Uniform honeycombs in hyperbolic space

Four compact regular hyperbolic honeycombs

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{4,3,5}	{3,5,3}
Poincaré ball model projections

In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

Hyperbolic uniform honeycomb families

Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups.

Compact uniform honeycomb families

The nine compact Coxeter groups are listed here with their Coxeter diagrams, in order of the relative volumes of their fundamental simplex domains.

These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is cited with the {3,5,3} family below. Only two families are related as a mirror-removal halving: [5,3^1,1] ↔ [5,3,4,1⁺].

Indexed	Fundamental simplex volume	Witt symbol	Coxeter notation	Commutator subgroup	Coxeter diagram	Honeycombs
H1	0.0358850633	${\bar{BH}}_3$	[5,3,4]	[(5,3)+,4,1+] = [5,31,1]+		15 forms, 2 regular
H2	0.0390502856	${\bar{J}}_3$	[3,5,3]	[3,5,3]+		9 forms, 1 regular
H3	0.0717701267	${\bar{DH}}_3$	[5,31,1]	[5,31,1]+		11 forms (7 overlap with [5,3,4] family, 4 are unique)
H4	0.0857701820	${\widehat{AB}}_3$	[(4,3,3,3)]	[(4,3,3,3)]+		9 forms
H5	0.0933255395	${\bar{K}}_3$	[5,3,5]	[5,3,5]+		9 forms, 1 regular
H6	0.2052887885	${\widehat{AH}}_3$	[(5,3,3,3)]	[(5,3,3,3)]+		9 forms
H7	0.2222287320	${\widehat{BB}}_3$	[(4,3)[2]]	[(4,3+,4,3+)]		6 forms
H8	0.3586534401	${\widehat{BH}}_3$	[(3,4,3,5)]	[(3,4,3,5)]+		9 forms
H9	0.5021308905	${\widehat{HH}}_3$	[(5,3)[2]]	[(5,3)[2]]+		6 forms

There are just two radical subgroups with nonsimplectic domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is [(4,3,4,3^*)], represented by Coxeter diagrams an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔ , which can be extended by restoring one mirror as . The other is [4,(3,5)^*], index 120 with a dodecahedral fundamental domain.

Paracompact hyperbolic uniform honeycombs

Further information: paracompact uniform honeycombs

There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity.

Hyperbolic paracompact group summary

Type	Coxeter groups
Linear graphs	| | | | | |
Tridental graphs	| |
Cyclic graphs	| | | | | | | |
Loop-n-tail graphs	| | |

Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains (double tetrahedra) as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, with the constraint that at least one node must be ringed across infinite order branches.

Dimension	Rank	Graphs
H3	5	, , , , , , , , , , , , , , , , , , , , , , , , , , ,

[3,5,3] family

There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or

One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron.

The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew polyhedrons: {4,10|3} and {10,4|3}.

#	Honeycomb name Coxeter diagram and Schläfli symbols	Cell counts/vertex and positions in honeycomb				Vertex figure	Picture
		0	1	2	3
1	icosahedral t0{3,5,3}				(12) (3.3.3.3.3)
2	rectified icosahedral t1{3,5,3}	(2) (5.5.5)			(3) (3.5.3.5)	100px	120px
3	truncated icosahedral t0,1{3,5,3}	(1) (5.5.5)			(3) (5.6.6)	100px	120px
4	cantellated icosahedral t0,2{3,5,3}	(1) (3.5.3.5)	(2) (4.4.3)		(2) (3.5.4.5)	100px	120px
5	runcinated icosahedral t0,3{3,5,3}	(1) (3.3.3.3.3)	(5) (4.4.3)	(5) (4.4.3)	(1) (3.3.3.3.3)		120px
6	bitruncated icosahedral t1,2{3,5,3}	(2) (3.10.10)			(2) (3.10.10)		120px
7	cantitruncated icosahedral t0,1,2{3,5,3}	(1) (3.10.10)	(1) (4.4.3)		(2) (4.6.10)	100px	120px
8	runcitruncated icosahedral t0,1,3{3,5,3}	(1) (3.5.4.5)	(1) (4.4.3)	(2) (4.4.6)	(1) (5.6.6)	100px	120px
9	omnitruncated icosahedral t0,1,2,3{3,5,3}	(1) (4.6.10)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.10)	100px	120px

#	Honeycomb name Coxeter diagram and Schläfli symbols	Cell counts/vertex and positions in honeycomb					Vertex figure	Picture
		0	1	2	3	Alt
[77]	partially diminished icosahedral pd{3,5,3}				(12) (3.3.3.5)	(4) (5.5.5)	100px	120px
Nonuniform	omnisnub icosahedral ht0,1,2,3{3,5,3}	(1) (3.3.3.3.5)	(1) (3.3.3.3	(1) (3.3.3.3)	(1) (3.3.3.3.5)	(4) +(3.3.3)	100px

[5,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or .

This family is related to the group [5,3^1,1] by a half symmetry [5,3,4,1⁺], or ↔ , when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔ .

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex				Vertex figure	Picture
		0	1	2	3
10	order-4 dodecahedral ↔	-	-	-	(8) (5.5.5)	100px	Error creating thumbnail: convert: improper image header `/usr/local/www/mediawiki/w/images/0/09/H3_534_CC_center.png' @ error/png.c/ReadPNGImage/4192. convert: no images defined `/tmp//transform_c95fbc3823aa.png' @ error/convert.c/ConvertImageCommand/3229. Error code: 1
11	rectified order-4 dodecahedral ↔	(2) (3.3.3.3)	-	-	(4) (3.5.3.5)	100px	120px
12	rectified order-5 cubic ↔	(5) (3.4.3.4)	-	-	(2) (3.3.3.3.3)	100px	120px
13	order-5 cubic	(20) (4.4.4)	-	-	-	100px
14	truncated order-4 dodecahedral ↔	(1) (3.3.3.3)	-	-	(4) (3.10.10)	100px	120px
15	bitruncated order-5 cubic ↔	(2) (4.6.6)	-	-	(2) (5.6.6)	100px	120px
16	truncated order-5 cubic	(5) (3.8.8)	-	-	(1) (3.3.3.3.3)	100px	120px
17	cantellated order-4 dodecahedral ↔	(1) (3.4.3.4)	(2) (4.4.4)	-	(2) (3.4.5.4)	100px	120px
18	cantellated order-5 cubic	(2) (3.4.4.4)	-	(2) (4.4.5)	(1) (3.5.3.5)	100px	120px
19	runcinated order-5 cubic	(1) (4.4.4)	(3) (4.4.4)	(3) (4.4.5)	(1) (5.5.5)	100px	120px
20	cantitruncated order-4 dodecahedral ↔	(1) (4.6.6)	(1) (4.4.4)	-	(2) (4.6.10)	100px	120px
21	cantitruncated order-5 cubic	(2) (4.6.8)	-	(1) (4.4.5)	(1) (5.6.6)	100px	120px
22	runcitruncated order-4 dodecahedral	(1) (3.4.4.4)	(1) (4.4.4)	(2) (4.4.10)	(1) (3.10.10)	100px	120px
23	runcitruncated order-5 cubic	(1) (3.8.8)	(2) (4.4.8)	(1) (4.4.5)	(1) (3.4.5.4)	100px	120px
24	omnitruncated order-5 cubic	(1) (4.6.8)	(1) (4.4.8)	(1) (4.4.10)	(1) (4.6.10)	100px	120px

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex					Vertex figure	Picture
		0	1	2	3	Alt
[34]	alternated order-5 cubic ↔	(20) (3.3.3)			(12) (3.3.3.3.3)		100px
[35]	cantic order-5 cubic ↔	(1) (3.5.3.5)	-	(2) (5.6.6)	(2) (3.6.6)		100px	120px
[36]	runcic order-5 cubic ↔	(1) (5.5.5)	-	(3) (3.4.5.4)	(1) (3.3.3)		100px	120px
[37]	runcicantic order-5 cubic ↔	(1) (3.10.10)	-	(2) (4.6.10)	(1) (3.6.6)		100px	120px
Nonuniform	snub rectified order-4 dodecahedral	(1) (3.3.3.3.3)	(1) (3.3.3)	-	(2) (3.3.3.3.5)	(4) +(3.3.3)	100px Irr. tridiminished icosahedron
Nonuniform	runcic snub rectified order-4 dodecahedral	(3.4.4.4)	(4.4.4.4)	-	(3.3.3.3.5)	+(3.3.3)
Nonuniform	omnisnub order-5 cubic	(1) (3.3.3.3.4)	(1) (3.3.3.4)	(1) (3.3.3.5)	(1) (3.3.3.3.5)	(4) +(3.3.3)	100px

[5,3,5] family

There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or

The bitruncated and runcinated forms (29 and 30) contain the faces of two regular skew polyhedrons: {4,6|5} and {6,4|5}.

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex				Vertex figure	Picture
		0	1	2	3
25	(Regular) Order-5 dodecahedral t0{5,3,5}				(20) (5.5.5)
26	rectified order-5 dodecahedral t1{5,3,5}	(2) (3.3.3.3.3)			(5) (3.5.3.5)	100px	120px
27	truncated order-5 dodecahedral t0,1{5,3,5}	(1) (3.3.3.3.3)			(5) (3.10.10)	100px	120px
28	cantellated order-5 dodecahedral t0,2{5,3,5}	(1) (3.5.3.5)	(2) (4.4.5)		(2) (3.5.4.5)	100px	120px
29	Runcinated order-5 dodecahedral t0,3{5,3,5}	(1) (5.5.5)	(3) (4.4.5)	(3) (4.4.5)	(1) (5.5.5)		120px
30	bitruncated order-5 dodecahedral t1,2{5,3,5}	(2) (5.6.6)			(2) (5.6.6)		120px
31	cantitruncated order-5 dodecahedral t0,1,2{5,3,5}	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.10)	100px	120px
32	runcitruncated order-5 dodecahedral t0,1,3{5,3,5}	(1) (3.5.4.5)	(1) (4.4.5)	(2) (4.4.10)	(1) (3.10.10)	100px	120px
33	omnitruncated order-5 dodecahedral t0,1,2,3{5,3,5}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.10)	(1) (4.6.10)	100px	120px

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex					Vertex figure	Picture
		0	1	2	3	Alt
Nonuniform	omnisnub order-5 dodecahedral ht0,1,2,3{5,3,5}	(1) (3.3.3.3.5)	(1) (3.3.3.5)	(1) (3.3.3.5)	(1) (3.3.3.3.5)	(4) +(3.3.3)	100px

[5,3^1,1] family

There are 11 forms (and only 4 not shared with [5,3,4] family), generated by ring permutations of the Coxeter group: [5,3^1,1] or . If the branch ring states match, an extended symmetry can double into the [5,3,4] family, ↔ .

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				vertex figure	Picture
		0	1	0'	3
34	alternated order-5 cubic ↔	-	-	(12) (3.3.3.3.3)	(20) (3.3.3)	100px
35	cantic order-5 cubic ↔	(1) (3.5.3.5)	-	(2) (5.6.6)	(2) (3.6.6)	100px	120px
36	runcic order-5 cubic ↔	(1) (5.5.5)	-	(3) (3.4.5.4)	(1) (3.3.3)	100px	120px
37	runcicantic order-5 cubic ↔	(1) (3.10.10)	-	(2) (4.6.10)	(1) (3.6.6)	100px	120px

#	Honeycomb name Coxeter diagram ↔	Cells by location (and count around each vertex)				vertex figure	Picture
		0	1	3	Alt
[10]	Order-4 dodecahedral ↔	(4) (5.5.5)	-	-		100px
[11]	rectified order-4 dodecahedral ↔	(2) (3.5.3.5)	-	(2) (3.3.3.3)		100px	120px
[12]	rectified order-5 cubic ↔	(1) (3.3.3.3.3)	-	(5) (3.4.3.4)		100px	120px
[15]	bitruncated order-5 cubic ↔	(1) (5.6.6)	-	(2) (4.6.6)		100px	120px
[14]	truncated order-4 dodecahedral ↔	(2) (3.10.10)	-	(1) (3.3.3.3)		100px	120px
[17]	cantellated order-4 dodecahedral ↔	(1) (3.4.5.4)	(2) (4.4.4)	(1) (3.4.3.4)		100px	120px
[20]	cantitruncated order-4 dodecahedral ↔	(1) (4.6.10)	(1) (4.4.4)	(1) (4.6.6)		100px	120px
Nonuniform	snub rectified order-4 dodecahedral ↔	(2) (3.3.3.3.5)	(1) (3.3.3)	(2) (3.3.3.3.3)	(4) +(3.3.3)	100px Irr. tridiminished icosahedron

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

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

The bitruncated and runcinated forms (41 and 42) contain the faces of two regular skew polyhedrons: {8,6|3} and {6,8|3}.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					vertex figure	Picture
		0	1	2	3	Alt
38	tetrahedral-cubic {(3,3,3,4)}	(4) (3.3.3)	-	(4) (4.4.4)	(6) (3.4.3.4)		100px	125px
39	tetrahedral-octahedral {(3,3,4,3)}	(12) (3.3.3.3)	(8) (3.3.3)	-	(8) (3.3.3.3)		100px	125px
40	cyclotruncated tetrahedral-cubic ct{(3,3,3,4)}	(3) (3.6.6)	(1) (3.3.3)	(1) (4.4.4)	(3) (4.6.6)		100px	125px
41	cyclotruncated cube-tetrahedron ct{(4,3,3,3)}	(1) (3.3.3)	(1) (3.3.3)	(3) (3.8.8)	(3) (3.8.8)			125px
42	cyclotruncated tetrahedral-octahedral ct{(3,3,4,3)}	(4) (3.6.6)	(4) (3.6.6)	(1) (3.3.3.3)	(1) (3.3.3.3)			125px
43	rectified tetrahedral-cubic r{(3,3,3,4)}	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.4.3.4)	(2) (3.4.4.4)		100px	125px
44	truncated tetrahedral-cubic t{(3,3,3,4)}	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.8.8)	(2) (4.6.8)		100px	125px
45	truncated tetrahedral-octahedral t{(3,3,4,3)}	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.4.4)	(1) (4.6.6)		100px	125px
46	omnitruncated tetrahedral-cubic tr{(3,3,3,4)}	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.8)	(1) (4.6.8)		100px	125px
Nonuniform	omnisnub tetrahedral-cubic sr{(3,3,3,4)}	(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(1) (3.3.3.3.4)	(1) (3.3.3.3.4)	(4) +(3.3.3)	100px

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

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

The bitruncated and runcinated forms (50 and 51) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				vertex figure	Picture
		0	1	2	3
47	tetrahedral-dodecahedral	(4) (3.3.3)	-	(4) (5.5.5)	(6) (3.5.3.5)	100px	120px
48	tetrahedral-icosahedral	(30) (3.3.3.3)	(20) (3.3.3)	-	(12) (3.3.3.3.3)	100px	120px
49	cyclotruncated tetrahedral-dodecahedral	(3) (3.6.6)	(1) (3.3.3)	(1) (5.5.5)	(3) (5.6.6)	100px	120px
52	rectified tetrahedral-dodecahedral	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.5.3.5)	(2) (3.4.5.4)	100px	120px
53	truncated tetrahedral-dodecahedral	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.10.10)	(2) (4.6.10)	100px	120px
54	truncated tetrahedral-icosahedral	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.5.4)	(1) (5.6.6)	100px	120px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)			vertex figure	Picture
		0,1	2,3	Alt
50	cyclotruncated dodecahedral-tetrahedral	(2) (3.3.3)	(6) (3.10.10)			120px
51	cyclotruncated tetrahedral-icosahedral	(10) (3.6.6)	(2) (3.3.3.3.3)			120px
55	omnitruncated tetrahedral-dodecahedral	(2) (4.6.6)	(2) (4.6.10)		100px	120px
Nonuniform	omnisnub tetrahedral-dodecahedral	(2) (3.3.3.3.3)	(2) (3.3.3.3.5)	(4) +(3.3.3)	100px

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

There are 6 forms, generated by ring permutations of the Coxeter group: . There are 4 extended symmetries possible based on the symmetry of the rings: , , , and .

This symmetry family is also related to a radical subgroup, index 6, ↔ , constructed by [(4,3,4,3^*)], and represents a trigonal trapezohedron fundamental domain.

The truncated forms (57 and 58) contain the faces of two regular skew polyhedrons: {6,6|4} and {8,8|3}.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				vertex figure	Pictures
		0	1	2	3
56	cubic-octahedral	(6) (3.3.3.3)	-	(8) (4.4.4)	(12) (3.4.3.4)	100px	120px
60	truncated cubic-octahedral	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.8.8)	(2) (4.6.8)	100px	120px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)			vertex figure	Picture
		0,3	1,2	Alt
57	cyclotruncated octahedral-cubic	(6) (4.6.6)	(2) (4.4.4)			120px
Nonuniform	cyclosnub octahedral-cubic	(4) (3.3.3.3.3)	(2) (3.3.3)	(4) +(3.3.3.3)	100px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)		vertex figure	Picture
		0,1	2,3
58	cyclotruncated cubic-octahedral	(2) (3.3.3.3)	(6) (3.8.8)		120px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)		vertex figure	Picture
		0,2	1,3
59	rectified cubic-octahedral	(2) (3.4.3.4)	(4) (3.4.4.4)	100px	120px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)		vertex figure	Picture
		0,1,2,3	Alt
61	omnitruncated cubic-octahedral	(4) (4.6.8)		100px	120px
Nonuniform	omnisnub cubic-octahedral	(4) (3.3.3.3.4)	(4) +(3.3.3)	100px

[(4,3,5,3)] family

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

The truncated forms (65 and 66) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				vertex figure	Picture
		0	1	2	3
62	octahedral-dodecahedral	(6) (3.3.3.3)	-	(8) (5.5.5)	(1) (3.5.3.5)	100px	120px
63	cubic-icosahedral	(30) (3.4.3.4)	(20) (4.4.4)	-	(12) (3.3.3.3.3)	100px	120px
64	cyclotruncated octahedral-dodecahedral	(3) (4.6.6)	(1) (4.4.4)	(1) (5.5.5)	(3) (5.6.6)		120px
67	rectified octahedral-dodecahedral	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (3.5.3.5)	(2) (3.4.5.4)	100px	120px
68	truncated octahedral-dodecahedral	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.10.10)	(2) (4.6.10)	100px	120px
69	truncated cubic-dodecahedral	(2) (4.6.8)	(1) (3.8.8)	(1) (3.4.5.4)	(1) (5.6.6)	100px	120px

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)			vertex figure	Picture
		0,1	2,3	Alt
65	cyclotruncated dodecahedral-octahedral	(2) (3.3.3.3)	(8) (3.10.10)			120px
66	cyclotruncated cubic-icosahedral	(10) (3.8.8)	(2) (3.3.3.3.3)		100px	120px
70	omnitruncated octahedral-dodecahedral	(2) (4.6.8)	(2) (4.6.10)		100px	120px
Nonuniform	omnisnub octahedral-dodecahedral	(2) (3.3.3.3.4)	(2) (3.3.3.3.5)	(4) +(3.3.3)	100px

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

There are 6 forms, generated by ring permutations of the Coxeter group: . There are 4 extended symmetries possible based on the symmetry of the rings: , , , and .

The truncated forms (72 and 73) contain the faces of two regular skew polyhedrons: {6,6|5} and {10,10|3}.

#	Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)					vertex figure	Picture
		0	1	2	3	Alt
71	dodecahedral-icosahedral	(12) (3.3.3.3.3)	-	(20) (5.5.5)	(30) (3.5.3.5)		100px	120px
72	cyclotruncated icosahedral-dodecahedral	(3) (5.6.6)	(1) (5.5.5)	(1) (5.5.5)	(3) (5.6.6)			120px
73	cyclotruncated dodecahedral-icosahedral	(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(3) (3.10.10)	(3) (3.10.10)			120px
74	rectified dodecahedral-icosahedral	(1) (3.5.3.5)	(2) (3.4.5.4)	(1) (3.5.3.5)	(2) (3.4.5.4)		100px	120px
75	truncated dodecahedral-icosahedral	(1) (5.6.6)	(1) (3.4.5.4)	(1) (3.10.10)	(2) (4.6.10)		100px	120px
76	omnitruncated dodecahedral-icosahedral	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.10)		100px	120px
Nonuniform	omnisnub dodecahedral-icosahedral	(1) (3.3.3.3.5)	(1) (3.3.3.3.5)	(1) (3.3.3.3.5)	(1) (3.3.3.3.5)	(4) +(3.3.3)	100px

Summary enumeration of compact uniform honeycombs

This is the complete enumeration of the 76 Wythoffian uniform honeycombs. The alternations are listed for completeness, but most are non-uniform.

Index	Coxeter group	Extended symmetry	Honeycombs		Chiral extended symmetry	Alternation honeycombs
H1	${\bar{BH}}_3$ [4,3,5]	[4,3,5]	15	| | | | | | | | | | | |	[1+,4,(3,5)+]	(2)	(= )
					[4,3,5]+	(1)
H2	${\bar{J}}_3$ [3,5,3]	[3,5,3]	6	| | | | |
		[2+[3,5,3]]	5	| |	[2+[3,5,3]]+	(1)
H3	${\bar{DH}}_3$ [5,31,1]	[5,31,1]	4	| | |
		[1[5,31,1]]=[5,3,4] ↔	(7)	| | | | | |	[1[5,31,1]]+ =[5,3,4]+	(1)
H4	${\widehat{AB}}_3$ [(4,3,3,3)]	[(4,3,3,3)]	6	| | | | |
		[2+[(4,3,3,3)]]	3	| |	[2+[(4,3,3,3)]]+	(1)
H5	${\bar{K}}_3$ [5,3,5]	[5,3,5]	6	| | | | |
		[2+[5,3,5]]	3	| |	[2+[5,3,5]]+	(1)
H6	${\widehat{AH}}_3$ [(5,3,3,3)]	[(5,3,3,3)]	6	| | | | |
		[2+[(5,3,3,3)]]	3	| |	[2+[(5,3,3,3)]]+	(1)
H7	${\widehat{BB}}_3$ [(3,4)[2]]	[(3,4)[2]]	2	|
		[2+[(3,4)[2]]]	1
		[2+[(3,4)[2]]]	1
		[2+[(3,4)[2]]]	1		[2+[(3+,4)[2]]]	(1)
		[(2,2)+[(3,4)[2]]]	1		[(2,2)+[(3,4)[2]]]+	(1)
H8	${\widehat{BH}}_3$ [(5,3,4,3)]	[(5,3,4,3)]	6	| | | | |
		[2+[(5,3,4,3)]]	3	| |	[2+[(5,3,4,3)]]+	(1)
H9	${\widehat{HH}}_3$ [(3,5)[2]]	[(3,5)[2]]	2	|
		[2+[(3,5)[2]]]	1
		[2+[(3,5)[2]]]	1
		[2+[(3,5)[2]]]	1
		[(2,2)+[(3,5)[2]]]	1		[(2,2)+[(3,5)[2]]]+	(1)

See also

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

References

• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space)
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II) [3]
• Coxeter Decompositions of Hyperbolic Tetrahedra, arXiv/PDF, A. Felikson, December 2002
• C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Canad. J. Math. 19, 1179–1186, 1967. PDF [4]
• Norman Johnson, Geometries and Transformations, Chapters 11,12,13, preprint 2011
• 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 [5]
• N.W. Johnson, R. Kellerhals, J.G. Ratcliffe,S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H³: p130. [6]
• Richard Klitzing, Hyperbolic honeycombs, H3 compact