Tessellation of convex uniform polyhedron cells
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 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 Coxeter Cell
Face
Edge
Vertex Dual
Coxeter
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}
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}
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}
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}
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
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
[3,41,1 ]
0.1526609324
[3+ ,41+ ,1+ ]
4
[3,6,3]
0.1691569344
[3+ ,6,3+ ]
8
[6,3,5]
0.1715016613
[1+ ,6,(3,5)+ ] = [5,3[3] ]+
15
[6,31,1 ]
0.2114461680
[1+ ,6,(31,1 )+ ] = [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
[41,1,1 ]
0.4579827971
[1+ ,41+ ,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] 3 can be represented by + ,4] : tetrahedron into a quadrilateral pyramid. Another pyramid is + ,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 + ,4)] = [((4,∞,3)),((3,∞,4))] or + ,4)] = [((4,∞,4)),((4,∞,4))] or
Another nonsimplectic half groups is
A radical nonsimplectic subgroup is
Linear graphs
[6,3,3] family
#
Honeycomb name Coxeter diagram : Schläfli symbol
Cells by location
Vertex figure
Picture
1
2
3
4
1
hexagonal -
-
-
(4)(6.6.6)
Tetrahedron
2
rectified hexagonal 1 {6,3,3} or r{6,3,3}
(2)(3.3.3)
-
-
(3)(3.6.3.6)
Triangular prism
3
rectified order-6 tetrahedral 1 {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 (∞)(3.3.3)
-
-
-
Triangular tiling
5
truncated hexagonal 0,1 {6,3,3} or t{6,3,3}
(1)(3.3.3)
-
-
(3)(3.12.12)
Triangular pyramid
6
cantellated hexagonal 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 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 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 1,2 {6,3,3} or 2t{6,3,3}
(2)(3.6.6)
-
-
(2)(6.6.6)
10
truncated order-6 tetrahedral 0,1 {3,3,6} or t{3,3,6}
(6)(3.6.6)
-
-
(1)(3.3.3.3.3.3)
11
cantitruncated hexagonal 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 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 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 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 0,1,2,3 {6,3,3}
(1)(4.6.6)
(1)(4.4.6)
(1)(4.4.12)
(1)(4.6.12)
[6,3,4] family
There are 15 forms, generated by ring permutations of the Coxeter group : [6,3,4] or
#
Name of honeycombCoxeter diagram Schläfli symbol
Cells by location and count per vertex
Vertex figure
Picture
0
1
2
3
16
(Regular) order-4 hexagonal
-
-
-
(8)(6.6.6)
(3.3.3.3)
17
rectified order-4 hexagonal 1 {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 1 {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 (20)(4.4.4)
-
-
-
(3.3.3.3.3.3)
20
truncated order-4 hexagonal 0,1 {6,3,4} or t{6,3,4}
(1)(3.3.3.3)
-
-
(4)(3.12.12)
21
bitruncated order-6 cubic 1,2 {6,3,4} or 2t{6,3,4}
(2)(4.6.6)
-
-
(2)(6.6.6)
22
truncated order-6 cubic 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 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 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 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 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 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 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 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 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 honeycombCoxeter diagram Schläfli symbol
Cells by location and count per vertex
Vertex figure
Picture
0
1
2
3
Alt
[87]
alternated order-6 cubic (3.3.3)
(3.3.3.3.3.3)
(3.6.3.6)
[88]
cantic order-6 cubic 2 {4,3,6}
(2)(3.6.6)
-
-
(1)(3.6.3.6)
(2)(6.6.6)
[89]
runcic order-6 cubic 3 {4,3,6}
(1)(3.3.3)
-
-
(1)(6.6.6)
(3)(3.4.6.4)
[90]
runcicantic order-6 cubic 2,3 {4,3,6}
(1)(3.6.6)
-
-
(1)(3.12.12)
(2)(4.6.12)
[141]
alternated order-4 hexagonal -
-
(3.3.3.3.3.3)
(3.3.3.3)
(4.6.6)
[142]
cantic order-4 hexagonal 1 {6,3,4}
(1)(3.4.3.4)
-
(2)(3.6.3.6)
(2)(4.6.6)
[143]
runcic order-4 hexagonal 3 {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 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 (3)
(1)
-
(1)
(3)
Nonuniform
bisnub order-6 cubic
(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
(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 cubic3 {4,3,6}
Nonuniform
snub rectified order-4 hexagonal
(3.3.3.3.3.3)
(3.3.3)
-
(3.3.3.3.6)
(3.3.3)
Nonuniform
runcisnub rectified order-4 hexagonal3 {6,3,4}
Nonuniform
omnisnub rectified order-6 cubic0,1,2,3 {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 nameCoxeter diagram Schläfli symbol
Cells by location
Vertex figure
Picture
0
1
2
3
31
order-5 hexagonal -
-
-
(20)(6)3
Icosahedron
32
rectified order-5 hexagonal 1 {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 1 {5,3,6} or r{5,3,6}
(5)(3.5.3.5)
-
-
(2)(3)6
(6.4.4)
34
order-6 dodecahedral (5.5.5)
-
-
-
(∞)(3)6
35
truncated order-5 hexagonal 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 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 0,3 {6,3,5}
(1)(5.5.5)
-
(6)(6.4.4)
(1)(6)3
38
cantellated order-6 dodecahedral 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 1,2 {6,3,5} or 2t{6,3,5}
(2)(5.6.6)
-
-
(2)(6)3
40
truncated order-6 dodecahedral 0,1 {5,3,6} or t{5,3,6}
(6)(3.10.10)
-
-
(1)(3)6
41
cantitruncated order-5 hexagonal 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 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 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 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 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 nameCoxeter diagram Schläfli symbol
Cells by location
Vertex figure
Picture
0
1
2
3
Alt
[145]
alternated order-5 hexagonal -
-
-
(20)(3)6
(12)(3)5
(5.6.6)
[146]
cantic order-5 hexagonal 2 {6,3,5}
(1)(3.5.3.5)
-
(2)(3.6.3.6)
(2)(5.6.6)
[147]
runcic order-5 hexagonal 3 {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 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
(3.3.5.3.5) -
(3.3.3.3) (3.3.3.3.3.3)
Nonuniform
omnisnub order-5 hexagonal0,1,2,3 {6,3,5}
(3.3.5.3.5) (3.3.3.5) (3.3.3.6) (3.3.6.3.6)
[6,3,6] family
There are 9 forms, generated by ring permutations of the Coxeter group : [6,3,6] or
#
Name of honeycombCoxeter diagram Schläfli symbol
Cells by location and count per vertex
Vertex figure
Picture
0
1
2
3
46
order-6 hexagonal -
-
-
(20)(6.6.6)
(3.3.3.3.3.3)
47
rectified order-6 hexagonal 1 {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 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 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 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 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 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 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 1,2 {6,3,6} or 2t{6,3,6}
(2)(6.6.6)
-
-
(2)(6.6.6)
Alternated forms
#
Name of honeycombCoxeter diagram Schläfli symbol
Cells by location and count per vertex
Vertex figure
Picture
0
1
2
3
Alt
[47]
rectified order-6 hexagonal (2)(3.3.3.3.3.3)
-
-
(6)(3.6.3.6)
(6.4.4)
[54]
triangular -
-
-
(3.3.3.3.3.3)
(3.3.3.3.3.3)
{6,3}
[55]
cantic order-6 hexagonal 2 {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 3 {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 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 (3.3.3.3.6)
-
-
(3.3.3.3.6)
(3.3.3)
(3.6.6)
Nonuniform
snub rectified order-6 hexagonal
(3.3.3.3.3.3)
(3.3.3.3)
-
(3.3.3.3.6)
(3.3.3)
Nonuniform
alternated runcinated order-6 hexagonal0,3 {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 hexagonal0,1,2,3 {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 nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Vertex figure
Picture
0
1
2
3
54
triangular -
-
-
(∞){3,6}
{6,3}
55
rectified triangular 1 {3,6,3} or r{3,6,3}
(2)(6)3
-
-
(3)(3.6)2
(3.4.4)
56
cantellated triangular 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 0,3 {3,6,3}
(1)(3)6
(6)(4.4.3)
(6)(4.4.3)
(1)(3)6
58
bitruncated triangular 1,2 {3,6,3} or 2t{3,6,3}
(2)(3.12.12)
-
-
(2)(3.12.12)
59
cantitruncated triangular 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 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 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 0,1 {3,6,3} or t{3,6,3} = {6,3,3}
(1)(6)3
-
-
(3)(6)3
{3,3}
Alternated forms
#
Honeycomb nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Vertex figure
Picture
0
1
2
3
Alt
[56]
cantellated triangular 2 {3,6,3}
(1)(3.6)2
-
-
(2)(3.6.4.6)
(3.4.4)
[60]
runcitruncated triangular 2,3 {3,6,3}
(1)(6)3
-
(1)(4.4.3)
(1)(3.6.4.6)
(2)(4.4.6)
[137]
alternated hexagonal (3)6 -
-
(3)6 (3)3
(3.6.6)
Scaliform
runcisnub triangular 3 {3,6,3}
r{6,3} -
(3.4.4) (3)6 tricup
Nonuniform
omnisnub triangular tiling honeycomb0,1,2,3 {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 nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Vertex figure
Picture
0
1
2
3
62
square -
-
-
(6)
Cube
63
rectified square 1 {4,4,3} or r{4,4,3}
(2)
-
-
(3)
Triangular prism
64
rectified order-4 octahedral 1 {3,4,4} or r{3,4,4}
(4)
-
-
(2)
65
order-4 octahedral (∞)
-
-
-
66
truncated square 0,1 {4,4,3} or t{4,4,3}
(1)
-
-
(3)
67
truncated order-4 octahedral 0,1 {3,4,4} or t{3,4,4}
(4)
-
-
(1)
68
bitruncated square 1,2 {4,4,3} or 2t{4,4,3}
(2)
-
-
(2)
69
cantellated square 0,2 {4,4,3} or rr{4,4,3}
(1)
(2)
-
(2)
70
cantellated order-4 octahedral 0,2 {3,4,4} or rr{3,4,4}
(2)
-
(2)
(1)
71
runcinated square 0,3 {4,4,3}
(1)
(3)
(3)
(1)
72
cantitruncated square 0,1,2 {4,4,3} or tr{4,4,3}
(1)
(1)
-
(2)
73
cantitruncated order-4 octahedral 0,1,2 {3,4,4} or tr{3,4,4}
(2)
-
(1)
(1)
74
runcitruncated square 0,1,3 {4,4,3}
(1)
(1)
(2)
(1)
75
runcitruncated order-4 octahedral 0,1,3 {3,4,4}
(1)
(2)
(1)
(1)
76
omnitruncated square 0,1,2,3 {4,4,3}
(1)
(1)
(1)
(1)
Alternated forms
#
Honeycomb nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Vertex figure
Picture
0
1
2
3
Alt
[83]
alternated square -
-
-
{4,3} (4.3.4.3)
[84]
cantic square 2 {4,4,3}
(3.4.3.4)
-
(3.8.8)
(4.8.8)
[85]
runcic square 3 {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 -
-
{}x{3}
157 -
-
{}x{6}
Scaliform snub order-4 octahedral -
-
{}v{4}
Scaliform runcisnub order-4 octahedral 3 {3,4,4}cup-4
152 snub square
-
-
{3,3}
Nonuniform
snub rectified order-4 octahedral
-
irr. {3,3}
Nonuniform
alternated runcitruncated square0,1,3 {3,4,4}
irr. {}v{4}
Nonuniform
omnisnub square0,1,2,3 {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 nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Symmetry
Vertex figure
Picture
0
1
2
3
77
order-4 square -
-
-
[4,4,4]
Cube
78
truncated order-4 square 0,1 {4,4,4} or t{4,4,4}-
-
[4,4,4]
79
bitruncated order-4 square 1,2 {4,4,4} or 2t{4,4,4}-
-
[[4,4,4]]
80
runcinated order-4 square 0,3 {4,4,4}[[4,4,4]]
81
runcitruncated order-4 square 0,1,3 {4,4,4}[4,4,4]
82
omnitruncated order-4 square 0,1,2,3 {4,4,4}[[4,4,4]]
[62]
square 1 {4,4,4} or r{4,4,4}-
-
[4,4,4]
Square tiling
[63]
rectified square 0,2 {4,4,4} or rr{4,4,4}-
[4,4,4]
[66]
truncated order-4 square 0,1,2 {4,4,4} or tr{4,4,4}-
[4,4,4]
Alternated constructions
#
Honeycomb nameCoxeter diagram Schläfli symbol
Cell counts/vertex
Symmetry
Vertex figure
Picture
0
1
2
3
Alt
[62]
Square (4.4.4.4)
-
-
(4.4.4.4)
[1+ ,4,4,4]
[63]
rectified square 2 {4,4,4}-
[4+ ,4,4]
[77]
order-4 square -
-
-
[1+ ,4,4,4]
Cube
[78]
truncated order-4 square (4.8.8)
-
(4.8.8)
-
(4.4.4.4)
[1+ ,4,4,4]
[79]
bitruncated order-4 square (4.8.8)
-
-
(4.8.8)
(4.8.8)
[1+ ,4,4,4]
[81]
runcitruncated order-4 square tiling 2,3 {4,4,4}[4,4,4]
[83]
alternated square -
-
[4,1+ ,4,4]
(4.3.4.3)
[104]
quarter order-4 square [[1+ ,4,4,4,1+ ]][4] ]]
153 alternated rectified square tiling -
[((2+ ,4,4)),4]
154 alternated runcinated order-4 square tiling0,3 {4,4,4}
[[(4,4,4,2+ )]]
Scaliform
snub order-4 square tiling -
-
[4+ ,4,4]
Nonuniform
runcic snub order-4 square tiling3 {4,4,4}
[4+ ,4,4]
Nonuniform
bisnub order-4 square tiling
-
-
[[4,4+ ,4]]
[152]
snub square tiling
-
[(4,4)+ ,4]
Nonuniform
alternated runcitruncated order-4 square tiling0,1,3 {4,4,4}
[((2,4)+ ,4,4)]
Nonuniform
omnisnub order-4 square tiling0,1,2,3 {4,4,4}
[[4,4,4]]+
Tridental graphs
[3,41,1 ] 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 :
[4,41,1 ] family
There are 7 forms, (all shared with [4,4,4] family), generated by ring permutations of the Coxeter group :
[6,31,1 ] family
There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group : [6,31,1 ] or
Cyclic graphs
[(4,4,3,3)] family
There are 11 forms, 4 unique to this family, generated by ring permutations of the Coxeter group :
[(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 :
[(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 :
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
[4,3[3] ] family
There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group : [4,3[3] ] or
[5,3[3] ] family
There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group : [5,3[3] ] or
[6,3[3] ] family
There are 11 forms, 4 unique, generated by ring permutations of the Coxeter group : [6,3[3] ] or
Multicyclic graphs
[3[ ]×[ ] ] family
There are 8 forms, 1 unique, generated by ring permutations of the Coxeter group :
[3[3,3] ] family
There are 4 forms, 0 unique, generated by ring permutations of the Coxeter group :
Summary enumerations by family
Linear graphs
Paracompact hyperbolic enumeration
Group
Extended
Honeycombs
Chiral
Alternation honeycombs
R
¯
3
{\displaystyle {\bar {R))_{3))
[4,4,3]
15
[1+ ,4,1+ ,4,3+ ]
(6)
[4,4,3]+
(1)
N
¯
3
{\displaystyle {\bar {N))_{3))
[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]]+
(1)
V
¯
3
{\displaystyle {\bar {V))_{3))
[6,3,3]
15
[1+ ,6,(3,3)+ ]
(2)
[6,3,3]+
(1)
B
V
¯
3
{\displaystyle {\bar {BV))_{3))
[6,3,4]
15
[1+ ,6,3+ ,4,1+ ]
(6)
[6,3,4]+
(1)
H
V
¯
3
{\displaystyle {\bar {HV))_{3))
[6,3,5]
15
[1+ ,6,(3,5)+ ]
(2)
[6,3,5]+
(1)
Y
¯
3
{\displaystyle {\bar {Y))_{3))
[3,6,3]
5
[3,6,3]
(1)
[2+ [3+ ,6,3+ ]]
(1)
[2+ [3,6,3]]
3
[2+ [3,6,3]]+
(1)
Z
¯
3
{\displaystyle {\bar {Z))_{3))
[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]]+
(1)
Tridental graphs
Paracompact hyperbolic enumeration
Group
Extended
Honeycombs
Chiral
Alternation honeycombs
D
V
¯
3
{\displaystyle {\bar {DV))_{3))
1,1 ][6,31,1 ]
4
[1[6,31,1 ]]=[6,3,4]
(7)
[1[1+ ,6,31,1 ]]+
(2)
[1[6,31,1 ]]+ =[6,3,4]+
(1)
O
¯
3
{\displaystyle {\bar {O))_{3))
1,1 ][3,41,1 ]
4
[3+ ,41,1 ]+
(2)
[1[3,41,1 ]]=[3,4,4]
(7)
[1[3+ ,41,1 ]]+
(2)
[1[3,41,1 ]]+
(1)
M
¯
3
{\displaystyle {\bar {M))_{3))
1,1,1 ][41,1,1 ]
0
(none)
[1[41,1,1 ]]=[4,4,4]
(4)
[1[1+ ,4,1+ ,41,1 ]]+ =[(4,1+ ,4,1+ ,4,2+ )]
(4)
[3[41,1,1 ]]=[4,4,3]
(3)
[3[1+ ,41,1,1 ]]+ =[1+ ,4,1+ ,4,3+ ]
(2)
[3[41,1,1 ]]+ =[4,4,3]+
(1)
Cyclic graphs
Paracompact hyperbolic enumeration
Group
Extended
Honeycombs
Chiral
Alternation honeycombs
C
R
^
3
{\displaystyle {\widehat {CR))_{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)]]+
(1)
R
R
^
3
{\displaystyle {\widehat {RR))_{3))
[4] ][4[4] ]
(none)
[2+ [4[4] ]]
1
[2+ [(4+ ,4)[2] ]]
(1)
[1[4[4] ]]=[4,41,1 ]
(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] ]]+ + [4,4,4]]+
(1)
A
V
^
3
{\displaystyle {\widehat {AV))_{3))
[(6,3,3,3)]
6
[2+ [(6,3,3,3)]]
3
[2+ [(6,3,3,3)]]+
(1)
B
V
^
3
{\displaystyle {\widehat {BV))_{3))
[(3,4,3,6)]
6
[(3+ ,4,3+ ,6)]
(1)
[2+ [(3,4,3,6)]]
3
[2+ [(3,4,3,6)]]+
(1)
H
V
^
3
{\displaystyle {\widehat {HV))_{3))
[(3,5,3,6)]
6
[2+ [(3,5,3,6)]]
3
[2+ [(3,5,3,6)]]+
(1)
V
V
^
3
{\displaystyle {\widehat {VV))_{3))
[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
Honeycombs
Chiral
Alternation honeycombs
B
R
^
3
{\displaystyle {\widehat {BR))_{3))
[(3,3,4,4)]
4
[1[(4,4,3,3)]]=[3,41,1 ]
(7)
[1[(3,3,4,1+ ,4)]]+ + ,41,1 ]+
(2)
[1[(3,3,4,4)]]+ 1,1 ]+
(1)
D
P
¯
3
{\displaystyle {\bar {DP))_{3))
[ ]x[ ] ][3[ ]x[ ] ]
1
[1[3[ ]x[ ] ]]=[6,31,1 ]
(2)
[1[3[ ]x[ ] ]]=[4,3[3] ]
(2)
[2[3[ ]x[ ] ]]=[6,3,4]
(3)
[2[3[ ]x[ ] ]]+ +
(1)
P
P
¯
3
{\displaystyle {\bar {PP))_{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] ]]+ +
(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
Honeycombs
Chiral
Alternation honeycombs
P
¯
3
{\displaystyle {\bar {P))_{3))
[3] ][3,3[3] ]
4
[1[3,3[3] ]]=[3,3,6]
(7)
[1[3,3[3] ]]+ +
(1)
B
P
¯
3
{\displaystyle {\bar {BP))_{3))
[3] ][4,3[3] ]
4
[1[4,3[3] ]]=[4,3,6]
(7)
[1+ ,4,(3[3] )+ ]
(2)
[4,3[3] ]+
(1)
H
P
¯
3
{\displaystyle {\bar {HP))_{3))
[3] ][5,3[3] ]
4
[1[5,3[3] ]]=[5,3,6]
(7)
[1[5,3[3] ]]+ +
(1)
V
P
¯
3
{\displaystyle {\bar {VP))_{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] ]]+ +
(1)
[1[6,3[3] ]]+ +
(1)
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 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 2002C. W. L. Garner, Regular Skew Polyhedra in Hyperbolic Three-Space Can. J. Math. 19, 1179-1186, 1967. PDF [1] Archived 2015-04-02 at the Wayback Machine
Norman Johnson , Geometries and Transformations , (2018) Chapters 11,12,13N. 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 [2] [3]
N.W. Johnson, R. Kellerhals , J.G. Ratcliffe, S.T. Tschantz, Commensurability classes of hyperbolic Coxeter groups , (2002) H3 : p130. [4]
Klitzing, Richard. "Hyperbolic honeycombs H3 paracompact" .
