# Paracompact uniform honeycombs

Jump to: navigation, search
 {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}

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.

## 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.

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} {3,4} {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.

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

Pyramidal hyperbolic paracompact group summary
Dimension Rank Graphs
H3 5

| | | |
| | | | |
| | | | | |
| | | | | | | | | | | |

## 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)

(3.10.10)
(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)

(3.3.3.4)

(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-6 dodecahedral

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 order-6 cubic

h2{4,4,3}

(3.4.3.4)
-
(3.8.8)

(4.8.8)
[85] runcic square

h3{4,4,3}

(3.4.3.4)
-
(3.8.8)

(4.8.8)
[86] runcicantic square

(4.6.6)
-
(3.4.4.4)

(4.8.8)
Nonsimplectic alternated rectified square

hr{4,4,3}
- - {}x{3}
Scaliform snub order-4 octahedral
= =
s{3,4,4}
- - irr. {}v{4}
Scaliform runcisnub order-4 octahedral

s3{3,4,4}
cup-4
Nonuniform snub square
=
s{4,4,3}
- - irr. {3,3}
Nonuniform snub rectified order-4 octahedral

sr{3,4,4}
- irr. {3,3}
Nonuniform alternated runcitruncated square

ht0,1,3{3,4,4}
irr. {}v{4}
Nonuniform omnisnub 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}