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. 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.

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

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	↔ sr{3,3,6}					Irr. (3.3.3)
Nonuniform	sr3{3,3,6}
Nonuniform	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	↔ 2s{4,3,6}	(3.3.3.3.3.3)	-	-	(3.3.3.3.6)	+(3.3.3)
Nonuniform
Nonuniform	↔ 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	sr3{4,3,6}
Nonuniform	↔ sr{6,3,4}	(3.3.3.3.3.3)	(3.3.3)	-	(3.3.3.3.6)	+(3.3.3)
Nonuniform	sr3{6,3,4}
Nonuniform	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	↔ sr{5,3,6}	(3.3.5.3.5)	-	(3.3.3.3)	(3.3.3.3.3.3)	irr. tet
Nonuniform	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	sr{6,3,6}	(3.3.3.3.3.3)	(3.3.3.3)	-	(3.3.3.3.6)	+(3.3.3)
Nonuniform	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	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	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)
Nonsimplectic	alternated rectified square ↔ hr{4,4,3}		-	-		{}x{3}
Scaliform	snub order-4 octahedral = = s{3,4,4}		-	-		{}v{4}
Scaliform	runcisnub order-4 octahedral s3{3,4,4}					cup-4
152	= s{4,4,3}		-	-		{3,3}
Nonuniform	sr{3,4,4}		-			irr. {3,3}
Nonuniform	ht0,1,3{3,4,4}					irr. {}v{4}
Nonuniform	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]]]
153	alternated rectified square tiling ↔ ↔ hrr{4,4,4}			-			[((2+,4,4)),4]
154	ht0,3{4,4,4}						[[(4,4,4,2+)]]
Scaliform	snub order-4 square tiling s{4,4,4}		-	-			[4+,4,4]
Nonuniform	s3{4,4,4}						[4+,4,4]
Nonuniform	2s{4,4,4}		-	-			[[4,4+,4]]
[152]	↔ sr{4,4,4}			-			[(4,4)+,4]
Nonuniform	ht0,1,3{4,4,4}						[((2,4)+,4,4)]
Nonuniform	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)
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
Scaliform	snub order-4 octahedral = = s{3,41,1}		-	-		irr. {}v{4}
Nonuniform	↔ 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)
[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			-
Nonsimplectic	( ↔ ) = ( ↔ )
Nonuniform	↔ ↔	(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)
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	↔
Nonuniform	↔	(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		(1) (3333)	(1) (444)	(4) (366)	(4) (466)
94		(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
Scaliform	snub order-4 octahedral = =		-	-		irr. {}v{4}
Nonuniform
Nonsimplectic	↔

[(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		(3.4.3.4)	(3.3.3.3)	-	(4.4.4.4)	(4.4.4.4)
97		(4) (3.8.8)	(1) (3.3.3.3)	(1) (4.4.4.4)	(4) (4.8.8)
98		(1) (4.4.4)	(1) (4.4.4)	(3) (4.8.8)	(3) (4.8.8)
99		(4) (4.6.6)	(4) (4.6.6)	(1) (4.4.4.4)	(1) (4.4.4.4)
100		(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (4.4.4.4)	(2) (4.4.4.4)
101		(1) (4.8.8)	(1) (3.4.4.4)	(2) (4.8.8)	(1) (4.8.8)
102		(2) (4.6.8	(1) (4.6.6)	(1) (4.4.4.4)	(1) (4.8.8)
103		(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
Nonsimplectic	↔	-				(3.4.4.4)
Nonuniform
Nonuniform
Nonuniform
Nonuniform		(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)
Nonsimplectic	↔		-
Nonsimplectic	↔
Nonuniform
Nonuniform	snub order-4 square
Nonuniform	↔	(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		(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		(3) (3.6.6)	(1) (3.3.3)	(1) (6.6.6)	(3) (6.6.6)
108		(1) (3.3.3)	(1) (3.3.3)	(4) (3.12.12)	(4) (3.12.12)
109		(6) (3.6.6)	(6) (3.6.6)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
110		(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
111		(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.12.12)	(2) (4.6.12)
112		(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.6.4)	(1) (6.6.6)
113		(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		(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		(3) (4.6.6)	(1) (4.4.4)	(1) (6.6.6)	(3) (6.6.6)
117		(1) (3.3.3.3)	(1) (3.3.3.3)	(4) (3.12.12)	(4) (3.12.12)
118		(6) (3.8.8)	(6) (3.8.8)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
119		(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
120		(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.12.12)	(2) (4.6.12)
121		(2) (4.6.8)	(1) (3.8.8)	(1) (3.4.6.4)	(1) (6.6.6)
122		(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		(3.3.3.3.3)	(3.3.3)	(3.3.3.3.3.3)	(3.3.3.3.3.3)	irr. {3,4}
Nonuniform		(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		(6) (3.3.3.3.3)	-	(8) (6.6.6)	(12) (3.6.3.6)	3.4.5.4
124		(30) (3.5.3.5)	(20) (5.5.5)	-	(12) (3.3.3.3.3.3)	(3.4.6.4)
125		(3) (5.6.6)	(1) (5.5.5)	(1) (6.6.6)	(3) (6.6.6)
126		(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(5) (3.12.12)	(5) (3.12.12)
127		(6) (3.10.10)	(6) (3.10.10)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
128		(1) (3.5.3.5)	(2) (3.4.5.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
129		(1) (5.6.6)	(1) (3.5.5.5)	(1) (3.12.12)	(2) (4.6.12)
130		(2) (4.6.10)	(1) (3.10.10)	(1) (3.4.6.4)	(1) (6.6.6)
131		(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		(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		(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)	(3) (3.12.12)	(3) (3.12.12)
134		(1) (3.6.3.6)	(2) (3.4.6.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
135		(1) (6.6.6)	(1) (3.4.6.4)	(1) (3.12.12)	(2) (4.6.12)
136		(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
Nonuniform
Nonuniform		(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	↔	(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	↔	(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	↔	(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	↔	(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	↔					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
${\displaystyle {\bar {R}}_{3}}$ [4,4,3]	[4,4,3]	15	| | | | | | | | | | | |	[1+,4,1+,4,3+]	(6)	(↔ ) (↔ ) | |
				[4,4,3]+	(1)
${\displaystyle {\bar {N}}_{3}}$ [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)
${\displaystyle {\bar {V}}_{3}}$ [6,3,3]	[6,3,3]	15	| | | | | | | | | | | |	[1+,6,(3,3)+]	(2)	(↔ )
				[6,3,3]+	(1)
${\displaystyle {\bar {BV}}_{3}}$ [6,3,4]	[6,3,4]	15	| | | | | | | | | | | |	[1+,6,3+,4,1+]	(6)	(↔ ) (↔ ) | |
				[6,3,4]+	(1)
${\displaystyle {\bar {HV}}_{3}}$ [6,3,5]	[6,3,5]	15	| | | | | | | | | | | |	[1+,6,(3,5)+]	(2)	(↔ )
				[6,3,5]+	(1)
${\displaystyle {\bar {Y}}_{3}}$ [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)
${\displaystyle {\bar {Z}}_{3}}$ [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
${\displaystyle {\bar {DV}}_{3}}$ [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)
${\displaystyle {\bar {O}}_{3}}$ [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)
${\displaystyle {\bar {M}}_{3}}$ [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
${\displaystyle {\widehat {CR}}_{3}}$ [(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)
${\displaystyle {\widehat {RR}}_{3}}$ [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)
${\displaystyle {\widehat {AV}}_{3}}$ [(6,3,3,3)]	[(6,3,3,3)]	6	| | | | |
	[2+[(6,3,3,3)]]	3	| |	[2+[(6,3,3,3)]]+	(1)
${\displaystyle {\widehat {BV}}_{3}}$ [(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)
${\displaystyle {\widehat {HV}}_{3}}$ [(3,5,3,6)]	[(3,5,3,6)]	6	| | | | |
	[2+[(3,5,3,6)]]	3	| |	[2+[(3,5,3,6)]]+	(1)
${\displaystyle {\widehat {VV}}_{3}}$ [(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
${\displaystyle {\widehat {BR}}_{3}}$ [(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)
${\displaystyle {\bar {DP}}_{3}}$ [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)
${\displaystyle {\bar {PP}}_{3}}$ [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
${\displaystyle {\bar {P}}_{3}}$ [3,3[3]]	[3,3[3]]	4	| | |
	[1[3,3[3]]]=[3,3,6] ↔	(7)	| | | | | |	[1[3,3[3]]]+ = [3,3,6]+	(1)
${\displaystyle {\bar {BP}}_{3}}$ [4,3[3]]	[4,3[3]]	4	| | |
	[1[4,3[3]]]=[4,3,6] ↔	(7)	| | | | | |	[1+,4,(3[3])+]	(2)	↔
				[4,3[3]]+	(1)
${\displaystyle {\bar {HP}}_{3}}$ [5,3[3]]	[5,3[3]]	4	| | |
	[1[5,3[3]]]=[5,3,6] ↔	(7)	| | | | | |	[1[5,3[3]]]+ = [5,3,6]+	(1)
${\displaystyle {\bar {VP}}_{3}}$ [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

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)
• 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 [1]
• 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 [2] [3]
• N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [4]
• Klitzing, Richard. "Hyperbolic honeycombs H3 paracompact".