1)

(3.6.3.6)
(2)

(3.4.6.4)
120 truncated octahedral-hexagonal
(1)

(4.6.6)
(1)

(3.4.4.4)
(1)

(3.12.12)
(2)

(4.6.12)
121 truncated cubic-triangular
(2)

(4.6.8)
(1)

(3.8.8)
(1)

(3.4.6.4)
(1)

(6.6.6)
122 omnitruncated octahedral-hexagonal
(1)

(4.6.8)
(1)

(4.6.8)
(1)

(4.6.12)
(1)

(4.6.12)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure
0
1
2
3
Alt
Nonuniform cyclosnub octahedral-hexagonal

(3.3.3.3.3)

(3.3.3)

(3.3.3.3.3.3)

(3.3.3.3.3.3)

irr. {3,4}
Nonuniform omnisnub octahedral-hexagonal

(3.3.3.3.4)

(3.3.3.3.4)

(3.3.3.3.6)

(3.3.3.3.6)

irr. {3,3}

[(6,3,5,3)] family

There are 9 forms, generated by ring permutations of the Coxeter group:

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
123 icosahedral-hexagonal
(6)

(3.3.3.3.3)
- (8)

(6.6.6)
(12)

(3.6.3.6)

3.4.5.4
124 dodecahedral-triangular
(30)

(3.5.3.5)
(20)

(5.5.5)
- (12)

(3.3.3.3.3.3)

(3.4.6.4)
125 cyclotruncated icosahedral-hexagonal
(3)

(5.6.6)
(1)

(5.5.5)
(1)

(6.6.6)
(3)

(6.6.6)
126 cyclotruncated hexagonal-icosahedral
(1)

(3.3.3.3.3)
(1)

(3.3.3.3.3)
(5)

(3.12.12)
(5)

(3.12.12)
127 cyclotruncated dodecahedral-triangular
(6)

(3.10.10)
(6)

(3.10.10)
(1)

(3.3.3.3.3.3)
(1)

(3.3.3.3.3.3)
128 rectified icosahedral-hexagonal
(1)

(3.5.3.5)
(2)

(3.4.5.4)
(1)

(3.6.3.6)
(2)

(3.4.6.4)
129 truncated icosahedral-hexagonal
(1)

(5.6.6)
(1)

(3.5.5.5)
(1)

(3.12.12)
(2)

(4.6.12)
130 truncated dodecahedral-triangular
(2)

(4.6.10)
(1)

(3.10.10)
(1)

(3.4.6.4)
(1)

(6.6.6)
131 omnitruncated icosahedral-hexagonal
(1)

(4.6.10)
(1)

(4.6.10)
(1)

(4.6.12)
(1)

(4.6.12)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
Alt
Nonuniform omnisnub icosahedral-hexagonal

(3.3.3.3.5)

(3.3.3.3.5)

(3.3.3.3.6)

(3.3.3.3.6)

+(3.3.3)

[(6,3,6,3)] family

There are 6 forms, generated by ring permutations of the Coxeter group: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
132 hexagonal-triangular

(3.3.3.3.3.3)
-
(6.6.6)

(3.6.3.6)

(3.4.6.4)
133 cyclotruncated hexagonal-triangular
(1)

(3.3.3.3.3.3)
(1)

(3.3.3.3.3.3)
(3)

(3.12.12)
(3)

(3.12.12)
134 cyclotruncated triangular-hexagonal
(1)

(3.6.3.6)
(2)

(3.4.6.4)
(1)

(3.6.3.6)
(2)

(3.4.6.4)
135 rectified hexagonal-triangular
(1)

(6.6.6)
(1)

(3.4.6.4)
(1)

(3.12.12)
(2)

(4.6.12)
136 truncated hexagonal-triangular
(1)

(4.6.12)
(1)

(4.6.12)
(1)

(4.6.12)
(1)

(4.6.12)
[16] order-4 hexagonal tiling

=
(3)

(6.6.6)
(1)

(6.6.6)
(1)

(6.6.6)
(3)

(6.6.6)

(3.3.3.3)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
Alt
[141] alternated order-4 hexagonal

(3.3.3.3.3.3)

(3.3.3.3.3.3)

(3.3.3.3.3.3)

(3.3.3.3.3.3)

+(3.3.3.3)

(4.6.6)
Nonuniform cyclocantisnub hexagonal-triangular
Nonuniform cycloruncicantisnub hexagonal-triangular
Nonuniform snub rectified hexagonal-triangular

(3.3.3.3.6)

(3.3.3.3.6)

(3.3.3.3.6)

(3.3.3.3.6)

+(3.3.3)

Loop-n-tail graphs

[3,3[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [3,3[3]] or . 7 are half symmetry forms of [3,3,6]: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
137 alternated hexagonal
() =
- -
(3.3.3)

(3.3.3.3.3.3)

(3.6.6)
138 cantic hexagonal
(1)

(3.3.3.3)
- (2)

(3.6.6)
(2)

(3.6.3.6)
139 runcic hexagonal
(1)

(4.4.4)
(1)

(4.4.3)
(3)

(3.4.3.4)
(1)

(3.3.3.3.3.3)
140 runcicantic hexagonal
(1)

(3.10.10)
(1)

(4.4.3)
(2)

(4.6.6)
(1)

(3.6.3.6)
[2] rectified hexagonal
(1)

(3.3.3)
- (1)

(3.3.3)
(6)

(3.6.3.6)

Triangular prism
[3] rectified order-6 tetrahedral
(2)

(3.3.3.3)
- (2)

(3.3.3.3)
(2)

(3.3.3.3.3.3)

Hexagonal prism
[4] order-6 tetrahedral
(4)

(4.4.4)
- (4)

(4.4.4)
-
[8] cantellated order-6 tetrahedral
(1)

(3.3.3.3)
(2)

(4.4.6)
(1)

(3.3.3.3)
(1)

(3.6.3.6)
[9] bitruncated order-6 tetrahedral
(1)

(3.6.6)
- (1)

(3.6.6)
(2)

(6.6.6)
[10] truncated order-6 tetrahedral
(2)

(3.10.10)
- (2)

(3.10.10)
(1)

(3.6.3.6)
[14] cantitruncated order-6 tetrahedral
(1)

(4.6.6)
(1)

(4.4.6)
(1)

(4.6.6)
(1)

(6.6.6)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure
0
1
0'
3
Alt
Nonuniform snub rectified order-6 tetrahedral

(3.3.3.3.3)

(3.3.3.3)

(3.3.3.3.3)

(3.3.3.3.3.3)

+(3.3.3)

[4,3[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [4,3[3]] or . 7 are half symmetry forms of [4,3,6]: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
141 alternated order-4 hexagonal
- -
(3.3.3.3)

(3.3.3.3.3.3)

(4.6.6)
142 cantic order-4 hexagonal
(1)

(3.4.3.4)
- (2)

(4.6.6)
(2)

(3.6.3.6)
143 runcic order-4 hexagonal
(1)

(4.4.4)
(1)

(4.4.3)
(3)

(3.4.4.4)
(1)

(3.3.3.3.3.3)
144 runcicantic order-4 hexagonal
(1)

(3.8.8)
(1)

(4.4.3)
(2)

(4.6.8)
(1)

(3.6.3.6)
[16] order-4 hexagonal
(4)

(4.4.4)
- (4)

(4.4.4)
-
[17] rectified order-4 hexagonal
(1)

(3.3.3.3)
- (1)

(3.3.3.3)
(6)

(3.6.3.6)
[18] rectified order-6 cubic
(2)

(3.4.3.4)
- (2)

(3.4.3.4)
(2)

(3.3.3.3.3.3)
[21] bitruncated order-4 hexagonal
(1)

(4.6.6)
- (1)

(4.6.6)
(2)

(6.6.6)
[22] truncated order-6 cubic
(2)

(3.8.8)
- (2)

(3.8.8)
(1)

(3.6.3.6)
[23] cantellated order-4 hexagonal
(1)

(3.4.4.4)
(2)

(4.4.6)
(1)

(3.4.4.4)
(1)

(3.6.3.6)
[26] cantitruncated order-4 hexagonal
(1)

(4.6.8)
(1)

(4.4.6)
(1)

(4.6.8)
(1)

(6.6.6)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure
0
1
0'
3
Alt
Nonuniform snub rectified order-4 hexagonal

(3.3.3.3.4)

(3.3.3.3)

(3.3.3.3.4)

(3.3.3.3.3.3)

+(3.3.3)

[5,3[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [5,3[3]] or . 7 are half symmetry forms of [5,3,6]: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
145 alternated order-5 hexagonal
- -
(3.3.3.3.3)

(3.3.3.3.3.3)

(3.6.3.6)
146 cantic order-5 hexagonal
(1)

(3.5.3.5)
- (2)

(5.6.6)
(2)

(3.6.3.6)
147 runcic order-5 hexagonal
(1)

(5.5.5)
(1)

(4.4.3)
(3)

(3.4.5.4)
(1)

(3.3.3.3.3.3)
148 runcicantic order-5 hexagonal
(1)

(3.10.10)
(1)

(4.4.3)
(2)

(4.6.10)
(1)

(3.6.3.6)
[32] rectified order-5 hexagonal
(1)

(3.3.3.3.3)
- (1)

(3.3.3.3.3)
(6)

(3.6.3.6)
[33] rectified order-6 dodecahedral
(2)

(3.5.3.5)
- (2)

(3.5.3.5)
(2)

(3.3.3.3.3.3)
[34] Order-5 hexagonal
(4)

(5.5.5)
- (4)

(5.5.5)
-
[35] truncated order-6 dodecahedral
(2)

(3.10.10)
- (2)

(3.10.10)
(1)

(3.6.3.6)
[38] cantellated order-5 hexagonal
(1)

(3.4.5.4)
(2)

(6.4.4)
(1)

(3.4.5.4)
(1)

(3.6.3.6)
[39] bitruncated order-5 hexagonal
(1)

(5.6.6)
- (1)

(5.6.6)
(2)

(6.6.6)
[44] cantitruncated order-5 hexagonal
(1)

(4.6.10)
(1)

(6.4.4)
(1)

(4.6.10)
(1)

(6.6.6)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
Alt
Nonuniform snub rectified order-5 hexagonal

(3.3.3.3.5)

(3.3.3)

(3.3.3.3.5)

(3.3.3.3.3.3)

+(3.3.3)

[6,3[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [6,3[3]] or . 7 are half symmetry forms of [6,3,6]: .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
149 runcic order-6 hexagonal
(1)

(6.6.6)
(1)

(4.4.3)
(3)

(3.4.6.4)
(1)

(3.3.3.3.3.3)
150 runcicantic order-6 hexagonal
(1)

(3.12.12)
(1)

(4.4.3)
(2)

(4.6.12)
(1)

(3.6.3.6)
[1] hexagonal
(1)

(6.6.6)
- (1)

(6.6.6)
(2)

(6.6.6)
[46] order-6 hexagonal
(4)

(6.6.6)
- (4)

(6.6.6)
-
[47] rectified order-6 hexagonal
(2)

(3.6.3.6)
- (2)

(3.6.3.6)
(2)

(3.3.3.3.3.3)
[47] rectified order-6 hexagonal
(1)

(3.3.3.3.3.3)
- (1)

(3.3.3.3.3.3)
(6)

(3.6.3.6)
[48] truncated order-6 hexagonal
(2)

(3.12.12)
- (2)

(3.12.12)
(1)

(3.3.3.3.3.3)
[49] cantellated order-6 hexagonal
(1)

(3.4.6.4)
(2)

(6.4.4)
(1)

(3.4.6.4)
(1)

(3.6.3.6)
[51] cantitruncated order-6 hexagonal
(1)

(4.6.12)
(1)

(6.4.4)
(1)

(4.6.12)
(1)

(6.6.6)
[54] triangular tiling honeycomb
( ) =
- -
(3.3.3.3.3.3)

(3.3.3.3.3.3)

(6.6.6)
[55] cantic order-6 hexagonal
( ) =
(1)

(3.6.3.6)
- (2)

(6.6.6)
(2)

(3.6.3.6)
Alternated forms
# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
1
0'
3
Alt
[54] triangular tiling honeycomb
( ) =

-
-
(6.6.6)
[137] alternated hexagonal
( ) = ( )

-


+(3.6.6)

(3.6.6)
[47] rectified order-6 hexagonal

(3.6.3.6)
-
(3.6.3.6)

(3.3.3.3.3.3)
[55] cantic order-6 hexagonal
( ) = ( ) =
(1)

(3.6.3.6)
- (2)

(6.6.6)
(2)

(3.6.3.6)
Nonuniform snub rectified order-6 hexagonal


(3.3.3.3.6)


(3.3.3.3)


(3.3.3.3.6)


(3.3.3.3.3.3)

+(3.3.3)

Multicyclic graphs

[3[ ]×[ ]] family

There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group: . Two are duplicated as , two as , and three as .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
151 Quarter order-4 hexagonal




[17] rectified order-4 hexagonal





(4.4.4)
[18] rectified order-6 cubic





(6.4.4)
[21] bitruncated order-6 cubic




[87] alternated order-6 cubic
-



(3.6.3.6)
[88] cantic order-6 cubic




[141] alternated order-4 hexagonal

-


(4.6.6)
[142] cantic order-4 hexagonal




# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
Vertex figure Picture
0
1
2
3
Alt
Nonuniform bisnub order-6 cubic





irr. {3,3}

[3[3,3]] family

There are 4 forms, 0 unique, generated by ring permutations of the Coxeter group: . They are repeated in four families: (index 2 subgroup), (index 4 subgroup), (index 6 subgroup), and (index 24 subgroup).

# Name
Coxeter diagram
0 1 2 3 vertex figure Picture
[1] hexagonal





{3,3}
[47] rectified order-6 hexagonal





t{2,3}
[54] triangular tiling honeycomb
( ) =

-


t{3[3]}
[55] rectified triangular





t{2,3}
# Name
Coxeter diagram
0 1 2 3 Alt vertex figure Picture
[137] alternated hexagonal
( ) =


s{3[3]}


s{3[3]}


s{3[3]}


s{3[3]}


{3,3}

(4.6.6)

Summary enumerations by family

Linear graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs

[4,4,3]
[4,4,3]
15 | | | |
| | | |
| | | |
[1+,4,1+,4,3+](6)
|
[4,4,3]+(1)

[4,4,4]
[4,4,4]
3 | [1+,4,1+,4,1+,4,1+](3)
[4,4,4]
(3) | [1+,4,1+,4,1+,4,1+](3)
[2+[4,4,4]]
3 | | [2+[(4,4+,4,2+)]](2)
[2+[4,4,4]]+(1)

[6,3,3]
[6,3,3]
15 | | | |
| | | |
| | | |
[1+,6,(3,3)+](2) (↔ )
[6,3,3]+(1)

[6,3,4]
[6,3,4]
15 | | | |
| | | |
| | | |
[1+,6,3+,4,1+](6)
|
[6,3,4]+(1)

[6,3,5]
[6,3,5]
15 | | | |
| | | |
| | | |
[1+,6,(3,5)+](2) (↔ )
[6,3,5]+(1)

[3,6,3]
[3,6,3]
5 | | |
[3,6,3]
(1) [2+[3+,6,3+]](1)
[2+[3,6,3]]
3 | [2+[3,6,3]]+(1)

[6,3,6]
[6,3,6]
6 |
| |
[1+,6,3+,6,1+](2) (↔ )
[2+[6,3,6]]
(1) [2+[(6,3+,6,2+)]](2)
[2+[6,3,6]]
2 |
[2+[6,3,6]]+(1)

Tridental graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs

[6,31,1]
[6,31,1]4 | |
[1[6,31,1]]=[6,3,4]
(7) | | | | | | [1[1+,6,31,1]]+(2) (↔ )
[1[6,31,1]]+=[6,3,4]+(1)

[3,41,1]
[3,41,1]4 | | [3+,41,1]+(2)
[1[3,41,1]]=[3,4,4]
(7) | | | | | | [1[3+,41,1]]+(2)
[1[3,41,1]]+(1)

[41,1,1]
[41,1,1]0 (none)
[1[41,1,1]]=[4,4,4]
(4) | | [1[1+,4,1+,41,1]]+=[(4,1+,4,1+,4,2+)](4) |
[3[41,1,1]]=[4,4,3]
(3) | | [3[1+,41,1,1]]+=[1+,4,1+,4,3+](2) (↔ )
[3[41,1,1]]+=[4,4,3]+(1)

Cyclic graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs

[(4,4,4,3)]
[(4,4,4,3)]6 | | | | [(4,1+,4,1+,4,3+)] (2)
[2+[(4,4,4,3)]]
3 | | [2+[(4,4+,4,3+)]] (2)
[2+[(4,4,4,3)]]+ (1)

[4[4]]
[4[4]] (none)
[2+[4[4]]]
1 [2+[(4+,4)[2]]] (1)
[1[4[4]]]=[4,41,1]
(2) [(1+,4)[4]] (2)
[2[4[4]]]=[4,4,4]
(1) [2+[(1+,4,4)[2]]] (1)
[(2+,4)[4[4]]]=[2+[4,4,4]]
=
(1) [(2+,4)[4[4]]]+
= [2+[4,4,4]]+
(1)

[(6,3,3,3)]
[(6,3,3,3)]6 | | | |
[2+[(6,3,3,3)]]
3 | [2+[(6,3,3,3)]]+(1)

[(3,4,3,6)]
[(3,4,3,6)]6 | | | | [(3+,4,3+,6)](1)
[2+[(3,4,3,6)]]
3 | [2+[(3,4,3,6)]]+(1)

[(3,5,3,6)]
[(3,5,3,6)]6 | | | |
[2+[(3,5,3,6)]]
3 | [2+[(3,5,3,6)]]+(1)

[(3,6)[2]]
[(3,6)[2]]2
[2+[(3,6)[2]]]
1
[2+[(3,6)[2]]]
1
[2+[(3,6)[2]]]
=
(1) [2+[(3+,6)[2]]](1)
[(2,2)+[(3,6)[2]]]
1 [(2,2)+[(3,6)[2]]]+(1)
Paracompact hyperbolic enumeration
Group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs

[(3,3,4,4)]
[(3,3,4,4)]4 | |
[1[(4,4,3,3)]]=[3,41,1]
(7) | | | | | | [1[(3,3,4,1+,4)]]+
= [3+,41,1]+
(2) (= )
[1[(3,3,4,4)]]+
= [3,41,1]+
(1)

[3[ ]x[ ]]
[3[ ]x[ ]]1
[1[3[ ]x[ ]]]=[6,31,1]
(2)
[1[3[ ]x[ ]]]=[4,3[3]]
(2)
[2[3[ ]x[ ]]]=[6,3,4]
(3) | [2[3[ ]x[ ]]]+
=[6,3,4]+
(1)

[3[3,3]]

[3[3,3]]0 (none)
[1[3[3,3]]]=[6,3[3]]
0 (none)
[3[3[3,3]]]=[3,6,3]
(2)
[2[3[3,3]]]=[6,3,6]
(1)
[(3,3)[3[3,3]]]=[6,3,3]
=
(1) [(3,3)[3[3,3]]]+
= [6,3,3]+
(1)

Loop-n-tail graphs

Symmetry in these graphs can be doubled by adding a mirror: [1[n,3[3]]] = [n,3,6]. Therefore ring-symmetry graphs are repeated in the linear graph families.

Paracompact hyperbolic enumeration
Group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs

[3,3[3]]
[3,3[3]]4 | |
[1[3,3[3]]]=[3,3,6]
(7) | | | | | [1[3,3[3]]]+
= [3,3,6]+
(1)

[4,3[3]]
[4,3[3]]4 | |
[1[4,3[3]]]=[4,3,6]
(7) | | | | | | [1+,4,(3[3])+](2)
[4,3[3]]+(1)

[5,3[3]]
[5,3[3]]4 | |
[1[5,3[3]]]=[5,3,6]
(7) | | | | | [1[5,3[3]]]+
= [5,3,6]+
(1)

[6,3[3]]
[6,3[3]]2
[6,3[3]] =(2) ( = )
[(3,3)[1+,6,3[3]]]=[6,3,3]
(1) [(3,3)[1+,6,3[3]]]+(1)
[1[6,3[3]]]=[6,3,6]
(6) | | | | | [3[1+,6,3[3]]]+
= [3,6,3]+
(1) (= )
[1[6,3[3]]]+
= [6,3,6]+
(1)

See also

Notes

References

  • James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine)
  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapter 16-17: Geometries on Three-manifolds I,II)
  • Coxeter Decompositions of Hyperbolic Tetrahedra, arXiv/PDF, A. Felikson, December 2002
  • C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Can. J. Math. 19, 1179-1186, 1967. PDF Archived 2015-04-02 at the Wayback Machine
  • Norman Johnson, Geometries and Transformations, (2018) Chapters 11,12,13
  • N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353
  • N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H3: p130.
  • Klitzing, Richard. "Hyperbolic honeycombs H3 paracompact".
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