Uniform honeycombs in hyperbolic space

Template:Short description Template:Unsolved

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.

Template:Clear

Four compact regular hyperbolic honeycombs
Poincaré ball model projections
Order-4 dodecahedral honeycomb Template:Math	Order-5 dodecahedral honeycomb Template:Math
Order-5 cubic honeycomb Template:Math	Icosahedral honeycomb Template:Math

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,^[1] in order of the relative volumes of their fundamental simplex domains.^[2]

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. Two known examples are 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^[2]	Witt symbol	Coxeter notation	Commutator subgroup	Coxeter diagram	Honeycombs
H₁	0.0358850633	${\bar{B H}}_{3}$	[5,3,4]	[(5,3)⁺,4,1⁺] = [5,3^1,1]⁺	Template:CDD	15 forms, 2 regular
H₂	0.0390502856	${\bar{J}}_{3}$	[3,5,3]	[3,5,3]⁺	Template:CDD	9 forms, 1 regular
H₃	0.0717701267	${\bar{D H}}_{3}$	[5,3^1,1]	[5,3^1,1]⁺	Template:CDD	11 forms (7 overlap with [5,3,4] family, 4 are unique)
H₄	0.0857701820	${\hat{A B}}_{3}$	[(4,3,3,3)]	[(4,3,3,3)]⁺	Template:CDD	9 forms
H₅	0.0933255395	${\bar{K}}_{3}$	[5,3,5]	[5,3,5]⁺	Template:CDD	9 forms, 1 regular
H₆	0.2052887885	${\hat{A H}}_{3}$	[(5,3,3,3)]	[(5,3,3,3)]⁺	Template:CDD	9 forms
H₇	0.2222287320	${\hat{B B}}_{3}$	[(4,3)^[2]]	[(4,3⁺,4,3⁺)]	Template:CDD	6 forms
H₈	0.3586534401	${\hat{B H}}_{3}$	[(3,4,3,5)]	[(3,4,3,5)]⁺	Template:CDD	9 forms
H₉	0.5021308905	${\hat{H H}}_{3}$	[(5,3)^[2]]	[(5,3)^[2]]⁺	Template:CDD	6 forms

There are just two radical subgroups with non-simplicial 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 Template:CDD an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔ Template:CDD, which can be extended by restoring one mirror as Template:CDD. The other is [4,(3,5)^*], index 120 with a dodecahedral fundamental domain.

Paracompact hyperbolic uniform honeycombs

Template:Further 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	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
Tridental graphs	Template:CDD \| Template:CDD \| Template:CDD
Cyclic graphs	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
Loop-n-tail graphs	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD

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	Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD

[3,5,3] family

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

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.^[3] Another is constructed with 2 antipodal vertices removed.^[4]

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
#	Honeycomb name Coxeter diagram and Schläfli symbols	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Vertex figure	Picture
1	icosahedral (ikhon) Template:CDD t₀{3,5,3}				(12) (3.3.3.3.3)
2	rectified icosahedral (rih) Template:CDD t₁{3,5,3}	(2) (5.5.5)			(3) (3.5.3.5)
3	truncated icosahedral (tih) Template:CDD t_0,1{3,5,3}	(1) (5.5.5)			(3) (5.6.6)
4	cantellated icosahedral (srih) Template:CDD t_0,2{3,5,3}	(1) (3.5.3.5)	(2) (4.4.3)		(2) (3.5.4.5)
5	runcinated icosahedral (spiddih) Template:CDD t_0,3{3,5,3}	(1) (3.3.3.3.3)	(5) (4.4.3)	(5) (4.4.3)	(1) (3.3.3.3.3)
6	bitruncated icosahedral (dih) Template:CDD t_1,2{3,5,3}	(2) (3.10.10)			(2) (3.10.10)
7	cantitruncated icosahedral (grih) Template:CDD t_0,1,2{3,5,3}	(1) (3.10.10)	(1) (4.4.3)		(2) (4.6.10)
8	runcitruncated icosahedral (prih) Template:CDD t_0,1,3{3,5,3}	(1) (3.5.4.5)	(1) (4.4.3)	(2) (4.4.6)	(1) (5.6.6)
9	omnitruncated icosahedral (gipiddih) Template:CDD t_0,1,2,3{3,5,3}	(1) (4.6.10)	(1) (4.4.6)	(1) (4.4.6)	(1) (4.6.10)

#	Honeycomb name Coxeter diagram and Schläfli symbols	Cell counts/vertex and positions in honeycomb					Vertex figure	Picture
#	Honeycomb name Coxeter diagram and Schläfli symbols	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
[77]	partially diminished icosahedral (pidih) pd{3,5,3}^[5]				(12) (3.3.3.5)	(4) (5.5.5)
[78]	semi-partially diminished icosahedral spd{3,5,3}^[4]				(6) (3.3.3.5) (6) (3.3.3.3.3)	(2) (5.5.5)
Nonuniform	omnisnub icosahedral (snih) Template:CDD ht_0,1,2,3{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)

[5,3,4] family

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

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

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex				Vertex figure	Picture
#	Name of honeycomb Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Vertex figure	Picture
10	order-4 dodecahedral (doehon) Template:CDD ↔ Template:CDD	-	-	-	(8) Template:CDD (5.5.5)
11	rectified order-4 dodecahedral (riddoh) Template:CDD ↔ Template:CDD	(2) Template:CDD (3.3.3.3)	-	-	(4) Template:CDD (3.5.3.5)
12	rectified order-5 cubic (ripech) Template:CDD ↔ Template:CDD	(5) Template:CDD (3.4.3.4)	-	-	(2) Template:CDD (3.3.3.3.3)
13	order-5 cubic (pechon) Template:CDD	(20) Template:CDD (4.4.4)	-	-	-
14	truncated order-4 dodecahedral (tiddoh) Template:CDD ↔ Template:CDD	(1) Template:CDD (3.3.3.3)	-	-	(4) Template:CDD (3.10.10)
15	bitruncated order-5 cubic (ciddoh) Template:CDD ↔ Template:CDD	(2) Template:CDD (4.6.6)	-	-	(2) Template:CDD (5.6.6)
16	truncated order-5 cubic (tipech) Template:CDD	(5) Template:CDD (3.8.8)	-	-	(1) Template:CDD (3.3.3.3.3)
17	cantellated order-4 dodecahedral (sriddoh) Template:CDD ↔ Template:CDD	(1) Template:CDD (3.4.3.4)	(2) Template:CDD (4.4.4)	-	(2) Template:CDD (3.4.5.4)
18	cantellated order-5 cubic (sripech) Template:CDD	(2) Template:CDD (3.4.4.4)	-	(2) Template:CDD (4.4.5)	(1) Template:CDD (3.5.3.5)
19	runcinated order-5 cubic (sidpicdoh) Template:CDD	(1) Template:CDD (4.4.4)	(3) Template:CDD (4.4.4)	(3) Template:CDD (4.4.5)	(1) Template:CDD (5.5.5)
20	cantitruncated order-4 dodecahedral (griddoh) Template:CDD ↔ Template:CDD	(1) Template:CDD (4.6.6)	(1) Template:CDD (4.4.4)	-	(2) Template:CDD (4.6.10)
21	cantitruncated order-5 cubic (gripech) Template:CDD	(2) Template:CDD (4.6.8)	-	(1) Template:CDD (4.4.5)	(1) Template:CDD (5.6.6)
22	runcitruncated order-4 dodecahedral (pripech) Template:CDD	(1) Template:CDD (3.4.4.4)	(1) Template:CDD (4.4.4)	(2) Template:CDD (4.4.10)	(1) Template:CDD (3.10.10)
23	runcitruncated order-5 cubic (priddoh) Template:CDD	(1) Template:CDD (3.8.8)	(2) Template:CDD (4.4.8)	(1) Template:CDD (4.4.5)	(1) Template:CDD (3.4.5.4)
24	omnitruncated order-5 cubic (gidpicdoh) Template:CDD	(1) Template:CDD (4.6.8)	(1) Template:CDD (4.4.8)	(1) Template:CDD (4.4.10)	(1) Template:CDD (4.6.10)

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

[5,3,5] family

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

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
#	Name of honeycomb Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Vertex figure	Picture
25	(Regular) Order-5 dodecahedral (pedhon) Template:CDD t₀{5,3,5}				(20) (5.5.5)
26	rectified order-5 dodecahedral (ripped) Template:CDD t₁{5,3,5}	(2) (3.3.3.3.3)			(5) (3.5.3.5)
27	truncated order-5 dodecahedral (tipped) Template:CDD t_0,1{5,3,5}	(1) (3.3.3.3.3)			(5) (3.10.10)
28	cantellated order-5 dodecahedral (sripped) Template:CDD t_0,2{5,3,5}	(1) (3.5.3.5)	(2) (4.4.5)		(2) (3.5.4.5)
29	Runcinated order-5 dodecahedral (spidded) Template:CDD t_0,3{5,3,5}	(1) (5.5.5)	(3) (4.4.5)	(3) (4.4.5)	(1) (5.5.5)
30	bitruncated order-5 dodecahedral (diddoh) Template:CDD t_1,2{5,3,5}	(2) (5.6.6)			(2) (5.6.6)
31	cantitruncated order-5 dodecahedral (gripped) Template:CDD t_0,1,2{5,3,5}	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.10)
32	runcitruncated order-5 dodecahedral (pripped) Template:CDD t_0,1,3{5,3,5}	(1) (3.5.4.5)	(1) (4.4.5)	(2) (4.4.10)	(1) (3.10.10)
33	omnitruncated order-5 dodecahedral (gipidded) Template:CDD t_0,1,2,3{5,3,5}	(1) (4.6.10)	(1) (4.4.10)	(1) (4.4.10)	(1) (4.6.10)

#	Name of honeycomb Coxeter diagram	Cells by location and count per vertex					Vertex figure	Picture
#	Name of honeycomb Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	Vertex figure	Picture
Nonuniform	omnisnub order-5 dodecahedral Template:CDD ht_0,1,2,3{5,3,5}	(1) Template:CDD (3.3.3.3.5)	(1) Template:CDD (3.3.3.5)	(1) Template:CDD (3.3.3.5)	(1) Template:CDD (3.3.3.3.5)	(4) +(3.3.3)

[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 Template:CDD. If the branch ring states match, an extended symmetry can double into the [5,3,4] family, Template:CDD ↔ Template:CDD.

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

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

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

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

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
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	vertex figure	Picture
38	tetrahedral-cubic (gadtatdic) Template:CDD {(3,3,3,4)}	(4) (3.3.3)	-	(4) (4.4.4)	(6) (3.4.3.4)
39	tetrahedral-octahedral (gacocaddit) Template:CDD {(3,3,4,3)}	(12) (3.3.3.3)	(8) (3.3.3)	-	(8) (3.3.3.3)
40	cyclotruncated tetrahedral-cubic (cytitch) Template:CDD ct{(3,3,3,4)}	(3) (3.6.6)	(1) (3.3.3)	(1) (4.4.4)	(3) (4.6.6)
41	cyclotruncated cube-tetrahedron (cyticth) Template:CDD ct{(4,3,3,3)}	(1) (3.3.3)	(1) (3.3.3)	(3) (3.8.8)	(3) (3.8.8)
42	cyclotruncated octahedral-tetrahedral (cytoth) Template:CDD ct{(3,3,4,3)}	(4) (3.6.6)	(4) (3.6.6)	(1) (3.3.3.3)	(1) (3.3.3.3)
43	rectified tetrahedral-cubic (ritch) Template:CDD 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)
44	truncated tetrahedral-cubic (titch) Template:CDD t{(3,3,3,4)}	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.8.8)	(2) (4.6.8)
45	truncated tetrahedral-octahedral (titdoh) Template:CDD t{(3,3,4,3)}	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.4.4)	(1) (4.6.6)
46	omnitruncated tetrahedral-cubic (otitch) Template:CDD tr{(3,3,3,4)}	(1) (4.6.6)	(1) (4.6.6)	(1) (4.6.8)	(1) (4.6.8)
Nonuniform	omnisnub tetrahedral-cubic Template:CDD 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)

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

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

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
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	vertex figure	Picture
47	tetrahedral-dodecahedral Template:CDD	(4) (3.3.3)	-	(4) (5.5.5)	(6) (3.5.3.5)
48	tetrahedral-icosahedral Template:CDD	(30) (3.3.3.3)	(20) (3.3.3)	-	(12) (3.3.3.3.3)
49	cyclotruncated tetrahedral-dodecahedral Template:CDD	(3) (3.6.6)	(1) (3.3.3)	(1) (5.5.5)	(3) (5.6.6)
52	rectified tetrahedral-dodecahedral Template:CDD	(1) (3.3.3.3)	(2) (3.4.3.4)	(1) (3.5.3.5)	(2) (3.4.5.4)
53	truncated tetrahedral-dodecahedral Template:CDD	(1) (3.6.6)	(1) (3.4.3.4)	(1) (3.10.10)	(2) (4.6.10)
54	truncated tetrahedral-icosahedral Template:CDD	(2) (4.6.6)	(1) (3.6.6)	(1) (3.4.5.4)	(1) (5.6.6)

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

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

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

This symmetry family is also related to a radical subgroup, index 6, Template:CDD ↔ Template:CDD, 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
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	vertex figure	Pictures
56	cubic-octahedral (cohon) Template:CDD	(6) (3.3.3.3)	-	(8) (4.4.4)	(12) (3.4.3.4)
60	truncated cubic-octahedral (tucoh) Template:CDD	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.8.8)	(2) (4.6.8)

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

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

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

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

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

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

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
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	vertex figure	Picture
62	octahedral-dodecahedral Template:CDD	(6) (3.3.3.3)	-	(8) (5.5.5)	(1) (3.5.3.5)
63	cubic-icosahedral Template:CDD	(30) (3.4.3.4)	(20) (4.4.4)	-	(12) (3.3.3.3.3)
64	cyclotruncated octahedral-dodecahedral Template:CDD	(3) (4.6.6)	(1) (4.4.4)	(1) (5.5.5)	(3) (5.6.6)
67	rectified octahedral-dodecahedral Template:CDD	(1) (3.4.3.4)	(2) (3.4.4.4)	(1) (3.5.3.5)	(2) (3.4.5.4)
68	truncated octahedral-dodecahedral Template:CDD	(1) (4.6.6)	(1) (3.4.4.4)	(1) (3.10.10)	(2) (4.6.10)
69	truncated cubic-dodecahedral Template:CDD	(2) (4.6.8)	(1) (3.8.8)	(1) (3.4.5.4)	(1) (5.6.6)

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

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

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

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
#	Honeycomb name Coxeter diagram	0 Template:CDD	1 Template:CDD	2 Template:CDD	3 Template:CDD	Alt	vertex figure	Picture
71	dodecahedral-icosahedral Template:CDD	(12) (3.3.3.3.3)	-	(20) (5.5.5)	(30) (3.5.3.5)
72	cyclotruncated icosahedral-dodecahedral Template:CDD	(3) (5.6.6)	(1) (5.5.5)	(1) (5.5.5)	(3) (5.6.6)
73	cyclotruncated dodecahedral-icosahedral Template:CDD	(1) (3.3.3.3.3)	(1) (3.3.3.3.3)	(3) (3.10.10)	(3) (3.10.10)
74	rectified dodecahedral-icosahedral Template:CDD	(1) (3.5.3.5)	(2) (3.4.5.4)	(1) (3.5.3.5)	(2) (3.4.5.4)
75	truncated dodecahedral-icosahedral Template:CDD	(1) (5.6.6)	(1) (3.4.5.4)	(1) (3.10.10)	(2) (4.6.10)
76	omnitruncated dodecahedral-icosahedral Template:CDD	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.10)	(1) (4.6.10)
Nonuniform	omnisnub dodecahedral-icosahedral Template:CDD	(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)

Other non-Wythoffians

There are infinitely many known non-Wythoffian uniform compact hyperbolic honeycombs, and there may be more undiscovered ones. Two have been listed above as diminishings of the icosahedral honeycomb {3,5,3}.^[6]

In 1997 Wendy Krieger discovered an infinite series of uniform hyperbolic honeycombs with pseudoicosahedral vertex figures, made from 8 cubes and 12 p-gonal prisms at a vertex for any integer p. In the case p = 4, all cells are cubes and the result is the order-5 cubic honeycomb. The case p = 2 degenerates to the Euclidean cubic honeycomb.^[6]

Another four known ones are related to noncompact families. The tessellation Template:CDD consists of truncated cubes Template:CDD and infinite order-8 triangular tilings Template:CDD. However the latter intersect the sphere at infinity orthogonally, having exactly the same curvature as the hyperbolic space, and can be replaced by mirror images of the remainder of the tessellation, resulting in a compact uniform honeycomb consisting only of the truncated cubes. (So they are analogous to the hemi-faces of spherical hemipolyhedra.)^[6]^[7] Something similar can be done with the tessellation Template:CDD consisting of small rhombicuboctahedra Template:CDD, infinite order-8 triangular tilings Template:CDD, and infinite order-8 square tilings Template:CDD. The order-8 square tilings already intersect the sphere at infinity orthogonally, and if the order-8 triangular tilings are augmented with a set of triangular prisms, the surface passing through their centre points also intersects the sphere at infinity orthogonally. After replacing with mirror images, the result is a compact honeycomb containing the small rhombicuboctahedra and the triangular prisms.^[8] Two more such constructions were discovered in 2023. The first one arises from the fact that Template:CDD and Template:CDD have the same circumradius; the former has truncated octahedra Template:CDD and order-6 square tilings Template:CDD, while the latter has cuboctahedra Template:CDD and order-6 square tilings Template:CDD. A compact uniform honeycomb is taken by discarding the order-6 square tilings they have in common, using only the truncated octahedra and cuboctahedra. The second one arises from a similar construction involving Template:CDD (which has small rhombicosidodecahedra Template:CDD, octahedra Template:CDD, and order-4 pentagonal tilings Template:CDD) and Template:CDD (which is the prism of the order-4 pentagonal tiling, having pentagonal prisms Template:CDD and order-4 pentagonal tilings Template:CDD). These two likewise have the same circumradius, and a compact uniform honeycomb is taken by using only the finite cells of both, discarding the order-4 pentagonal tilings they have in common.^[9]

Another non-Wythoffian was discovered in 2021. It has as vertex figure a snub cube with 8 vertices removed and contains two octahedra and eight snub cubes at each vertex.^[6] Subsequently Krieger found a non-Wythoffian with a snub cube as the vertex figure, containing 32 tetrahedra and 6 octahedra at each vertex, and that the truncated and rectified versions of this honeycomb are still uniform. In 2022, Richard Klitzing generalised this construction to use any snub Template:CDD as vertex figure: the result is compact for p=4 or 5 (with a snub cube or snub dodecahedral vertex figure respectively), paracompact for p=6 (with a snub trihexagonal tiling as the vertex figure), and hypercompact for p>6. Again, the truncated and rectified versions of these honeycombs are still uniform.^[6]

There are also other forms based on parallelepiped domains. Two known forms generalise the cubic-octahedral honeycomb, having distorted small rhombicuboctahedral vertex figures. One form has small rhombicuboctahedra, cuboctahedra, and cubes; another has small rhombicosidodecahedra, icosidodecahedra, and cubes. (The version with tetrahedral-symmetry polyhedra is the cubic-octahedral honeycomb, using cuboctahedra, octahedra, and cubes).^[9]

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	${\bar{B H}}_{3}$ [4,3,5] Template:CDD	[4,3,5] Template:CDD	15	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1⁺,4,(3,5)⁺]	(2)	Template:CDD (= Template:CDD) Template:CDD
H₁	${\bar{B H}}_{3}$ [4,3,5] Template:CDD	[4,3,5] Template:CDD	15		[4,3,5]⁺	(1)	Template:CDD
H₂	${\bar{J}}_{3}$ [3,5,3] Template:CDD	[3,5,3] Template:CDD	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₂	${\bar{J}}_{3}$ [3,5,3] Template:CDD	[2⁺[3,5,3]] Template:CDD	5	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[3,5,3]]⁺	(1)	Template:CDD
H₃	${\bar{D H}}_{3}$ [5,3^1,1] Template:CDD	[5,3^1,1] Template:CDD	4	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₃	${\bar{D H}}_{3}$ [5,3^1,1] Template:CDD	[1[5,3^1,1]]=[5,3,4] Template:CDD ↔ Template:CDD	(7)	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD	[1[5,3^1,1]]⁺ =[5,3,4]⁺	(1)	Template:CDD
H₄	${\hat{A B}}_{3}$ [(4,3,3,3)] Template:CDD	[(4,3,3,3)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₄	${\hat{A B}}_{3}$ [(4,3,3,3)] Template:CDD	[2⁺[(4,3,3,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(4,3,3,3)]]⁺	(1)	Template:CDD
H₅	${\bar{K}}_{3}$ [5,3,5] Template:CDD	[5,3,5] Template:CDD	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₅	${\bar{K}}_{3}$ [5,3,5] Template:CDD	[2⁺[5,3,5]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[5,3,5]]⁺	(1)	Template:CDD
H₆	${\hat{A H}}_{3}$ [(5,3,3,3)] Template:CDD	[(5,3,3,3)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₆	${\hat{A H}}_{3}$ [(5,3,3,3)] Template:CDD	[2⁺[(5,3,3,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(5,3,3,3)]]⁺	(1)	Template:CDD
H₇	${\hat{B B}}_{3}$ [(3,4)^[2]] Template:CDD	[(3,4)^[2]]	2	Template:CDD \| Template:CDD
		[2⁺[(3,4)^[2]]] Template:CDD	1	Template:CDD
		[2⁺[(3,4)^[2]]] Template:CDD	1	Template:CDD
		[2⁺[(3,4)^[2]]] Template:CDD	1	Template:CDD	[2⁺[(3⁺,4)^[2]]]	(1)	Template:CDD
		[(2,2)⁺[(3,4)^[2]]] Template:CDD	1	Template:CDD	[(2,2)⁺[(3,4)^[2]]]⁺	(1)	Template:CDD
H₈	${\hat{B H}}_{3}$ [(5,3,4,3)] Template:CDD	[(5,3,4,3)]	6	Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD \| Template:CDD
H₈	${\hat{B H}}_{3}$ [(5,3,4,3)] Template:CDD	[2⁺[(5,3,4,3)]] Template:CDD	3	Template:CDD \| Template:CDD \| Template:CDD	[2⁺[(5,3,4,3)]]⁺	(1)	Template:CDD
H₉	${\hat{H H}}_{3}$ [(3,5)^[2]] Template:CDD	[(3,5)^[2]]	2	Template:CDD \| Template:CDD
		[2⁺[(3,5)^[2]]] Template:CDD	1	Template:CDD
		[2⁺[(3,5)^[2]]] Template:CDD	1	Template:CDD
		[2⁺[(3,5)^[2]]] Template:CDD	1	Template:CDD
		[(2,2)⁺[(3,5)^[2]]] Template:CDD	1	Template:CDD	[(2,2)⁺[(3,5)^[2]]]⁺	(1)	Template:CDD

Notes

Template:Reflist

References

J. Humphreys (1990), Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29
H.S.M. Coxeter (1954), "Regular Honeycombs in Hyperbolic Space" Proceedings of the International Congress of Mathematicians, vol. 3, North-Holland, pp. 155–169. Reprinted as Ch. 10 in Coxeter (1999), The Beauty of Geometry: Twelve Essays, Dover, Template:Isbn
H.S.M. Coxeter (1973), Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
J. Weeks The Shape of Space, 2nd ed. Template:Isbn, Chapters 16–17: Geometries on Three-manifolds I, II
A. Felikson (2002), "Coxeter Decompositions of Hyperbolic Tetrahedra" (preprint) Template:Arxiv
C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Can. J. Math. 19, 1179–1186, 1967. PDF [1] Template:Webarchive
N. W. Johnson (2018), Geometries and Transformations, Chapters 11–13
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz (1999), The size of a hyperbolic Coxeter simplex, Transformation Groups, Volume 4, Issue 4, pp 329–353 [2]
N. W. Johnson, R. Kellerhals, J.G. Ratcliffe, S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups H³: p130. [3]
Template:KlitzingPolytopes

↑ Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [4]
↑ ^2.0 ^2.1 Felikson, 2002
↑ Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [5]
↑ ^4.0 ^4.1 Template:Cite web
↑ Template:Cite web
↑ ^6.0 ^6.1 ^6.2 ^6.3 ^6.4 Template:Cite web
↑ Template:Cite web
↑ Template:Cite web
↑ ^9.0 ^9.1 Template:Cite web

[1] Humphreys, 1990, page 141, 6.9 List of hyperbolic Coxeter groups, figure 2 [4]

[Felikson,_2002-2] 2.0 ^2.1 Felikson, 2002

[3] Wendy Y. Krieger, Walls and bridges: The view from six dimensions, Symmetry: Culture and Science Volume 16, Number 2, pages 171–192 (2005) [5]

[klitzing-4] 4.0 ^4.1 Template:Cite web

[5] Template:Cite web

[klitzing2-6] 6.0 ^6.1 ^6.2 ^6.3 ^6.4 Template:Cite web

[klitzing3-7] Template:Cite web

[klitzing4-8] Template:Cite web

[klitzing5-9] 9.0 ^9.1 Template:Cite web

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Uniform honeycombs in hyperbolic space

Contents

Hyperbolic uniform honeycomb families

Compact uniform honeycomb families

Paracompact hyperbolic uniform honeycombs

[3,5,3] family

[5,3,4] family

[5,3,5] family

[5,3^1,1] family

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

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

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

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

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

Other non-Wythoffians

Summary enumeration of compact uniform honeycombs

See also

Notes

References

Navigation menu

Uniform honeycombs in hyperbolic space

Hyperbolic uniform honeycomb families

Compact uniform honeycomb families

Paracompact hyperbolic uniform honeycombs

[3,5,3] family

[5,3,4] family

[5,3,5] family

[5,31,1] family

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

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

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

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

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

Other non-Wythoffians

Summary enumeration of compact uniform honeycombs

See also

Notes

References

Navigation menu

Search

[5,3^1,1] family