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}	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}	[6,3,3]
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}	[4,4,3]
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}	[6,3,4]
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}	[6,3,5]
Order-6 hexagonal tiling honeycomb	{6,3,6}	{6,3}	{6}	{6}	{3,6}	Self-dual	[6,3,6]

Coxeter groups of paracompact uniform honeycombs


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

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

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

Tetrahedral hyperbolic paracompact group summary
Coxeter group	Simplex volume	Commutator subgroup	Unique honeycomb count
[6,3,3]	0.0422892336	[1⁺,6,(3,3)⁺] = [3,3^[3]]⁺	15
[4,4,3]	0.0763304662	[1⁺,4,1⁺,4,3⁺]	15
[3,3^[3]]	0.0845784672	[3,3^[3]]⁺	4
[6,3,4]	0.1057230840	[1⁺,6,3⁺,4,1⁺] = [3^[]x[]]⁺	15
[3,4^1,1]	0.1526609324	[3⁺,4^1⁺,1⁺]	4
[3,6,3]	0.1691569344	[3⁺,6,3⁺]	8
[6,3,5]	0.1715016613	[1⁺,6,(3,5)⁺] = [5,3^[3]]⁺	15
[6,3^1,1]	0.2114461680	[1⁺,6,(3^1,1)⁺] = [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
[4^1,1,1]	0.4579827971	[1⁺,4^{1⁺,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
#	Honeycomb name Coxeter diagram: Schläfli symbol	1	2	3	4	Vertex figure	Picture
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 t_0,1{6,3,3} or t{6,3,3}	(1) (3.3.3)	-	-	(3) (3.12.12)	Triangular pyramid
6	cantellated hexagonal t_0,2{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 t_0,3{6,3,3}	(1) (3.3.3)	(3) (4.4.3)	(3) (4.4.6)	(1) (6.6.6)
8	cantellated order-6 tetrahedral t_0,2{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 t_1,2{6,3,3} or 2t{6,3,3}	(2) (3.6.6)	-	-	(2) (6.6.6)
10	truncated order-6 tetrahedral t_0,1{3,3,6} or t{3,3,6}	(6) (3.6.6)	-	-	(1) (3.3.3.3.3.3)
11	cantitruncated hexagonal t_0,1,2{6,3,3} or tr{6,3,3}	(1) (3.6.6)	(1) (4.4.3)	-	(2) (4.6.12)
12	runcitruncated hexagonal t_0,1,3{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 t_0,1,3{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 t_0,1,2{3,3,6} or tr{3,3,6}	(2) (4.6.6)	-	(1) (4.4.6)	(1) (6.6.6)
15	omnitruncated hexagonal t_0,1,2,3{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
#	Honeycomb name Coxeter diagram: Schläfli symbol	1	2	3	4	Alt	Vertex figure	Picture
[137]	alternated hexagonal ( ↔ ) =	-	-		(4) (3.3.3.3.3.3)	(4) (3.3.3)	(3.6.6)
[138]	cantic hexagonal ↔	(1) (3.3.3.3)	-		(2) (3.6.3.6)	(2) (3.6.6)
[139]	runcic hexagonal ↔	(1) (4.4.4)	(1) (4.4.3)		(1) (3.3.3.3.3.3)	(3) (3.4.3.4)
[140]	runcicantic hexagonal ↔	(1) (3.6.6)	(1) (4.4.3)		(1) (3.6.3.6)	(2) (4.6.6)
Nonuniform	snub rectified order-6 tetrahedral ↔ sr{3,3,6}					Irr. (3.3.3)
Nonuniform	cantic snub order-6 tetrahedral sr₃{3,3,6}
Nonuniform	omnisnub order-6 tetrahedral ht_0,1,2,3{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
#	Name of honeycomb Coxeter diagram Schläfli symbol	0	1	2	3	Vertex figure	Picture
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 t_0,1{6,3,4} or t{6,3,4}	(1) (3.3.3.3)	-	-	(4) (3.12.12)
21	bitruncated order-6 cubic t_1,2{6,3,4} or 2t{6,3,4}	(2) (4.6.6)	-	-	(2) (6.6.6)
22	truncated order-6 cubic t_0,1{4,3,6} or t{4,3,6}	(6) (3.8.8)	-	-	(1) (3.3.3.3.3.3)
23	cantellated order-4 hexagonal t_0,2{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 t_0,2{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 t_0,3{6,3,4}	(1) (4.4.4)	(3) (4.4.4)	(3) (4.4.6)	(1) (6.6.6)
26	cantitruncated order-4 hexagonal t_0,1,2{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 t_0,1,2{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 t_0,1,3{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 t_0,1,3{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 t_0,1,2,3{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
#	Name of honeycomb Coxeter diagram Schläfli symbol	0	1	2	3	Alt	Vertex figure	Picture
[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 ↔ h_2,3{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 ↔ h_2,3{6,3,4}	(1) (3.8.8)	(1) (4.4.3)		(1) (3.6.3.6)	(2) (4.6.8)
[151]	quarter order-4 hexagonal ↔ q{6,3,4}	(3)	(1)	-	(1)	(3)
Nonuniform	bisnub order-6 cubic ↔ 2s{4,3,6}	(3.3.3.3.3.3)	-	-	(3.3.3.3.6)	+(3.3.3)
Nonuniform	runcic bisnub order-6 cubic
Nonuniform	snub rectified order-6 cubic ↔ sr{4,3,6}	(3.3.3.3.3)	(3.3.3)	(3.3.3.3)	(3.3.3.3.6)	+(3.3.3)
Nonuniform	runcic snub rectified order-6 cubic sr₃{4,3,6}
Nonuniform	snub rectified order-4 hexagonal ↔ sr{6,3,4}	(3.3.3.3.3.3)	(3.3.3)	-	(3.3.3.3.6)	+(3.3.3)
Nonuniform	runcisnub rectified order-4 hexagonal sr₃{6,3,4}
Nonuniform	omnisnub rectified order-6 cubic ht_0,1,2,3{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
#	Honeycomb name Coxeter diagram Schläfli symbol	0	1	2	3	Vertex figure	Picture
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 t_0,1{6,3,5} or t{6,3,5}	(1) (3.3.3.3.3)	-	-	(5) 3.12.12
36	cantellated order-5 hexagonal t_0,2{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 t_0,3{6,3,5}	(1) (5.5.5)	-	(6) (6.4.4)	(1) (6)³
38	cantellated order-6 dodecahedral t_0,2{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 t_1,2{6,3,5} or 2t{6,3,5}	(2) (5.6.6)	-	-	(2) (6)³
40	truncated order-6 dodecahedral t_0,1{5,3,6} or t{5,3,6}	(6) (3.10.10)	-	-	(1) (3)⁶
41	cantitruncated order-5 hexagonal t_0,1,2{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 t_0,1,3{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 t_0,1,3{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 t_0,1,2{5,3,6} or tr{5,3,6}	(1) (4.6.10)	-	(2) (6.4.4)	(1) (6)³
45	omnitruncated order-6 dodecahedral t_0,1,2,3{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
#	Honeycomb name Coxeter diagram Schläfli symbol	0	1	2	3	Alt	Vertex figure	Picture
[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 ↔ h_2,3{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 ht_0,1,2,3{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
#	Name of honeycomb Coxeter diagram Schläfli symbol	0	1	2	3	Vertex figure	Picture
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 t_0,1{6,3,6} or t{6,3,6}	(1) (3.3.3.3.3.3)	-	-	(6) (3.12.12)
49	cantellated order-6 hexagonal t_0,2{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 t_0,3{6,3,6}	(1) (6.6.6)	(3) (4.4.6)	(3) (4.4.6)	(1) (6.6.6)
51	cantitruncated order-6 hexagonal t_0,1,2{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 t_0,1,3{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 t_0,1,2,3{6,3,6}	(1) (4.6.12)	(1) (4.4.12)	(1) (4.4.12)	(1) (4.6.12)
[1]	bitruncated order-6 hexagonal ↔ ↔ t_1,2{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
#	Name of honeycomb Coxeter diagram Schläfli symbol	0	1	2	3	Alt	Vertex figure	Picture
[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 ↔ h_2,3{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 ht_0,3{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 ht_0,1,2,3{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Vertex figure	Picture
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 t_0,2{3,6,3} or rr{3,6,3}	(1) (3.6)²	(2) (4.4.3)	-	(2) (3.6.4.6)
57	runcinated triangular t_0,3{3,6,3}	(1) (3)⁶	(6) (4.4.3)	(6) (4.4.3)	(1) (3)⁶
58	bitruncated triangular t_1,2{3,6,3} or 2t{3,6,3}	(2) (3.12.12)	-	-	(2) (3.12.12)
59	cantitruncated triangular t_0,1,2{3,6,3} or tr{3,6,3}	(1) (3.12.12)	(1) (4.4.3)	-	(2) (4.6.12)
60	runcitruncated triangular t_0,1,3{3,6,3}	(1) (3.6.4.6)	(1) (4.4.3)	(2) (4.4.6)	(1) (6)³
61	omnitruncated triangular t_0,1,2,3{3,6,3}	(1) (4.6.12)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.12)
[1]	truncated triangular ↔ ↔ t_0,1{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Alt	Vertex figure	Picture
[56]	cantellated triangular = s₂{3,6,3}	(1) (3.6)²	-	-	(2) (3.6.4.6)	(3.4.4)
[60]	runcitruncated triangular = s_2,3{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 ht_0,1,2,3{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Vertex figure	Picture
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 = t_0,1{4,4,3} or t{4,4,3}	(1)	-	-	(3)
67	truncated order-4 octahedral t_0,1{3,4,4} or t{3,4,4}	(4)	-	-	(1)
68	bitruncated square t_1,2{4,4,3} or 2t{4,4,3}	(2)	-	-	(2)
69	cantellated square t_0,2{4,4,3} or rr{4,4,3}	(1)	(2)	-	(2)
70	cantellated order-4 octahedral t_0,2{3,4,4} or rr{3,4,4}	(2)	-	(2)	(1)
71	runcinated square t_0,3{4,4,3}	(1)	(3)	(3)	(1)
72	cantitruncated square t_0,1,2{4,4,3} or tr{4,4,3}	(1)	(1)	-	(2)
73	cantitruncated order-4 octahedral t_0,1,2{3,4,4} or tr{3,4,4}	(2)	-	(1)	(1)
74	runcitruncated square t_0,1,3{4,4,3}	(1)	(1)	(2)	(1)
75	runcitruncated order-4 octahedral t_0,1,3{3,4,4}	(1)	(2)	(1)	(1)
76	omnitruncated square t_0,1,2,3{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Alt	Vertex figure	Picture
[83]	alternated square ↔ h{4,4,3}	-	-	-		{4,3}	(4.3.4.3)
[84]	cantic square ↔ h₂{4,4,3}	(3.4.3.4)	-	(3.8.8)		(4.8.8)
[85]	runcic square ↔ h₃{4,4,3}	(3.3.3.3)	-	(3.4.4.4)		(4.4.4)
[86]	runcicantic square ↔	(4.6.6)	-	(3.4.4.4)		(4.8.8)
[153]	alternated rectified square ↔ hr{4,4,3}		-	-		{}x{3}
157			-	-		{}x{6}
Scaliform	snub order-4 octahedral = = s{3,4,4}		-	-		{}v{4}
Scaliform	runcisnub order-4 octahedral s₃{3,4,4}					cup-4
152	snub square = s{4,4,3}		-	-		{3,3}
Nonuniform	snub rectified order-4 octahedral sr{3,4,4}		-			irr. {3,3}
Nonuniform	alternated runcitruncated square ht_0,1,3{3,4,4}					irr. {}v{4}
Nonuniform	omnisnub square ht_0,1,2,3{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Symmetry	Vertex figure	Picture
77	order-4 square {4,4,4}	-	-	-		[4,4,4]	Cube
78	truncated order-4 square t_0,1{4,4,4} or t{4,4,4}		-	-		[4,4,4]
79	bitruncated order-4 square t_1,2{4,4,4} or 2t{4,4,4}		-	-		[[4,4,4]]
80	runcinated order-4 square t_0,3{4,4,4}					[[4,4,4]]
81	runcitruncated order-4 square t_0,1,3{4,4,4}					[4,4,4]
82	omnitruncated order-4 square t_0,1,2,3{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 ↔ t_0,2{4,4,4} or rr{4,4,4}			-		[4,4,4]
[66]	truncated order-4 square ↔ t_0,1,2{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
#	Honeycomb name Coxeter diagram and Schläfli symbol	0	1	2	3	Alt	Symmetry	Vertex figure	Picture
[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 = s_2,3{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	alternated runcinated order-4 square tiling ht_0,3{4,4,4}						[[(4,4,4,2⁺)]]
Scaliform	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]]
[152]	snub square tiling ↔ sr{4,4,4}			-			[(4,4)⁺,4]
Nonuniform	alternated runcitruncated order-4 square tiling ht_0,1,3{4,4,4}						[((2,4)⁺,4,4)]
Nonuniform	omnisnub order-4 square tiling ht_0,1,2,3{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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	Vertex figure	Picture
Scaliform	snub order-4 octahedral = = s{3,4^1,1}		-	-		irr. {}v{4}
Nonuniform	snub rectified order-4 octahedral ↔ sr{3,4^1,1}	(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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Vertex figure	Picture
[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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	Vertex figure	Picture
[77]	order-4 square ( ↔ ↔ ) =		-		-	Cube
[78]	truncated order-4 square ( ↔ ) = ( ↔ )
[83]	Alternated square ↔		-
Scaliform	Snub order-4 square		-
Nonuniform			-
Nonuniform			-
[153]	( ↔ ) = ( ↔ )
Nonuniform	Snub square ↔ ↔	(3.3.4.3.4)	(3.3.3)	(3.3.4.3.4)	(3.3.4.3.4)	+(3.3.3)

[6,3^1,1] 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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	Vertex figure	Picture
[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
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
91	tetrahedral-square	-	(6) (444)	(8) (333)	(12) (3434)	(3444)
92	cyclotruncated square-tetrahedral	(444)	(488)	(333)	(388)
93	cyclotruncated tetrahedral-square	(1) (3333)	(1) (444)	(4) (366)	(4) (466)
94	truncated tetrahedral-square	(1) (3444)	(1) (488)	(1) (366)	(2) (468)
[64]	( ↔ ) = rectified order-4 octahedral	(3434)	(4444)	(3434)	(3434)
[65]	( ↔ ) = order-4 octahedral	(3333)	-	(3333)	(3333)
[67]	( ↔ ) = truncated order-4 octahedral	(466)	(4444)	(3434)	(466)
[83]	alternated square ( ↔ ) =	(444)	(4444)	-	(444)	(4.3.4.3)
[84]	cantic square ( ↔ ) =	(388)	(488)	(3434)	(388)
[85]	runcic square ( ↔ ) =	(3444)	(3434)	(3333)	(3444)
[86]	runcicantic square ( ↔ ) =	(468)	(488)	(466)	(468)

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

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
95	cubic-square	(8) (4.4.4)	-	(6) (4.4.4.4)	(12) (4.4.4.4)	(3.4.4.4)
96	octahedral-square	(3.4.3.4)	(3.3.3.3)	-	(4.4.4.4)	(4.4.4.4)
97	cyclotruncated cubic-square	(4) (3.8.8)	(1) (3.3.3.3)	(1) (4.4.4.4)	(4) (4.8.8)
98	cyclotruncated square-cubic	(1) (4.4.4)	(1) (4.4.4)	(3) (4.8.8)	(3) (4.8.8)
99	cyclotruncated octahedral-square	(4) (4.6.6)	(4) (4.6.6)	(1) (4.4.4.4)	(1) (4.4.4.4)
100	rectified cubic-square	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (4.4.4.4)	(2) (4.4.4.4)
101	truncated cubic-square	(1) (4.8.8)	(1) (3.4.4.4)	(2) (4.8.8)	(1) (4.8.8)
102	truncated octahedral-square	(2) (4.6.8	(1) (4.6.6)	(1) (4.4.4.4)	(1) (4.8.8)
103	omnitruncated octahedral-square	(1) (4.6.8)	(1) (4.6.8)	(1) (4.8.8)	(1) (4.8.8)

Alternated forms
#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure
156	alternated cubic-square ↔	-				(3.4.4.4)
Nonuniform	snub octahedral-square
Nonuniform	cyclosnub square-cubic
Nonuniform	cyclosnub octahedral-square
Nonuniform	omnisnub cubic-square	(3.3.3.3.4)	(3.3.3.3.4)	(3.3.4.3.4)	(3.3.4.3.4)	+(3.3.3)

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure
[83]	alternated square ( ↔ ↔ ) =	(6) (4.4.4.4)	(6) (4.4.4.4)	(6) (4.4.4.4)	(6) (4.4.4.4)	(8) (4.4.4)	(4.3.4.3)
[77]	alternated order-4 square ↔		-
Nonsimplectic	cantic order-4 square ↔
Nonuniform	cyclosnub square
Nonuniform	snub order-4 square
Nonuniform	bisnub order-4 square ↔	(3.3.4.3.4)	(3.3.4.3.4)	(3.3.4.3.4)	(3.3.4.3.4)	+(3.3.3)

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure
105	tetrahedral-hexagonal	(4) (3.3.3)	-	(4) (6.6.6)	(6) (3.6.3.6)	(3.4.3.4)
106	tetrahedral-triangular	(3.3.3.3)	(3.3.3)	-	(3.3.3.3.3.3)	(3.4.6.4)
107	cyclotruncated tetrahedral-hexagonal	(3) (3.6.6)	(1) (3.3.3)	(1) (6.6.6)	(3) (6.6.6)
108	cyclotruncated hexagonal-tetrahedral	(1) (3.3.3)	(1) (3.3.3)	(4) (3.12.12)	(4) (3.12.12)
109	cyclotruncated tetrahedral-triangular	(6) (3.6.6)	(6) (3.6.6)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
110	rectified tetrahedral-hexagonal	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
111	truncated tetrahedral-hexagonal	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.12.12)	(2) (4.6.12)
112	truncated tetrahedral-triangular	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.6.4)	(1) (6.6.6)
113	omnitruncated tetrahedral-hexagonal	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.12)	(1) (4.6.12)

Alternated forms
#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure
Nonuniform	omnisnub tetrahedral-hexagonal	(3.3.3.3.3)	(3.3.3.3.3)	(3.3.3.3.6)	(3.3.3.3.6)	+(3.3.3)

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure
114	octahedral-hexagonal	(6) (3.3.3.3)	-	(8) (6.6.6)	(12) (3.6.3.6)
115	cubic-triangular	(∞) (3.4.3.4)	(∞) (4.4.4)	-	(∞) (3.3.3.3.3.3)	(3.4.6.4)
116	cyclotruncated octahedral-hexagonal	(3) (4.6.6)	(1) (4.4.4)	(1) (6.6.6)	(3) (6.6.6)
117	cyclotruncated hexagonal-octahedral	(1) (3.3.3.3)	(1) (3.3.3.3)	(4) (3.12.12)	(4) (3.12.12)
118	cyclotruncated cubic-triangular	(6) (3.8.8)	(6) (3.8.8)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
119	rectified octahedral-hexagonal	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
120	truncated octahedral-hexagonal	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.12.12)	(2) (4.6.12)
121	truncated cubic-triangular	(2) (4.6.8)	(1) (3.8.8)	(1) (3.4.6.4)	(1) (6.6.6)
122	omnitruncated octahedral-hexagonal	(1) (4.6.8)	(1) (4.6.8)	(1) (4.6.12)	(1) (4.6.12)

Alternated forms
#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure
Nonuniform	cyclosnub octahedral-hexagonal	(3.3.3.3.3)	(3.3.3)	(3.3.3.3.3.3)	(3.3.3.3.3.3)	irr. {3,4}
Nonuniform	omnisnub octahedral-hexagonal	(3.3.3.3.4)	(3.3.3.3.4)	(3.3.3.3.6)	(3.3.3.3.6)	irr. {3,3}

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
123	icosahedral-hexagonal	(6) (3.3.3.3.3)	-	(8) (6.6.6)	(12) (3.6.3.6)	3.4.5.4
124	dodecahedral-triangular	(30) (3.5.3.5)	(20) (5.5.5)	-	(12) (3.3.3.3.3.3)	(3.4.6.4)
125	cyclotruncated icosahedral-hexagonal	(3) (5.6.6)	(1) (5.5.5)	(1) (6.6.6)	(3) (6.6.6)
126	cyclotruncated hexagonal-icosahedral	(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(5) (3.12.12)	(5) (3.12.12)
127	cyclotruncated dodecahedral-triangular	(6) (3.10.10)	(6) (3.10.10)	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)
128	rectified icosahedral-hexagonal	(1) (3.5.3.5)	(2) (3.4.5.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
129	truncated icosahedral-hexagonal	(1) (5.6.6)	(1) (3.5.5.5)	(1) (3.12.12)	(2) (4.6.12)
130	truncated dodecahedral-triangular	(2) (4.6.10)	(1) (3.10.10)	(1) (3.4.6.4)	(1) (6.6.6)
131	omnitruncated icosahedral-hexagonal	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.12)	(1) (4.6.12)

Alternated forms
#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure	Picture
Nonuniform	omnisnub icosahedral-hexagonal	(3.3.3.3.5)	(3.3.3.3.5)	(3.3.3.3.6)	(3.3.3.3.6)	+(3.3.3)

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

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
132	hexagonal-triangular	(3.3.3.3.3.3)	-	(6.6.6)	(3.6.3.6)	(3.4.6.4)
133	cyclotruncated hexagonal-triangular	(1) (3.3.3.3.3.3)	(1) (3.3.3.3.3.3)	(3) (3.12.12)	(3) (3.12.12)
134	cyclotruncated triangular-hexagonal	(1) (3.6.3.6)	(2) (3.4.6.4)	(1) (3.6.3.6)	(2) (3.4.6.4)
135	rectified hexagonal-triangular	(1) (6.6.6)	(1) (3.4.6.4)	(1) (3.12.12)	(2) (4.6.12)
136	truncated hexagonal-triangular	(1) (4.6.12)	(1) (4.6.12)	(1) (4.6.12)	(1) (4.6.12)
[16]	order-4 hexagonal tiling =	(3) (6.6.6)	(1) (6.6.6)	(1) (6.6.6)	(3) (6.6.6)	(3.3.3.3)

Alternated forms
#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure	Picture
[141]	alternated order-4 hexagonal ↔ ↔ ↔	(3.3.3.3.3.3)	(3.3.3.3.3.3)	(3.3.3.3.3.3)	(3.3.3.3.3.3)	+(3.3.3.3)	(4.6.6)
Nonuniform	cyclocantisnub hexagonal-triangular
Nonuniform	cycloruncicantisnub hexagonal-triangular
Nonuniform	snub rectified hexagonal-triangular	(3.3.3.3.6)	(3.3.3.3.6)	(3.3.3.3.6)	(3.3.3.3.6)	+(3.3.3)

Loop-n-tail graphs

[3,3^[3]] family

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	0'	3	vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	vertex figure
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	vertex figure
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	0'	3	Alt	vertex figure	Picture
[54]	triangular tiling honeycomb ( ↔ ↔ ) =		-		-		(6.6.6)
[137]	alternated hexagonal ( ↔ ) = ( ↔ )		-			+(3.6.6)	(3.6.6)
[47]	rectified order-6 hexagonal ↔ ↔ ↔	(3.6.3.6)	-	(3.6.3.6)	(3.3.3.3.3.3)
[55]	cantic order-6 hexagonal ( ↔ ) = ( ↔ ) =	(1) (3.6.3.6)	-	(2) (6.6.6)	(2) (3.6.3.6)
Nonuniform	snub rectified order-6 hexagonal ↔	(3.3.3.3.6)	(3.3.3.3)	(3.3.3.3.6)	(3.3.3.3.3.3)	+(3.3.3)

Multicyclic graphs

[3^{[ ]×[ ]}] family

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

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure	Picture
#	Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure	Picture
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
#	Honeycomb name Coxeter diagram	0	1	2	3	Alt	Vertex figure	Picture
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	1	vertex figure
[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
${\bar {R}}_{3}$ [4,4,3]	[4,4,3]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[1⁺,4,1⁺,4,3⁺]	(6)	\|
${\bar {R}}_{3}$ [4,4,3]	[4,4,3]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[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]]	3	\| \|	[2⁺[4,4,4]]⁺	(1)
${\bar {V}}_{3}$ [6,3,3]	[6,3,3]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[1⁺,6,(3,3)⁺]	(2)	(↔ )
${\bar {V}}_{3}$ [6,3,3]	[6,3,3]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[6,3,3]⁺	(1)
${\bar {BV}}_{3}$ [6,3,4]	[6,3,4]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[1⁺,6,3⁺,4,1⁺]	(6)	\|
${\bar {BV}}_{3}$ [6,3,4]	[6,3,4]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[6,3,4]⁺	(1)
${\bar {HV}}_{3}$ [6,3,5]	[6,3,5]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[1⁺,6,(3,5)⁺]	(2)	(↔ )
${\bar {HV}}_{3}$ [6,3,5]	[6,3,5]	15	\| \| \| \| \| \| \| \| \| \| \| \|	[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⁺,6,2⁺)]]	(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
${\bar {DV}}_{3}$ [6,3^1,1]	[6,3^1,1]	4	\| \|
	[1[6,3^1,1]]=[6,3,4] ↔	(7)	\| \| \| \| \| \|	[1[1⁺,6,3^1,1]]⁺	(2)	(↔ )
	[1[6,3^1,1]]=[6,3,4] ↔	(7)	\| \| \| \| \| \|	[1[6,3^1,1]]⁺=[6,3,4]⁺	(1)
${\bar {O}}_{3}$ [3,4^1,1]	[3,4^1,1]	4	\| \|	[3⁺,4^1,1]⁺	(2)	↔
	[1[3,4^1,1]]=[3,4,4] ↔	(7)	\| \| \| \| \| \|	[1[3⁺,4^1,1]]⁺	(2)
	[1[3,4^1,1]]=[3,4,4] ↔	(7)	\| \| \| \| \| \|	[1[3,4^1,1]]⁺	(1)
${\bar {M}}_{3}$ [4^1,1,1]	[4^1,1,1]	0	(none)
	[1[4^1,1,1]]=[4,4,4] ↔	(4)	\| \|	[1[1⁺,4,1⁺,4^1,1]]⁺=[(4,1⁺,4,1⁺,4,2⁺)]	(4)	\|
	[3[4^1,1,1]]=[4,4,3] ↔	(3)	\| \|	[3[1⁺,4^1,1,1]]⁺=[1⁺,4,1⁺,4,3⁺]	(2)	(↔ )
	[3[4^1,1,1]]=[4,4,3] ↔	(3)	\| \|	[3[4^1,1,1]]⁺=[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)]]	3	\| \|	[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]]]=[4,4^1,1] ↔	(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	\| \| \| \|
${\widehat {AV}}_{3}$ [(6,3,3,3)]	[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)
${\widehat {BV}}_{3}$ [(3,4,3,6)]	[2⁺[(3,4,3,6)]]	3	\|	[2⁺[(3,4,3,6)]]⁺	(1)
${\widehat {HV}}_{3}$ [(3,5,3,6)]	[(3,5,3,6)]	6	\| \| \| \|
${\widehat {HV}}_{3}$ [(3,5,3,6)]	[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)]]=[3,4^1,1] ↔	(7)	\| \| \| \| \| \|	[1[(3,3,4,1⁺,4)]]⁺ = [3⁺,4^1,1]⁺	(2)	(= )
	[1[(4,4,3,3)]]=[3,4^1,1] ↔	(7)	\| \| \| \| \| \|	[1[(3,3,4,4)]]⁺ = [3,4^1,1]⁺	(1)
${\bar {DP}}_{3}$ [3^{[ ]x[ ]}]	[3^{[ ]x[ ]}]	1
	[1[3^{[ ]x[ ]}]]=[6,3^1,1] ↔	(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	\| \|
${\bar {P}}_{3}$ [3,3^[3]]	[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)	↔
	[1[4,3^[3]]]=[4,3,6] ↔	(7)	\| \| \| \| \| \|	[4,3^[3]]⁺	(1)
${\bar {HP}}_{3}$ [5,3^[3]]	[5,3^[3]]	4	\| \|
${\bar {HP}}_{3}$ [5,3^[3]]	[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[6,3^[3]]]⁺ = [6,3,6]⁺	(1)

Notes

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

References

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

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

[1]

Regular paracompact honeycombs

Coxeter groups of paracompact uniform honeycombs

Linear graphs

[6,3,3] family

[6,3,4] family

[6,3,5] family

[6,3,6] family

[3,6,3] family

[4,4,3] family

[4,4,4] family

Tridental graphs

[3,41,1] family

[4,41,1] family

[6,31,1] family

Cyclic graphs

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

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

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

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

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

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

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

Loop-n-tail graphs

[3,3[3]] family

[4,3[3]] family

[5,3[3]] family

[6,3[3]] family

Multicyclic graphs

[3[ ]×[ ]] family

[3[3,3]] family

Summary enumerations by family

Linear graphs

Tridental graphs

Cyclic graphs

Loop-n-tail graphs

See also

Notes

References

[3,4^1,1] family

[4,4^1,1] family

[6,3^1,1] family

[3,3^[3]] family

[4,3^[3]] family

[5,3^[3]] family

[6,3^[3]] family

[3^{[ ]×[ ]}] family

[3^[3,3]] family