Uniform honeycombs in hyperbolic space

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.

Four compact regular hyperbolic honeycombs

Order-4 dodecahedral honeycomb {5,3,4}	Order-5 dodecahedral honeycomb {5,3,5}
Order-5 cubic honeycomb {4,3,5}	Icosahedral honeycomb {3,5,3}
Poincaré ball model projections

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.

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	${\displaystyle {\bar {BH}}_{3}}$	[5,3,4]	[(5,3)+,4,1+] = [5,31,1]+		15 forms, 2 regular
H2	0.0390502856	${\displaystyle {\bar {J}}_{3}}$	[3,5,3]	[3,5,3]+		9 forms, 1 regular
H3	0.0717701267	${\displaystyle {\bar {DH}}_{3}}$	[5,31,1]	[5,31,1]+		11 forms (7 overlap with [5,3,4] family, 4 are unique)
H4	0.0857701820	${\displaystyle {\widehat {AB}}_{3}}$	[(4,3,3,3)]	[(4,3,3,3)]+		9 forms
H5	0.0933255395	${\displaystyle {\bar {K}}_{3}}$	[5,3,5]	[5,3,5]+		9 forms, 1 regular
H6	0.2052887885	${\displaystyle {\widehat {AH}}_{3}}$	[(5,3,3,3)]	[(5,3,3,3)]+		9 forms
H7	0.2222287320	${\displaystyle {\widehat {BB}}_{3}}$	[(4,3)[2]]	[(4,3+,4,3+)]		6 forms
H8	0.3586534401	${\displaystyle {\widehat {BH}}_{3}}$	[(3,4,3,5)]	[(3,4,3,5)]+		9 forms
H9	0.5021308905	${\displaystyle {\widehat {HH}}_{3}}$	[(5,3)[2]]	[(5,3)[2]]+		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 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.

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.

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.^[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
		0	1	2	3
1	icosahedral (ikhon) t0{3,5,3}				(12) (3.3.3.3.3)
2	rectified icosahedral (rih) t1{3,5,3}	(2) (5.5.5)			(3) (3.5.3.5)
3	truncated icosahedral (tih) t0,1{3,5,3}	(1) (5.5.5)			(3) (5.6.6)
4	cantellated icosahedral (srih) t0,2{3,5,3}	(1) (3.5.3.5)	(2) (4.4.3)		(2) (3.5.4.5)
5	runcinated icosahedral (spiddih) 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)
6	bitruncated icosahedral (dih) t1,2{3,5,3}	(2) (3.10.10)			(2) (3.10.10)
7	cantitruncated icosahedral (grih) t0,1,2{3,5,3}	(1) (3.10.10)	(1) (4.4.3)		(2) (4.6.10)
8	runcitruncated icosahedral (prih) t0,1,3{3,5,3}	(1) (3.5.4.5)	(1) (4.4.3)	(2) (4.4.6)	(1) (5.6.6)
9	omnitruncated icosahedral (gipiddih) t0,1,2,3{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
		0	1	2	3	Alt
[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) 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)

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 (doehon) ↔	-	-	-	(8) (5.5.5)
11	rectified order-4 dodecahedral (riddoh) ↔	(2) (3.3.3.3)	-	-	(4) (3.5.3.5)
12	rectified order-5 cubic (ripech) ↔	(5) (3.4.3.4)	-	-	(2) (3.3.3.3.3)
13	order-5 cubic (pechon)	(20) (4.4.4)	-	-	-
14	truncated order-4 dodecahedral (tiddoh) ↔	(1) (3.3.3.3)	-	-	(4) (3.10.10)
15	bitruncated order-5 cubic (ciddoh) ↔	(2) (4.6.6)	-	-	(2) (5.6.6)
16	truncated order-5 cubic (tipech)	(5) (3.8.8)	-	-	(1) (3.3.3.3.3)
17	cantellated order-4 dodecahedral (sriddoh) ↔	(1) (3.4.3.4)	(2) (4.4.4)	-	(2) (3.4.5.4)
18	cantellated order-5 cubic (sripech)	(2) (3.4.4.4)	-	(2) (4.4.5)	(1) (3.5.3.5)
19	runcinated order-5 cubic (sidpicdoh)	(1) (4.4.4)	(3) (4.4.4)	(3) (4.4.5)	(1) (5.5.5)
20	cantitruncated order-4 dodecahedral (griddoh) ↔	(1) (4.6.6)	(1) (4.4.4)	-	(2) (4.6.10)
21	cantitruncated order-5 cubic (gripech)	(2) (4.6.8)	-	(1) (4.4.5)	(1) (5.6.6)
22	runcitruncated order-4 dodecahedral (pripech)	(1) (3.4.4.4)	(1) (4.4.4)	(2) (4.4.10)	(1) (3.10.10)
23	runcitruncated order-5 cubic (priddoh)	(1) (3.8.8)	(2) (4.4.8)	(1) (4.4.5)	(1) (3.4.5.4)
24	omnitruncated order-5 cubic (gidpicdoh)	(1) (4.6.8)	(1) (4.4.8)	(1) (4.4.10)	(1) (4.6.10)

#	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 (apech) ↔	(20) (3.3.3)			(12) (3.3.3.3.3)
[35]	cantic order-5 cubic (tapech) ↔	(1) (3.5.3.5)	-	(2) (5.6.6)	(2) (3.6.6)
[36]	runcic order-5 cubic (birapech) ↔	(1) (5.5.5)	-	(3) (3.4.5.4)	(1) (3.3.3)
[37]	runcicantic order-5 cubic (bitapech) ↔	(1) (3.10.10)	-	(2) (4.6.10)	(1) (3.6.6)
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)	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)

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 (pedhon) t0{5,3,5}				(20) (5.5.5)
26	rectified order-5 dodecahedral (ripped) t1{5,3,5}	(2) (3.3.3.3.3)			(5) (3.5.3.5)
27	truncated order-5 dodecahedral (tipped) t0,1{5,3,5}	(1) (3.3.3.3.3)			(5) (3.10.10)
28	cantellated order-5 dodecahedral (sripped) t0,2{5,3,5}	(1) (3.5.3.5)	(2) (4.4.5)		(2) (3.5.4.5)
29	Runcinated order-5 dodecahedral (spidded) t0,3{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) t1,2{5,3,5}	(2) (5.6.6)			(2) (5.6.6)
31	cantitruncated order-5 dodecahedral (gripped) t0,1,2{5,3,5}	(1) (5.6.6)	(1) (4.4.5)		(2) (4.6.10)
32	runcitruncated order-5 dodecahedral (pripped) t0,1,3{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) t0,1,2,3{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
		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)

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 (apech) ↔	-	-	(12) (3.3.3.3.3)	(20) (3.3.3)
35	cantic order-5 cubic (tapech) ↔	(1) (3.5.3.5)	-	(2) (5.6.6)	(2) (3.6.6)
36	runcic order-5 cubic (birapech) ↔	(1) (5.5.5)	-	(3) (3.4.5.4)	(1) (3.3.3)
37	runcicantic order-5 cubic (bitapech) ↔	(1) (3.10.10)	-	(2) (4.6.10)	(1) (3.6.6)

#	Honeycomb name Coxeter diagram ↔	Cells by location (and count around each vertex)				vertex figure	Picture
		0	1	3	Alt
[10]	Order-4 dodecahedral (doehon) ↔	(4) (5.5.5)	-	-
[11]	rectified order-4 dodecahedral (riddoh) ↔	(2) (3.5.3.5)	-	(2) (3.3.3.3)
[12]	rectified order-5 cubic (ripech) ↔	(1) (3.3.3.3.3)	-	(5) (3.4.3.4)
[15]	bitruncated order-5 cubic (ciddoh) ↔	(1) (5.6.6)	-	(2) (4.6.6)
[14]	truncated order-4 dodecahedral (tiddoh) ↔	(2) (3.10.10)	-	(1) (3.3.3.3)
[17]	cantellated order-4 dodecahedral (sriddoh) ↔	(1) (3.4.5.4)	(2) (4.4.4)	(1) (3.4.3.4)
[20]	cantitruncated order-4 dodecahedral (griddoh) ↔	(1) × × €4.95 / month Favourites List Audio Listining Color Mode Monthly New Updates Fullname * Email * Credit or debit card Subscribe A PHP Error was encountered Severity: Notice Message: Undefined variable: userid Filename: user/popup_modal.php Line Number: 68 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_modal.php Line: 68 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 478 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once "> Subscribe Now × €4.95 € 9.90 / Modal title × Close Save changes × Uniform_honeycombs_in_hyperbolic_space" id="cancelSubscription" name="cancelSubscription" class="mb-0"> A PHP Error was encountered Severity: Notice Message: Undefined variable: user_membership Filename: user/popup_modal.php Line Number: 192 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_modal.php Line: 192 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 478 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once A PHP Error was encountered Severity: Warning Message: Invalid argument supplied for foreach() Filename: user/popup_modal.php Line Number: 192 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_modal.php Line: 192 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 478 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once A PHP Error was encountered Severity: Notice Message: Undefined variable: userid Filename: user/popup_modal.php Line Number: 208 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_modal.php Line: 208 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 478 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once "> Harry Potter Books × A PHP Error was encountered Severity: Notice Message: Undefined variable: products Filename: user/popup_harry_book.php Line Number: 24 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_harry_book.php Line: 24 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 479 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once A PHP Error was encountered Severity: Warning Message: Invalid argument supplied for foreach() Filename: user/popup_harry_book.php Line Number: 24 Backtrace: File: /var/www/wikizero/data/www/wikizero.com/application/views/user/popup_harry_book.php Line: 24 Function: _error_handler File: /var/www/wikizero/data/www/wikizero.com/application/views/page/index.php Line: 479 Function: view File: /var/www/wikizero/data/www/wikizero.com/application/controllers/Main.php Line: 107 Function: view File: /var/www/wikizero/data/www/wikizero.com/index.php Line: 315 Function: require_once