Paracompact uniform honeycombs

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}

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}	{3,5}	{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 notation	Simplex volume	Commutator subgroup	Coxeter diagram	Unique honeycomb count
[6,3,3]	0.0422892336	[1+,6,(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+]		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)+]		15
[6,31,1]	0.2114461680	[1+,6,(31,1)+]		4
[4,3[3]]	0.2114461680	[1+,4,3[3]]+		4
[4,4,4]	0.2289913985	[4+,4+,4+]+		6
[6,3,6]	0.2537354016	[1+,6,3+,6,1+]		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]]		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. 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 ↔ File:CDel branch c2-3.png.

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	Error creating thumbnail: convert: improper image header `/usr/local/www/mediawiki/w/images/7/7a/H3_633_FC_boundary.png' @ error/png.c/ReadPNGImage/4192. convert: no images defined `/tmp//transform_900e83319bf3.png' @ error/convert.c/ConvertImageCommand/3229. Error code: 1
2	rectified hexagonal t1{6,3,3} or r{6,3,3}	(2) (3.3.3)	-	-	(3) (3.6.3.6)	80px Triangular prism	100px
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)	80px Hexagonal prism	100px
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)	80px 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)	80px
7	runcinated hexagonal t0,3{6,3,3}	(1) (3.3.3)	(3) (4.4.3)	(3) (4.4.6)	(1) (6.6.6)	80px
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)	80px
9	bitruncated hexagonal t1,2{6,3,3} or 2t{6,3,3}	(2) (3.6.6)	-	-	(2) (6.6.6)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px

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)	80px
[139]	runcic hexagonal ↔	(1) (4.4.4)	(1) (4.4.3)		(1) (3.3.3.3.3.3)	(3) (3.4.3.4)	80px
[140]	runcicantic hexagonal ↔	(1) (3.10.10)	(1) (4.4.3)		(1) (3.6.3.6)	(2) (4.6.6)	80px
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)	80px (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)	80px (4.4.4)	100px
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)	80px (6.4.4)	100px
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)	80px
21	bitruncated order-6 cubic t1,2{6,3,4} or 2t{6,3,4}	(2) (4.6.6)	-	-	(2) (6.6.6)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px

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)	80px
[89]	runcic order-6 cubic ↔ h3{4,3,6}	(1) (3.3.3)	-	-	(1) (6.6.6)	(3) (3.4.6.4)	80px
[90]	runcicantic order-6 cubic ↔ h2,3{4,3,6}	(1) (3.6.6)	-	-	(1) (3.12.12)	(2) (4.6.12)	80px
[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)	80px
[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)	80px
[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)	80px
[151]	quarter order-4 hexagonal ↔ q{6,3,4}	(3)	(1)	-	(1)	(3)	80px
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	80px 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	80px (5.4.4)	100px
33	rectified order-6 dodecahedral t1{5,3,6} or r{5,3,6}	(5) (3.5.3.5)	-	-	(2) (3)6	80px (6.4.4)	100px
34	order-6 dodecahedral {5,3,6}	(5.5.5)	-	-	-	(∞) (3)6
35	truncated order-6 dodecahedral t0,1{6,3,5} or t{6,3,5}	(1) (3.3.3.3.3)	-	-	(5) 3.12.12	80px
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	80px
37	runcinated order-6 dodecahedral t0,3{6,3,5}	(1) (5.5.5)	-	(6) (6.4.4)	(1) (6)3	80px
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	80px
39	bitruncated order-6 dodecahedral t1,2{6,3,5} or 2t{6,3,5}	(2) (5.6.6)	-	-	(2) (6)3	80px
40	truncated order-6 dodecahedral t0,1{5,3,6} or t{5,3,6}	(6) (3.10.10)	-	-	(1) (3)6	80px
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	80px
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	80px
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	80px
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	80px
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	80px

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)	80px
[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)	80px
[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)	80px
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)	80px (6.4.4)	100px
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)	80px
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)	80px
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)	80px
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)	80px
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)	80px
[1]	bitruncated order-6 hexagonal ↔ ↔ t1,2{6,3,6} or 2t{6,3,6}	(2) (6.6.6)	-	-	(2) (6.6.6)	80px

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)		80px (6.4.4)	100px
[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)		80px	100px
[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)		80px
[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)		80px
[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	80px (3.4.4)	100px
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)	80px
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)	80px
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	80px
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)	80px
[1]	truncated triangular ↔ ↔ t0,1{3,6,3} or t{3,6,3} = {6,3,3}	(1) (6)3	-	-	(3) (6)3	80px {3,3}	Error creating thumbnail: convert: improper image header `/usr/local/www/mediawiki/w/images/7/7a/H3_633_FC_boundary.png' @ error/png.c/ReadPNGImage/4192. convert: no images defined `/tmp//transform_50c77ee0f652.png' @ error/convert.c/ConvertImageCommand/3229. Error code: 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
[56]	cantellated triangular = s2{3,6,3}	(1) (3.6)2	-	-	(2) (3.6.4.6)	(3.4.4)	80px
[60]	runcitruncated triangular = s2,3{3,6,3}	(1) (6)3	-	(1) (4.4.3)	(1) 40px (3.6.4.6)	(2) (4.4.6)	80px
[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)	80px Cube
63	rectified square = t1{4,4,3} or r{4,4,3}	(2)	-	-	(3)	80px Triangular prism	100px
64	rectified order-4 octahedral t1{3,4,4} or r{3,4,4}	(4)	-	-	(2)	80px	100px
65	order-4 octahedral {3,4,4}	(∞)	-	-	-
66	truncated square = t0,1{4,4,3} or t{4,4,3}	(1)	-	-	(3)	80px
67	truncated order-4 octahedral t0,1{3,4,4} or t{3,4,4}	(4)	-	-	(1)	80px
68	bitruncated square t1,2{4,4,3} or 2t{4,4,3}	(2)	-	-	(2)	80px
69	cantellated square t0,2{4,4,3} or rr{4,4,3}	(1)	(2)	-	(2)	80px
70	cantellated order-4 octahedral t0,2{3,4,4} or rr{3,4,4}	(2)	-	(2)	(1)	80px
71	runcinated square t0,3{4,4,3}	(1)	(3)	(3)	(1)	80px
72	cantitruncated square t0,1,2{4,4,3} or tr{4,4,3}	(1)	(1)	-	(2)	80px
73	cantitruncated order-4 octahedral t0,1,2{3,4,4} or tr{3,4,4}	(2)	-	(1)	(1)	80px
74	runcitruncated square t0,1,3{4,4,3}	(1)	(1)	(2)	(1)	80px
75	runcitruncated order-4 octahedral t0,1,3{3,4,4}	(1)	(2)	(1)	(1)	80px
76	omnitruncated square t0,1,2,3{4,4,3}	(1)	(1)	(1)	(1)	80px

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)	80px
[85]	runcic square ↔ h3{4,4,3}	(3.4.3.4)	-	(3.8.8)		(4.8.8)	80px
[86]	runcicantic square ↔	(4.6.6)	-	(3.4.4.4)		(4.8.8)	80px
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]	80px
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]	80px
82	omnitruncated order-4 square t0,1,2,3{4,4,4}					[[4,4,4]]	80px
[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]	80px	100px
[66]	truncated order-4 square ↔ t0,1,2{4,4,4} or tr{4,4,4}			-		[4,4,4]	80px

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]	80px	100px
[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]	80px
[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]	80px
[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]]]
Nonsimplectic	alternated rectified square tiling ↔ ↔ hrr{4,4,4}			-			[((2+,4,4)),4]
Nonsimplectic	alternated runcinated order-4 square tiling ht0,3{4,4,4}						[[(4,4,4,2+)]]
Nonuniform	snub order-4 square tiling s{4,4,4}		-	-			[4+,4,4]
Nonuniform	runcic snub order-4 square tiling s3{4,4,4}						[4+,4,4]
Nonuniform	bisnub order-4 square tiling 2s{4,4,4}		-	-			[[4,4+,4]]
Nonuniform	snub square tiling ↔ sr{4,4,4}			-			[(4,4)+,4]
Nonuniform	alternated runcitruncated order-4 square tiling ht0,1,3{4,4,4}						[((2,4)+,4,4)]
Nonuniform	omnisnub order-4 square tiling ht0,1,2,3{4,4,4}						[[4,4,4]]+

Tridental graphs

[3,4^1,1] 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)	80px
85	runcic square ↔	(4.4.4.4)	-	(3.4.4.4)	(4.4.4.4)	80px
86	runcicantic square ↔	(4.6.6)	-	(3.4.4.4)	(4.8.8)	80px
[63]	rectified square ↔	(4.4.4)	-	(4.4.4)	(4.4.4.4)	80px	100px
[64]	rectified order-4 octahedral ↔	(3.4.3.4)	-	(3.4.3.4)	(4.4.4.4)	80px	100px
[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)	80px
[68]	bitruncated square ↔	(3.8.8)	-	(3.8.8)	(4.8.8)	80px
[70]	cantellated order-4 octahedral ↔	(3.4.4.4)	(4.4.4)	(3.4.4.4)	(4.4.4.4)	80px
[73]	cantitruncated order-4 octahedral ↔	(4.6.8)	(4.4.4)	(4.6.8)	(4.8.8)	80px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure	Picture
		0	1	0'	3	Alt
Scaliform	snub 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)

[4,4^1,1] 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)	80px	100px
[66]	truncated square ( ↔ ) =	(4.8.8)	(4.4.4)	(4.8.8)	(4.8.8)	80px
[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)	80px
[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 ( ↔ ) = ( ↔ )						80px
[83]	Alternated square ↔		-
Nonuniform	Snub order-4 square		-
Nonuniform			-
Nonuniform			-
Nonsimplectic	( ↔ ) = ( ↔ )
Nonuniform	Snub square ↔ ↔	(3.3.4.3.4)	(3.3.3)	(3.3.4.3.4)	(3.3.4.3.4)	+(3.3.3)

[6,3^1,1] family

There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group: [6,3^1,1] 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)	80px
89	runcic order-6 cubic ↔	(1) (6.6.6)	-	(3) (3.4.6.4)	(1) (3.3.3)	80px
90	runcicantic order-6 cubic ↔	(1) (3.12.12)	-	(2) (4.6.12)	(1) (3.6.6)	80px
[16]	order-4 hexagonal ↔	(4) (6.6.6)	-	(4) (6.6.6)	-	80px (3.3.3.3)
[17]	rectified order-4 hexagonal ↔	(2) (3.6.3.6)	-	(2) (3.6.3.6)	(2) (3.3.3.3)	80px	100px
[18]	rectified order-6 cubic ↔	(1) (3.3.3.3.3)	-	(1) (3.3.3.3.3)	(6) (3.4.3.4)	80px	100px
[20]	truncated order-4 hexagonal ↔	(2) (3.12.12)	-	(2) (3.12.12)	(1) (3.3.3.3)	80px
[21]	bitruncated order-6 cubic ↔	(1) (6.6.6)	-	(1) (6.6.6)	(2) (4.6.6)	80px
[24]	cantellated order-6 cubic ↔	(1) (3.4.6.4)	(2) (4.4.4)	(1) (3.4.6.4)	(1) (3.4.3.4)	80px
[27]	cantitruncated order-6 cubic ↔	(1) (4.6.12)	(1) (4.4.4)	(1) (4.6.12)	(1) (4.6.6)	80px

#	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)	80px
93	cyclotruncated tetrahedral-square	(1) (3333)	(1) (444)	(4) (366)	(4) (466)	80px
94	truncated tetrahedral-square	(1) (3444)	(1) (488)	(1) (366)	(2) (468)	80px
[64]	( ↔ ) = rectified order-4 octahedral	(3434)	(4444)	(3434)	(3434)	80px	100px
[65]	( ↔ ) = order-4 octahedral	(3333)	-	(3333)	(3333)
[67]	( ↔ ) = truncated order-4 octahedral	(466)	(4444)	(3434)	(466)	80px
[83]	alternated square ( ↔ ) =	(444)	(4444)	-	(444)	(4.3.4.3)
[84]	cantic square ( ↔ ) =	(388)	(488)	(3434)	(388)	80px
[85]	runcic square ( ↔ ) =	(3444)	(3434)	(3333)	(3444)	80px
[86]	runcicantic square ( ↔ ) =	(468)	(488)	(466)	(468)	80px

#	Honeycomb name Coxeter diagram	Cells by location					Vertex figure	Picture
		0	1	2	3	Alt
Scaliform	snub order-4 octahedral = =		-	-		irr. {}v{4}
Nonuniform
Nonsimplectic	alternated tetrahedral-square ↔

[(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)	80px
100	rectified cubic-square	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (4.4.4.4)	(2) (4.4.4.4)	80px
101	truncated cubic-square	(1) (4.8.8)	(1) (3.4.4.4)	(2) (4.8.8)	(1) (4.8.8)	80px
102	truncated octahedral-square	(2) (4.6.8	(1) (4.6.6)	(1) (4.4.4.4)	(1) (4.8.8)	80px
103	omnitruncated octahedral-square	(1) (4.6.8)	(1) (4.6.8)	(1) (4.8.8)	(1) (4.8.8)	80px

Alternated forms

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
		0	1	2	3	Alt
Nonsimplectic	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)	80px
[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)	80px
[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)
Nonsimplectic	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)	80px
110	rectified tetrahedral-hexagonal	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.6.3.6)	(2) (3.4.6.4)	80px
111	truncated tetrahedral-hexagonal	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.12.12)	(2) (4.6.12)	80px
112	truncated tetrahedral-triangular	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.6.4)	(1) (6.6.6)	80px
113	omnitruncated tetrahedral-hexagonal	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.12)	(1) (4.6.12)	80px

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)	80px

[(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)	80px
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)	80px
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)	80px
119	rectified octahedral-hexagonal	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (3.6.3.6)	(2) (3.4.6.4)	80px
120	truncated octahedral-hexagonal	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.12.12)	(2) (4.6.12)	80px
121	truncated cubic-triangular	(2) (4.6.8)	(1) (3.8.8)	(1) (3.4.6.4)	(1) (6.6.6)	80px
122	omnitruncated octahedral-hexagonal	(1) (4.6.8)	(1) (4.6.8)	(1) (4.6.12)	(1) (4.6.12)	80px

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)	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}	80px

[(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)	80px
128	rectified icosahedral-hexagonal	(1) (3.5.3.5)	(2) (3.4.5.4)	(1) (3.6.3.6)	(2) (3.4.6.4)	80px
129	truncated icosahedral-hexagonal	(1) (5.6.6)	(1) (3.5.5.5)	(1) (3.12.12)	(2) (4.6.12)	80px
130	truncated dodecahedral-triangular	(2) (4.6.10)	(1) (3.10.10)	(1) (3.4.6.4)	(1) (6.6.6)	80px
131	omnitruncated icosahedral-hexagonal	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.12)	(1) (4.6.12)	80px

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)	80px

[(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)	80px
135	rectified hexagonal-triangular	(1) (6.6.6)	(1) (3.4.6.4)	(1) (3.12.12)	(2) (4.6.12)	80px
136	truncated hexagonal-triangular	(1) (4.6.12)	(1) (4.6.12)	(1) (4.6.12)	(1) (4.6.12)	80px
[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)	80px

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)	80px
139	runcic hexagonal ↔	(1) (4.4.4)	(1) (4.4.3)	(3) (3.4.3.4)	(1) (3.3.3.3.3.3)	80px
140	runcicantic hexagonal ↔	(1) (3.10.10)	(1) (4.4.3)	(2) (4.6.6)	(1) (3.6.3.6)	80px
[2]	rectified hexagonal ↔	(1) (3.3.3)	-	(1) (3.3.3)	(6) (3.6.3.6)	80px Triangular prism	100px
[3]	rectified order-6 tetrahedral ↔	(2) (3.3.3.3)	-	(2) (3.3.3.3)	(2) (3.3.3.3.3.3)	80px Hexagonal prism	100px
[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)	80px
[9]	bitruncated order-6 tetrahedral ↔	(1) (3.6.6)	-	(1) (3.6.6)	(2) (6.6.6)	80px
[10]	truncated order-6 tetrahedral ↔	(2) (3.10.10)	-	(2) (3.10.10)	(1) (3.6.3.6)	80px
[14]	cantitruncated order-6 tetrahedral ↔	(1) (4.6.6)	(1) (4.4.6)	(1) (4.6.6)	(1) (6.6.6)	80px

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)	80px
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)	80px
144	runcicantic order-4 hexagonal ↔	(1) (3.8.8)	(1) (4.4.3)	(2) (4.6.8)	(1) (3.6.3.6)	80px
[16]	order-4 hexagonal ↔	(4) (4.4.4)	-	(4) (4.4.4)	-	80px
[17]	rectified order-4 hexagonal ↔	(1) (3.3.3.3)	-	(1) (3.3.3.3)	(6) (3.6.3.6)	80px	100px
[18]	rectified order-6 cubic ↔	(2) (3.4.3.4)	-	(2) (3.4.3.4)	(2) (3.3.3.3.3.3)	80px	100px
[21]	bitruncated order-4 hexagonal ↔	(1) (4.6.6)	-	(1) (4.6.6)	(2) (6.6.6)	80px
[22]	truncated order-6 cubic ↔	(2) (3.8.8)	-	(2) (3.8.8)	(1) (3.6.3.6)	80px
[23]	cantellated order-4 hexagonal ↔	(1) (3.4.4.4)	(2) (4.4.6)	(1) (3.4.4.4)	(1) (3.6.3.6)	80px
[26]	cantitruncated order-4 hexagonal ↔	(1) (4.6.8)	(1) (4.4.6)	(1) (4.6.8)	(1) (6.6.6)	80px

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)	80px
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)	80px
148	runcicantic order-5 hexagonal ↔	(1) (3.10.10)	(1) (4.4.3)	(2) (4.6.10)	(1) (3.6.3.6)	80px
[32]	rectified order-5 hexagonal ↔	(1) (3.3.3.3.3)	-	(1) (3.3.3.3.3)	(6) (3.6.3.6)	80px	100px
[33]	rectified order-6 dodecahedral ↔	(2) (3.5.3.5)	-	(2) (3.5.3.5)	(2) (3.3.3.3.3.3)	80px	100px
[34]	Order-5 hexagonal ↔	(4) (5.5.5)	-	(4) (5.5.5)	-	80px
[35]	truncated order-6 dodecahedral ↔	(2) (3.10.10)	-	(2) (3.10.10)	(1) (3.6.3.6)	80px
[38]	cantellated order-5 hexagonal ↔	(1) (3.4.5.4)	(2) (6.4.4)	(1) (3.4.5.4)	(1) (3.6.3.6)	80px
[39]	bitruncated order-5 hexagonal ↔	(1) (5.6.6)	-	(1) (5.6.6)	(2) (6.6.6)	80px
[44]	cantitruncated order-5 hexagonal ↔	(1) (4.6.10)	(1) (6.4.4)	(1) (4.6.10)	(1) (6.6.6)	80px

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)	80px
150	runcicantic order-6 hexagonal ↔	(1) (3.12.12)	(1) (4.4.3)	(2) (4.6.12)	(1) (3.6.3.6)	80px
[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)	80px	100px
[47]	rectified order-6 hexagonal ↔	(1) (3.3.3.3.3.3)	-	(1) (3.3.3.3.3.3)	(6) (3.6.3.6)	80px	100px
[48]	truncated order-6 hexagonal ↔	(2) (3.12.12)	-	(2) (3.12.12)	(1) (3.6.3.6)	80px
[49]	cantellated order-6 hexagonal ↔	(1) (3.4.6.4)	(2) (6.4.4)	(1) (3.4.6.4)	(1) (3.6.3.6)	80px
[51]	cantitruncated order-6 hexagonal ↔	(1) (4.6.12)	(1) (6.4.4)	(1) (4.6.12)	(1) (6.6.6)	80px
[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)	80px	100px

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)		80px
[55]	cantic order-6 hexagonal ( ↔ ) = ( ↔ ) =	(1) (3.6.3.6)	-	(2) (6.6.6)	(2) (3.6.3.6)		80px	100px
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 ↔					80px
[17]	rectified order-4 hexagonal ↔ ↔ ↔					60px (4.4.4)	100px
[18]	rectified order-6 cubic ↔ ↔ ↔					60px (6.4.4)	100px
[21]	bitruncated order-6 cubic ↔ ↔ ↔					60px
[87]	alternated order-6 cubic ↔ ↔	-				(3.6.3.6)
[88]	cantic order-6 cubic ↔ ↔					60px
[141]	alternated order-4 hexagonal ↔ ↔		-			(4.6.6)
[142]	cantic order-4 hexagonal ↔ ↔					80px

#	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}	Error creating thumbnail: convert: improper image header `/usr/local/www/mediawiki/w/images/7/7a/H3_633_FC_boundary.png' @ error/png.c/ReadPNGImage/4192. convert: no images defined `/tmp//transform_d0a56588e785.png' @ error/convert.c/ConvertImageCommand/3229. Error code: 1
[47]	rectified order-6 hexagonal ↔					60px t{2,3}	100px
[54]	triangular tiling honeycomb ( ↔ ) =		-			t{3[3]}
[55]	rectified triangular ↔					60px t{2,3}	100px

#	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
${\bar{R}}_3$ [4,4,3]	[4,4,3]	15	| | | | | | | | | | | |	[1+,4,1+,4,3+]	(6)	(↔ ) (↔ ) | |
				[4,4,3]+	(1)
${\bar{N}}_3$ [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)
${\bar{V}}_3$ [6,3,3]	[6,3,3]	15	| | | | | | | | | | | |	[1+,6,(3,3)+]	(2)	(↔ )
				[6,3,3]+	(1)
${\bar{BV}}_3$ [6,3,4]	[6,3,4]	15	| | | | | | | | | | | |	[1+,6,3+,4,1+]	(6)	(↔ ) (↔ ) | |
				[6,3,4]+	(1)
${\bar{HV}}_3$ [6,3,5]	[6,3,5]	15	| | | | | | | | | | | |	[1+,6,(3,5)+]	(2)	(↔ )
				[6,3,5]+	(1)
${\bar{Y}}_3$ [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)
${\bar{Z}}_3$ [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
${\bar{DV}}_3$ [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)
${\bar{O}}_3$ [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)
${\bar{M}}_3$ [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
${\widehat{CR}}_3$ [(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)
${\widehat{RR}}_3$ [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)
${\widehat{AV}}_3$ [(6,3,3,3)]	[(6,3,3,3)]	6	| | | | |
	[2+[(6,3,3,3)]]	3	| |	[2+[(6,3,3,3)]]+	(1)
${\widehat{BV}}_3$ [(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)
${\widehat{HV}}_3$ [(3,5,3,6)]	[(3,5,3,6)]	6	| | | | |
	[2+[(3,5,3,6)]]	3	| |	[2+[(3,5,3,6)]]+	(1)
${\widehat{VV}}_3$ [(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
${\widehat{BR}}_3$ [(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)
${\bar{DP}}_3$ [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)
${\bar{PP}}_3$ [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
${\bar{P}}_3$ [3,3[3]]	[3,3[3]]	4	| | |
	[1[3,3[3]]]=[3,3,6] ↔	(7)	| | | | | |	[1[3,3[3]]]+ = [3,3,6]+	(1)
${\bar{BP}}_3$ [4,3[3]]	[4,3[3]]	4	| | |
	[1[4,3[3]]]=[4,3,6] ↔	(7)	| | | | | |	[1+,4,(3[3])+]	(2)	↔
				[4,3[3]]+	(1)
${\bar{HP}}_3$ [5,3[3]]	[5,3[3]]	4	| | |
	[1[5,3[3]]]=[5,3,6] ↔	(7)	| | | | | |	[1[5,3[3]]]+ = [5,3,6]+	(1)
${\bar{VP}}_3$ [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

• Uniform tilings in hyperbolic plane
• List of regular polytopes#Tessellations of hyperbolic 3-space

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)
• 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 Canad. J. Math. 19, 1179-1186, 1967. PDF [1]
• Norman Johnson, Geometries and Transformations, (2015) 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 [2] [3]
• N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [4]
• Richard Klitzing, Hyperbolic honeycombs, H3 paracompact

• P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)