Paracompact uniform honeycombs explained

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,,,,,,,,,, and, and are shown below. Four have finite Ideal polyhedral cells:,,, and .

NameSchläfli
Symbol
Coxeter
Cell
type
Face
type
Edge
figure
Vertex
figure
DualCoxeter
group
Order-6 tetrahedral honeycomb[6,3,3]
Hexagonal tiling honeycomb
Order-4 octahedral honeycomb[4,4,3]
Square tiling honeycomb
Triangular tiling honeycombSelf-dual[3,6,3]
Order-6 cubic honeycomb[6,3,4]
Order-4 hexagonal tiling honeycomb
Order-4 square tiling honeycombSelf-dual[4,4,4]
Order-6 dodecahedral honeycomb[6,3,5]
Order-5 hexagonal tiling honeycomb
Order-6 hexagonal tiling honeycombSelf-dual[6,3,6]

Coxeter groups of paracompact uniform honeycombs

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.

Coxeter group!Simplex
volume!Commutator subgroup!Unique honeycomb count
[6,3,3]0.0422892336[1<sup>+</sup>,6,(3,3)<sup>+</sup>] = [3,3<sup>[3]]+ 15
[4,4,3]0.0763304662[1<sup>+</sup>,4,1<sup>+</sup>,4,3<sup>+</sup>] 15
[3,3<sup>[3]]0.0845784672[3,3<sup>[3]]+ 4
[6,3,4]0.1057230840[1<sup>+</sup>,6,3<sup>+</sup>,4,1<sup>+</sup>] = [3<sup>[]x[]]+ 15
[3,4<sup>1,1</sup>]0.1526609324[3<sup>+</sup>,4<sup>1<sup>+</sup>,1<sup>+</sup></sup>] 4
[3,6,3]0.1691569344[3<sup>+</sup>,6,3<sup>+</sup>] 8
[6,3,5]0.1715016613[1<sup>+</sup>,6,(3,5)<sup>+</sup>] = [5,3<sup>[3]]+ 15
[6,3<sup>1,1</sup>]0.2114461680[1<sup>+</sup>,6,(3<sup>1,1</sup>)<sup>+</sup>] = [3<sup>[]x[]]+ 4
[4,3<sup>[3]]0.2114461680[1<sup>+</sup>,4,3<sup>[3]]+ = [3<sup>[]x[]]+4
[4,4,4]0.2289913985[4<sup>+</sup>,4<sup>+</sup>,4<sup>+</sup>]+ 6
[6,3,6]0.2537354016[1<sup>+</sup>,6,3<sup>+</sup>,6,1<sup>+</sup>] = [3<sup>[3,3]]+8
[(4,4,3,3)]0.3053218647[(4,1<sup>+</sup>,4,(3,3)<sup>+</sup>)]4
[5,3<sup>[3]]0.3430033226[5,3<sup>[3]]+4
[(6,3,3,3)]0.3641071004[(6,3,3,3)]+9
[3<sup>[]x[]]0.4228923360[3<sup>[]x[]]+1
[4<sup>1,1,1</sup>]0.4579827971[1<sup>+</sup>,4<sup>1<sup>+</sup>,1<sup>+</sup>,1<sup>+</sup></sup>]0
[6,3<sup>[3]]0.5074708032[1<sup>+</sup>,6,3<sup>[3]] = [3<sup>[3,3]]+2
[(6,3,4,3)]0.5258402692[(6,3<sup>+</sup>,4,3<sup>+</sup>)]9
[(4,4,4,3)]0.5562821156[(4,1<sup>+</sup>,4,1<sup>+</sup>,4,3<sup>+</sup>)]9
[(6,3,5,3)]0.6729858045[(6,3,5,3)]+9
[(6,3,6,3)]0.8457846720[(6,3<sup>+</sup>,6,3<sup>+</sup>)] 5
[(4,4,4,4)]0.9159655942[(4<sup>+</sup>,4<sup>+</sup>,4<sup>+</sup>,4<sup>+</sup>)]1
[3<sup>[3,3]]1.014916064[3<sup>[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 [&infin;,3,3,&infin;] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1<sup>+</sup>,4] : = . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is or, constructed as [4,4,1<sup>+</sup>,4] = [&infin;,4,4,&infin;] : = .

Removing a mirror from some of the cyclic hyperbolic Coxeter graphs become bow-tie graphs: [(3,3,4,1<sup>+</sup>,4)] = [((3,&infin;,3)),((3,&infin;,3))] or, [(3,4,4,1<sup>+</sup>,4)] = [((4,&infin;,3)),((3,&infin;,4))] or, [(4,4,4,1<sup>+</sup>,4)] = [((4,&infin;,4)),((4,&infin;,4))] or . =, =, = .

Another nonsimplectic half groups is ↔ .

A radical nonsimplectic subgroup is ↔, which can be doubled into a triangular prism domain as ↔ .

Linear graphs

[6,3,3] family

Alternated forms
Honeycomb name
Coxeter diagram:
Schläfli symbol		Cells by location
(and count around each vertex)		Vertex figurePicture
1
2
3
4
Alt
[137]alternated hexagonal
(↔) = 		--(4)

(3.3.3.3.3.3)		(4)

(3.3.3)
(3.6.6)
[138]cantic hexagonal
↔ 		(1)

(3.3.3.3)		-(2)

(3.6.3.6)		(2)

(3.6.6)
[139]runcic hexagonal
↔ 		(1)

(4.4.4)		(1)

(4.4.3)		(1)

(3.3.3.3.3.3)		(3)

(3.4.3.4)
[140]runcicantic hexagonal
↔ 		(1)

(3.6.6)		(1)

(4.4.3)		(1)

(3.6.3.6)		(2)

(4.6.6)
Nonuniformsnub rectified order-6 tetrahedral
↔
sr
Irr. (3.3.3)
Nonuniformcantic snub order-6 tetrahedral

sr3
Nonuniformomnisnub order-6 tetrahedral

ht0,1,2,3
Irr. (3.3.3)

[6,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or

[6,3,6] family

There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or

[3,6,3] family

There are 9 forms, generated by ring permutations of the Coxeter group: [3,6,3] or

[4,4,3] family

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

Alternated forms
Honeycomb name
Coxeter diagram
and Schläfli symbol		Cell counts/vertex
and positions in honeycomb		Vertex figurePicture
0
1
2
3
Alt
[83] alternated square
↔
h 		- - -
[84] cantic square
↔
h2
(3.4.3.4)		-
(3.8.8)
(4.8.8)
[85] runcic square
↔
h3
(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 		- -
157 - -
snub order-4 octahedral
= =
s 		- -
runcisnub order-4 octahedral

s3
152 snub square
=
s
- -
Nonuniform snub rectified order-4 octahedral

sr 		-
Nonuniform alternated runcitruncated square

ht0,1,3
Nonuniform omnisnub square

ht0,1,2,3



[4,4,4] family

There are 9 forms, generated by ring permutations of the Coxeter group: [4,4,4] or .

Tridental graphs

[3,4<sup>1,1</sup>] 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 figurePicture
0
1
0'
3
Alt
snub order-4 octahedral
= =
s 		- -
Nonuniformsnub rectified order-4 octahedral
↔
sr
(3.3.3.3.4)
(3.3.3)
(3.3.3.3.4)
(3.3.4.3.4)
+(3.3.3)

[4,4<sup>1,1</sup>] 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
(and count around each vertex)		Vertex figurePicture
0
1
0'
3
Alt
[77]order-4 square
(↔ ↔) = 		--

Cube
[78]truncated order-4 square
(↔) = (↔)
[83]Alternated square
↔ 		-

- align=center BGCOLOR="#e0f0f0"Nonsimplectic-
Nonsimplectic ↔ -
ScaliformSnub order-4 square
-
Nonuniform-
Nonuniform-
[153](↔)
= (↔)
NonuniformSnub square
↔ ↔

(3.3.4.3.4)

(3.3.3)

(3.3.4.3.4)

(3.3.4.3.4)
+(3.3.3)

[6,3<sup>1,1</sup>] 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<sup>1,1</sup>] or .

Honeycomb name
Coxeter diagram		Cells by location
(and count around each vertex)		Vertex figurePicture
0
1
0'
3
Alt
[141]alternated order-4 hexagonal
↔ ↔ ↔
(4.6.6)
Nonuniformbisnub order-4 hexagonal
↔
Nonuniformsnub 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 locationVertex figurePicture
0
1
2
3
Alt
snub order-4 octahedral
= = 		- -
Nonuniform
155 alternated tetrahedral-square
↔

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

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

[(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 ↔ .

!rowspan=2

Honeycomb name
Coxeter diagram		Cells by location
(and count around each vertex)		Vertex figure
0
1
2
3
Alt
[83]alternated square
(↔ ↔) = 		(6)

(4.4.4.4)
(6)

(4.4.4.4)
(6)

(4.4.4.4)
(6)

(4.4.4.4)
(8)

(4.4.4)
(4.3.4.3)
[77]alternated order-4 square
↔
-

Nonsimplecticcantic order-4 square
↔



Nonuniformcyclosnub square




Nonuniformsnub order-4 square




Nonuniformbisnub 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: .

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

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

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

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

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

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

Alternated forms
Honeycomb name
Coxeter diagram		Cells by location
(and count around each vertex)		Vertex figurePicture
0
1
2
3
Alt
[141]alternated order-4 hexagonal
↔ ↔ ↔
(3.3.3.3.3.3)
(3.3.3.3.3.3)
(3.3.3.3.3.3)
(3.3.3.3.3.3)
+(3.3.3.3)
(4.6.6)
Nonuniformcyclocantisnub hexagonal-triangular
Nonuniformcycloruncicantisnub hexagonal-triangular
Nonuniformsnub 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<sup>[3]] family

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

[4,3<sup>[3]] family

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

[5,3<sup>[3]] family

There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group: [5,3<sup>[3]] or . 7 are half symmetry forms of [5,3,6]: ↔ .

[6,3<sup>[3]] family

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

Multicyclic graphs

[3<sup>[ ]×[]] family

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

[3<sup>[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).

Summary enumerations by family

Linear graphs

Honeycombs!Chiral
extended
symmetry!colspan=2
Alternation honeycombs

{\bar{R}}3


[4,4,3]
[4,4,3]
15

[1<sup>+</sup>,4,1<sup>+</sup>,4,3<sup>+</sup>](6) (↔)
(↔)

[4,4,3]+(1)

{\bar{N}}3


[4,4,4]
[4,4,4]
3[1<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>](3) (↔ =)
[4,4,4]
↔ 		(3)[1<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>](3) (↔)
[2<sup>+</sup>[4,4,4]]
3[2<sup>+</sup>[(4,4<sup>+</sup>,4,2<sup>+</sup>)]](2)
[2<sup>+</sup>[4,4,4]]+(1)

{\bar{V}}3


[6,3,3]
[6,3,3]
15

[1<sup>+</sup>,6,(3,3)<sup>+</sup>](2) (↔)
[6,3,3]+(1)

{\bar{BV}}3


[6,3,4]
[6,3,4]
15

[1<sup>+</sup>,6,3<sup>+</sup>,4,1<sup>+</sup>](6) (↔)
(↔)

[6,3,4]+(1)

{\bar{HV}}3


[6,3,5]
[6,3,5]
15

[1<sup>+</sup>,6,(3,5)<sup>+</sup>](2) (↔)
[6,3,5]+(1)

{\bar{Y}}3


[3,6,3]
[3,6,3]
5
[3,6,3]
↔ 		(1)[2<sup>+</sup>[3<sup>+</sup>,6,3<sup>+</sup>]] (1)
[2<sup>+</sup>[3,6,3]]
3[2<sup>+</sup>[3,6,3]]+ (1)

{\bar{Z}}3


[6,3,6]
[6,3,6]
6
[1<sup>+</sup>,6,3<sup>+</sup>,6,1<sup>+</sup>](2) (↔)
[2<sup>+</sup>[6,3,6]]
↔ 		(1)[2<sup>+</sup>[(6,3<sup>+</sup>,6,2<sup>+</sup>)]](2)
[2<sup>+</sup>[6,3,6]]
2
[2<sup>+</sup>[6,3,6]]+(1)

Tridental graphs

Honeycombs!Chiral
extended
symmetry!colspan=2
Alternation honeycombs

{\bar{DV}}3


[6,3<sup>1,1</sup>]
[6,3<sup>1,1</sup>]4
[1[6,3<sup>1,1</sup>]]=[6,3,4]
↔ 		(7)[1[1<sup>+</sup>,6,3<sup>1,1</sup>]]+(2) (↔)
[1[6,3<sup>1,1</sup>]]+=[6,3,4]+(1)

{\bar{O}}3


[3,4<sup>1,1</sup>]
[3,4<sup>1,1</sup>]4[3<sup>+</sup>,4<sup>1,1</sup>]+(2) ↔
[1[3,4<sup>1,1</sup>]]=[3,4,4]
↔ 		(7)[1[3<sup>+</sup>,4<sup>1,1</sup>]]+(2)
[1[3,4<sup>1,1</sup>]]+(1)

{\bar{M}}3


[4<sup>1,1,1</sup>]
[4<sup>1,1,1</sup>]0(none)
[1[4<sup>1,1,1</sup>]]=[4,4,4]
↔ 		(4)[1[1<sup>+</sup>,4,1<sup>+</sup>,4<sup>1,1</sup>]]+=[(4,1<sup>+</sup>,4,1<sup>+</sup>,4,2<sup>+</sup>)](4) (↔)
[3[4<sup>1,1,1</sup>]]=[4,4,3]
↔ 		(3)[3[1<sup>+</sup>,4<sup>1,1,1</sup>]]+=[1<sup>+</sup>,4,1<sup>+</sup>,4,3<sup>+</sup>](2) (↔)
[3[4<sup>1,1,1</sup>]]+=[4,4,3]+(1)

Cyclic graphs

Honeycombs!Chiral
extended
symmetry!colspan=2
Alternation honeycombs

{\widehat{CR}}3


[(4,4,4,3)]
[(4,4,4,3)] 6[(4,1<sup>+</sup>,4,1<sup>+</sup>,4,3<sup>+</sup>)](2) ↔
[2<sup>+</sup>[(4,4,4,3)]]
3[2<sup>+</sup>[(4,4<sup>+</sup>,4,3<sup>+</sup>)]](2)
[2<sup>+</sup>[(4,4,4,3)]]+(1)

{\widehat{RR}}3


[4<sup>[4]]
[4<sup>[4]](none)
[2<sup>+</sup>[4<sup>[4]]]
1[2<sup>+</sup>[(4<sup>+</sup>,4)<sup>[2]]](1)
[1[4<sup>[4]]]=[4,4<sup>1,1</sup>]
↔ 		(2)[(1<sup>+</sup>,4)<sup>[4]](2) ↔
[2[4<sup>[4]]]=[4,4,4]
↔ 		(1)[2<sup>+</sup>[(1<sup>+</sup>,4,4)<sup>[2]]](1)
[(2<sup>+</sup>,4)[4<sup>[4]]]=[2<sup>+</sup>[4,4,4]]
= 		(1)[(2<sup>+</sup>,4)[4<sup>[4]]]+
= [2<sup>+</sup>[4,4,4]]+		(1)

{\widehat{AV}}3


[(6,3,3,3)]
[(6,3,3,3)] 6
[2<sup>+</sup>[(6,3,3,3)]]
3[2<sup>+</sup>[(6,3,3,3)]]+ (1)

{\widehat{BV}}3


[(3,4,3,6)]
[(3,4,3,6)] 6[(3<sup>+</sup>,4,3<sup>+</sup>,6)](1)
[2<sup>+</sup>[(3,4,3,6)]]
3[2<sup>+</sup>[(3,4,3,6)]]+ (1)

{\widehat{HV}}3


[(3,5,3,6)]
[(3,5,3,6)] 6
[2<sup>+</sup>[(3,5,3,6)]]
3[2<sup>+</sup>[(3,5,3,6)]]+ (1)

{\widehat{VV}}3


[(3,6)<sup>[2]]
[(3,6)<sup>[2]] 2
[2<sup>+</sup>[(3,6)<sup>[2]]]
1
[2<sup>+</sup>[(3,6)<sup>[2]]]
1
[2<sup>+</sup>[(3,6)<sup>[2]]]
= 		(1)[2<sup>+</sup>[(3<sup>+</sup>,6)<sup>[2]]](1)
[(2,2)<sup>+</sup>[(3,6)<sup>[2]]]
1[(2,2)<sup>+</sup>[(3,6)<sup>[2]]]+(1)
Honeycombs!Chiral
extended
symmetry!colspan=2
Alternation honeycombs

{\widehat{BR}}3


[(3,3,4,4)]
[(3,3,4,4)] 4
[1[(4,4,3,3)]]=[3,4<sup>1,1</sup>]
↔ 		(7)[1[(3,3,4,1<sup>+</sup>,4)]]+
= [3<sup>+</sup>,4<sup>1,1</sup>]+		(2) (=)
[1[(3,3,4,4)]]+
= [3,4<sup>1,1</sup>]+ 		(1)

{\bar{DP}}3


[3<sup>[ ]x[]]
[3<sup>[ ]x[]] 1
[1[3<sup>[ ]x[]]]=[6,3<sup>1,1</sup>]
↔ 		(2)
[1[3<sup>[ ]x[]]]=[4,3<sup>[3]]
↔ 		(2)
[2[3<sup>[ ]x[]]]=[6,3,4]
↔ 		(3)[2[3<sup>[ ]x[]]]+
=[6,3,4]+		(1)

{\bar{PP}}3


[3<sup>[3,3]]

[3<sup>[3,3]]0(none)
[1[3<sup>[3,3]]]=[6,3<sup>[3]]
↔ 		0(none)
[3[3<sup>[3,3]]]=[3,6,3]
↔ 		(2)
[2[3<sup>[3,3]]]=[6,3,6]
↔ 		(1)
[(3,3)[3<sup>[3,3]]]=[6,3,3]
= 		(1)[(3,3)[3<sup>[3,3]]]+
= [6,3,3]+ 		(1)

Loop-n-tail graphs

Symmetry in these graphs can be doubled by adding a mirror: [1[''n'',3<sup>[3]]] = [''n'',3,6]. Therefore ring-symmetry graphs are repeated in the linear graph families.

Honeycombs!Chiral
extended
symmetry!colspan=2
Alternation honeycombs

{\bar{P}}3


[3,3<sup>[3]]
[3,3<sup>[3]]4
[1[3,3<sup>[3]]]=[3,3,6]
↔ 		(7)[1[3,3<sup>[3]]]+
= [3,3,6]+		(1)

{\bar{BP}}3


[4,3<sup>[3]]
[4,3<sup>[3]]4
[1[4,3<sup>[3]]]=[4,3,6]
↔ 		(7)[1<sup>+</sup>,4,(3<sup>[3])+](2) ↔
[4,3<sup>[3]]+(1)

{\bar{HP}}3


[5,3<sup>[3]]
[5,3<sup>[3]]4
[1[5,3<sup>[3]]]=[5,3,6]
↔ 		(7)[1[5,3<sup>[3]]]+
= [5,3,6]+		(1)

{\bar{VP}}3


[6,3<sup>[3]]
[6,3<sup>[3]]2
[6,3<sup>[3]] = (2)(↔) (=)
[(3,3)[1<sup>+</sup>,6,3<sup>[3]]]=[6,3,3]
↔ ↔ 		(1)[(3,3)[1<sup>+</sup>,6,3<sup>[3]]]+(1)
[1[6,3<sup>[3]]]=[6,3,6]
↔ 		(6)[3[1<sup>+</sup>,6,3<sup>[3]]]+
= [3,6,3]+		(1) ↔ (=)
[1[6,3<sup>[3]]]+
= [6,3,6]+		(1)

See also

References

Notes and References

  1. https://arxiv.org/abs/math/0301133.pdf P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)

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