Paracompact uniform honeycombs

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Short description: Tessellation of convex uniform polyhedron cells
Example paracompact regular honeycombs
100px
{3,3,6}
 100px
{6,3,3}
 100px
{4,3,6}
 100px
{6,3,4}
100px
{5,3,6}
 100px
{6,3,5}
 100px
{6,3,6}
 100px
{3,6,3}
100px
{4,4,3}
 100px
{3,4,4}
 100px
{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 two 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 {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
Symbol
{p,q,r} 		Coxeter
 Cell
type
{p,q} 		Face
type
{p} 		Edge
figure
{r} 		Vertex
figure
{q,r} 		Dual Coxeter
group
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

250px 160px
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. Seven uniform honeycombs that arise here as alternations have been numbered 152 to 158, after the 151 Wythoffian forms not requiring alternation for their construction.

Tetrahedral hyperbolic paracompact group summary
Coxeter group Simplex
volume 		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] The smallest paracompact form in H3 can be represented by or , or [∞,3,3,∞] which can be constructed by a mirror removal of paracompact hyperbolic group [3,4,4] as [3,4,1+,4] : = . The doubled fundamental domain changes from a tetrahedron into a quadrilateral pyramid. Another pyramid is or , constructed as [4,4,1+,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 , [(3,4,4,1+,4)] = [((4,∞,3)),((3,∞,4))] or , [(4,4,4,1+,4)] = [((4,∞,4)),((4,∞,4))] or . = , = , = .

Another nonsimplectic half groups is ↔ .

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

Pyramidal hyperbolic paracompact group summary
Dimension Rank Graphs
H3 5

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Linear graphs

[6,3,3] family

# Honeycomb name
Coxeter diagram:
Schläfli symbol 		Cells by location
(and count around each vertex) 		Vertex figure Picture
1
 2
 3
 4
1 hexagonal (hexah)

{6,3,3} 		- - - (4)
40px
(6.6.6) 		80px
Tetrahedron 		120px
2 rectified hexagonal (rihexah)

t1{6,3,3} or r{6,3,3} 		(2)
40px
(3.3.3) 		- - (3)
40px
(3.6.3.6) 		80px
Triangular prism 		120px
3 rectified order-6 tetrahedral (rath)

t1{3,3,6} or r{3,3,6} 		(6)
40px
(3.3.3.3) 		- - (2)
40px
(3.3.3.3.3.3) 		80px
Hexagonal prism 		120px
4 order-6 tetrahedral (thon)

{3,3,6} 		(∞)
40px
(3.3.3) 		- - - 40px
Triangular tiling 		120px
5 truncated hexagonal (thexah)

t0,1{6,3,3} or t{6,3,3} 		(1)
40px
(3.3.3) 		- - (3)
40px
(3.12.12) 		80px
Triangular pyramid 		120px
6 cantellated hexagonal (srihexah)

t0,2{6,3,3} or rr{6,3,3} 		(1)
40px
3.3.3.3 		(2)
40px
(4.4.3) 		- (2)
40px
(3.4.6.4) 		80px 120px
7 runcinated hexagonal (sidpithexah)

t0,3{6,3,3} 		(1)
40px
(3.3.3) 		(3)
40px
(4.4.3) 		(3)
40px
(4.4.6) 		(1)
40px
(6.6.6) 		80px 120px
8 cantellated order-6 tetrahedral (srath)

t0,2{3,3,6} or rr{3,3,6} 		(1)
40px
(3.4.3.4) 		- (2)
40px
(4.4.6) 		(2)
40px
(3.6.3.6) 		80px 120px
9 bitruncated hexagonal (tehexah)

t1,2{6,3,3} or 2t{6,3,3} 		(2)
40px
(3.6.6) 		- - (2)
40px
(6.6.6) 		80px 120px
10 truncated order-6 tetrahedral (tath)

t0,1{3,3,6} or t{3,3,6} 		(6)
40px
(3.6.6) 		- - (1)
40px
(3.3.3.3.3.3) 		80px 120px
11 cantitruncated hexagonal (grihexah)

t0,1,2{6,3,3} or tr{6,3,3} 		(1)
40px
(3.6.6) 		(1)
40px
(4.4.3) 		- (2)
40px
(4.6.12) 		80px 120px
12 runcitruncated hexagonal (prath)

t0,1,3{6,3,3} 		(1)
40px
(3.4.3.4) 		(2)
40px
(4.4.3) 		(1)
40px
(4.4.12) 		(1)
40px
(3.12.12) 		80px 120px
13 runcitruncated order-6 tetrahedral (prihexah)

t0,1,3{3,3,6} 		(1)
40px
(3.6.6) 		(1)
40px
(4.4.6) 		(2)
40px
(4.4.6) 		(1)
40px
(3.4.6.4) 		80px 120px
14 cantitruncated order-6 tetrahedral (grath)

t0,1,2{3,3,6} or tr{3,3,6} 		(2)
40px
(4.6.6) 		- (1)
40px
(4.4.6) 		(1)
40px
(6.6.6) 		80px 120px
15 omnitruncated hexagonal (gidpithexah)

t0,1,2,3{6,3,3} 		(1)
40px
(4.6.6) 		(1)
40px
(4.4.6) 		(1)
40px
(4.4.12) 		(1)
40px
(4.6.12) 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram:
Schläfli symbol 		Cells by location
(and count around each vertex) 		Vertex figure Picture
1
 2
 3
 4
 Alt
[137] alternated hexagonal (ahexah)
( ↔ ) = 		- - (4)
40px
(3.3.3.3.3.3) 		(4)
40px
(3.3.3) 		40px
(3.6.6)
[138] cantic hexagonal (tahexah)
↔ 		(1)
40px
(3.3.3.3) 		- (2)
40px
(3.6.3.6) 		(2)
40px
(3.6.6) 		80px
[139] runcic hexagonal (birahexah)
↔ 		(1)
40px
(4.4.4) 		(1)
40px
(4.4.3) 		(1)
40px
(3.3.3.3.3.3) 		(3)
40px
(3.4.3.4) 		80px
[140] runcicantic hexagonal (bitahexah)
↔ 		(1)
40px
(3.6.6) 		(1)
40px
(4.4.3) 		(1)
40px
(3.6.3.6) 		(2)
40px
(4.6.6) 		80px
Nonuniform snub rectified order-6 tetrahedral
↔
sr{3,3,6} 		40px 40px 40px
Irr. (3.3.3) 		80px
Nonuniform cantic snub order-6 tetrahedral

sr3{3,3,6}
Nonuniform omnisnub order-6 tetrahedral

ht0,1,2,3{6,3,3} 		40px 40px 40px
Irr. (3.3.3)

[6,3,4] family

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

# Name of honeycomb
Coxeter diagram
Schläfli symbol 		Cells by location and count per vertex Vertex figure Picture
0
 1
 2
 3
16 (Regular) order-4 hexagonal (shexah)

{6,3,4} 		- - - (8)

40px
(6.6.6) 		80px
(3.3.3.3) 		120px
17 rectified order-4 hexagonal (rishexah)

t1{6,3,4} or r{6,3,4} 		(2)

40px
(3.3.3.3) 		- - (4)

40px
(3.6.3.6) 		80px
(4.4.4) 		120px
18 rectified order-6 cubic (rihach)

t1{4,3,6} or r{4,3,6} 		(6)

40px
(3.4.3.4) 		- - (2)

40px
(3.3.3.3.3.3) 		80px
(6.4.4) 		120px
19 order-6 cubic (hachon)

{4,3,6} 		(20)

40px
(4.4.4) 		- - - 40px
(3.3.3.3.3.3) 		120px
20 truncated order-4 hexagonal (tishexah)

t0,1{6,3,4} or t{6,3,4} 		(1)

40px
(3.3.3.3) 		- - (4)

40px
(3.12.12) 		80px 120px
21 bitruncated order-6 cubic (chexah)

t1,2{6,3,4} or 2t{6,3,4} 		(2)

40px
(4.6.6) 		- - (2)

40px
(6.6.6) 		80px 120px
22 truncated order-6 cubic (thach)

t0,1{4,3,6} or t{4,3,6} 		(6)

40px
(3.8.8) 		- - (1)

40px
(3.3.3.3.3.3) 		80px 120px
23 cantellated order-4 hexagonal (srishexah)

t0,2{6,3,4} or rr{6,3,4} 		(1)

40px
(3.4.3.4) 		(2)

40px
(4.4.4) 		- (2)

40px
(3.4.6.4) 		80px 120px
24 cantellated order-6 cubic (srihach)

t0,2{4,3,6} or rr{4,3,6} 		(2)

40px
(3.4.4.4) 		- (2)

40px
(4.4.6) 		(1)

40px
(3.6.3.6) 		80px 120px
25 runcinated order-6 cubic (sidpichexah)

t0,3{6,3,4} 		(1)

40px
(4.4.4) 		(3)

40px
(4.4.4) 		(3)

40px
(4.4.6) 		(1)

40px
(6.6.6) 		80px 120px
26 cantitruncated order-4 hexagonal (grishexah)

t0,1,2{6,3,4} or tr{6,3,4} 		(1)

40px
(4.6.6) 		(1)

40px
(4.4.4) 		- (2)

40px
(4.6.12) 		80px 120px
27 cantitruncated order-6 cubic (grihach)

t0,1,2{4,3,6} or tr{4,3,6} 		(2)

40px
(4.6.8) 		- (1)

40px
(4.4.6) 		(1)

40px
(6.6.6) 		80px 120px
28 runcitruncated order-4 hexagonal (prihach)

t0,1,3{6,3,4} 		(1)

40px
(3.4.4.4) 		(1)

40px
(4.4.4) 		(2)

40px
(4.4.12) 		(1)

40px
(3.12.12) 		80px 120px
29 runcitruncated order-6 cubic (prishexah)

t0,1,3{4,3,6} 		(1)

40px
(3.8.8) 		(2)

40px
(4.4.8) 		(1)

40px
(4.4.6) 		(1)

40px
(3.4.6.4) 		80px 120px
30 omnitruncated order-6 cubic (gidpichexah)

t0,1,2,3{6,3,4} 		(1)

40px
(4.6.8) 		(1)

40px
(4.4.8) 		(1)

40px
(4.4.12) 		(1)

40px
(4.6.12) 		80px 120px
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol 		Cells by location and count per vertex Vertex figure Picture
0
 1
 2
 3
 Alt
[87] alternated order-6 cubic (ahach)
↔
h{4,3,6} 		40px
(3.3.3) 		    40px
(3.3.3.3.3.3)
40px
(3.6.3.6)
[88] cantic order-6 cubic (tachach)
↔
h2{4,3,6} 		(2)
40px
(3.6.6) 		- - (1)
40px
(3.6.3.6) 		(2)
40px
(6.6.6) 		80px
[89] runcic order-6 cubic (birachach)
↔
h3{4,3,6} 		(1)
40px
(3.3.3) 		- - (1)
40px
(6.6.6) 		(3)
40px
(3.4.6.4) 		80px
[90] runcicantic order-6 cubic (bitachach)
↔
h2,3{4,3,6} 		(1)
40px
(3.6.6) 		- - (1)
40px
(3.12.12) 		(2)
40px
(4.6.12) 		80px
[141] alternated order-4 hexagonal (ashexah)
↔ ↔
h{6,3,4} 		- - 40px
(3.3.3.3.3.3) 		40px
(3.3.3.3) 		40px
(4.6.6)
[142] cantic order-4 hexagonal (tashexah)
↔ ↔
h1{6,3,4} 		(1)
40px
(3.4.3.4) 		- (2)
40px
(3.6.3.6) 		(2)
40px
(4.6.6) 		80px
[143] runcic order-4 hexagonal (birashexah)
↔
h3{6,3,4} 		(1)
40px
(4.4.4) 		(1)
40px
(4.4.3) 		(1)
40px
(3.3.3.3.3.3) 		(3)
40px
(3.4.4.4) 		80px
[144] runcicantic order-4 hexagonal (bitashexah)
↔
h2,3{6,3,4} 		(1)
40px
(3.8.8) 		(1)
40px
(4.4.3) 		(1)
40px
(3.6.3.6) 		(2)
40px
(4.6.8) 		80px
[151] quarter order-4 hexagonal (quishexah)
↔
q{6,3,4} 		(3)
40px 		(1)
40px 		- (1)
40px 		(3)
40px 		80px
Nonuniform bisnub order-6 cubic
↔
2s{4,3,6}
40px
(3.3.3.3.3) 		- -
40px
(3.3.3.3.3.3) 		40px
+(3.3.3) 		80px
Nonuniform runcic bisnub order-6 cubic
Nonuniform snub rectified order-6 cubic
↔
sr{4,3,6}
40px
(3.3.3.3.3)
40px
(3.3.3)
40px
(3.3.3.3)
40px
(3.3.3.3.6) 		40px
+(3.3.3)
Nonuniform runcic snub rectified order-6 cubic

sr3{4,3,6}
Nonuniform snub rectified order-4 hexagonal
↔
sr{6,3,4}
40px
(3.3.3.3.3.3)
40px
(3.3.3) 		-
40px
(3.3.3.3.6) 		40px
+(3.3.3)
Nonuniform runcisnub rectified order-4 hexagonal

sr3{6,3,4}
Nonuniform omnisnub rectified order-6 cubic

ht0,1,2,3{6,3,4}
40px
(3.3.3.3.4)
40px
(3.3.3.4)
40px
(3.3.3.6)
40px
(3.3.3.3.6) 		40px
+(3.3.3)

[6,3,5] family

# Honeycomb name
Coxeter diagram
Schläfli symbol 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
31 order-5 hexagonal (phexah)

{6,3,5} 		- - - (20)
50px
(6)3 		80px
Icosahedron 		120px
32 rectified order-5 hexagonal (riphexah)

t1{6,3,5} or r{6,3,5} 		(2)
40px
(3.3.3.3.3) 		- - (5)
50px
(3.6)2 		80px
(5.4.4) 		120px
33 rectified order-6 dodecahedral (rihed)

t1{5,3,6} or r{5,3,6} 		(5)
40px
(3.5.3.5) 		- - (2)
50px
(3)6 		80px
(6.4.4) 		120px
34 order-6 dodecahedral (hedhon)

{5,3,6} 		40px
(5.5.5) 		- - - (∞)
50px
(3)6 		120px
35 truncated order-5 hexagonal (tiphexah)

t0,1{6,3,5} or t{6,3,5} 		(1)
40px
(3.3.3.3.3) 		- - (5)
50px
3.12.12 		80px 120px
36 cantellated order-5 hexagonal (sriphexah)

t0,2{6,3,5} or rr{6,3,5} 		(1)
40px
(3.5.3.5) 		(2)
40px
(5.4.4) 		- (2)
50px
3.4.6.4 		80px 120px
37 runcinated order-6 dodecahedral (sidpidohexah)

t0,3{6,3,5} 		(1)
40px
(5.5.5) 		- (6)
40px
(6.4.4) 		(1)
50px
(6)3 		80px 120px
38 cantellated order-6 dodecahedral (srihed)

t0,2{5,3,6} or rr{5,3,6} 		(2)
40px
(4.3.4.5) 		- (2)
40px
(6.4.4) 		(1)
50px
(3.6)2 		80px 120px
39 bitruncated order-6 dodecahedral (dohexah)

t1,2{6,3,5} or 2t{6,3,5} 		(2)
40px
(5.6.6) 		- - (2)
50px
(6)3 		80px 120px
40 truncated order-6 dodecahedral (thed)

t0,1{5,3,6} or t{5,3,6} 		(6)
40px
(3.10.10) 		- - (1)
50px
(3)6 		80px 120px
41 cantitruncated order-5 hexagonal (griphexah)

t0,1,2{6,3,5} or tr{6,3,5} 		(1)
40px
(5.6.6) 		(1)
40px
(5.4.4) 		- (2)
50px
4.6.10 		80px 120px
42 runcitruncated order-5 hexagonal (prihed)

t0,1,3{6,3,5} 		(1)
40px
(4.3.4.5) 		(1)
40px
(5.4.4) 		(2)
40px
(12.4.4) 		(1)
50px
3.12.12 		80px 120px
43 runcitruncated order-6 dodecahedral (priphaxh)

t0,1,3{5,3,6} 		(1)
40px
(3.10.10) 		(1)
40px
(10.4.4) 		(2)
40px
(6.4.4) 		(1)
50px
3.4.6.4 		80px 120px
44 cantitruncated order-6 dodecahedral (grihed)

t0,1,2{5,3,6} or tr{5,3,6} 		(1)
40px
(4.6.10) 		- (2)
40px
(6.4.4) 		(1)
50px
(6)3 		80px 120px
45 omnitruncated order-6 dodecahedral (gidpidohaxh)

t0,1,2,3{6,3,5} 		(1)
40px
(4.6.10) 		(1)
40px
(10.4.4) 		(1)
40px
(12.4.4) 		(1)
50px
4.6.12 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram
Schläfli symbol 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
 Alt
[145] alternated order-5 hexagonal (aphexah)
↔
h{6,3,5} 		- - - (20)
50px
(3)6 		(12)
40px
(3)5 		40px
(5.6.6)
[146] cantic order-5 hexagonal (taphexah)
↔
h2{6,3,5} 		(1)
40px
(3.5.3.5) 		- (2)
40px
(3.6.3.6) 		(2)
40px
(5.6.6) 		80px
[147] runcic order-5 hexagonal (biraphexah)
↔
h3{6,3,5} 		(1)
40px
(5.5.5) 		(1)
40px
(4.4.3) 		(1)
40px
(3.3.3.3.3.3) 		(3)
40px
(3.4.5.4) 		80px
[148] runcicantic order-5 hexagonal (bitaphexah)
↔
h2,3{6,3,5} 		(1)
40px
(3.10.10) 		(1)
40px
(4.4.3) 		(1)
40px
(3.6.3.6) 		(2)
40px
(4.6.10) 		80px
Nonuniform snub rectified order-6 dodecahedral
↔
sr{5,3,6} 		40px
(3.3.5.3.5)
 - 40px
(3.3.3.3)
 40px
(3.3.3.3.3.3)
 40px
irr. tet
Nonuniform omnisnub order-5 hexagonal

ht0,1,2,3{6,3,5} 		40px
(3.3.5.3.5)
 40px
(3.3.3.5)
 40px
(3.3.3.6)
 40px
(3.3.6.3.6)
 40px
irr. tet

[6,3,6] family

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

# Name of honeycomb
Coxeter diagram
Schläfli symbol 		Cells by location and count per vertex Vertex figure Picture
0
 1
 2
 3
46 order-6 hexagonal (hihexah)

{6,3,6} 		- - - (20)
40px
(6.6.6) 		40px
(3.3.3.3.3.3) 		120px
47 rectified order-6 hexagonal (rihihexah)

t1{6,3,6} or r{6,3,6} 		(2)
40px
(3.3.3.3.3.3) 		- - (6)
40px
(3.6.3.6) 		80px
(6.4.4) 		120px
48 truncated order-6 hexagonal (thihexah)

t0,1{6,3,6} or t{6,3,6} 		(1)
40px
(3.3.3.3.3.3) 		- - (6)
40px
(3.12.12) 		80px 120px
49 cantellated order-6 hexagonal (srihihexah)

t0,2{6,3,6} or rr{6,3,6} 		(1)
40px
(3.6.3.6) 		(2)
40px
(4.4.6) 		- (2)
40px
(3.6.4.6) 		80px 120px
50 Runcinated order-6 hexagonal (spiddihexah)

t0,3{6,3,6} 		(1)
40px
(6.6.6) 		(3)
40px
(4.4.6) 		(3)
40px
(4.4.6) 		(1)
40px
(6.6.6) 		80px 120px
51 cantitruncated order-6 hexagonal (grihihexah)

t0,1,2{6,3,6} or tr{6,3,6} 		(1)
40px
(6.6.6) 		(1)
40px
(4.4.6) 		- (2)
40px
(4.6.12) 		80px 120px
52 runcitruncated order-6 hexagonal (prihihexah)

t0,1,3{6,3,6} 		(1)
40px
(3.6.4.6) 		(1)
40px
(4.4.6) 		(2)
40px
(4.4.12) 		(1)
40px
(3.12.12) 		80px 120px
53 omnitruncated order-6 hexagonal (gidpiddihexah)

t0,1,2,3{6,3,6} 		(1)
40px
(4.6.12) 		(1)
40px
(4.4.12) 		(1)
40px
(4.4.12) 		(1)
40px
(4.6.12) 		80px 120px
[1] bitruncated order-6 hexagonal (hexah)
↔ ↔
t1,2{6,3,6} or 2t{6,3,6} 		(2)
40px
(6.6.6) 		- - (2)
40px
(6.6.6) 		80px 120px
Alternated forms
# Name of honeycomb
Coxeter diagram
Schläfli symbol 		Cells by location and count per vertex Vertex figure Picture
0
 1
 2
 3
 Alt
[47] rectified order-6 hexagonal (rihihexah)
↔ ↔
q{6,3,6} = r{6,3,6} 		(2)
40px
(3.3.3.3.3.3) 		- - (6)
40px
(3.6.3.6) 		80px
(6.4.4) 		120px
[54] triangular (trah)
( ↔ ) =
h{6,3,6} = {3,6,3} 		- - -
40px
(3.3.3.3.3.3)
40px
(3.3.3.3.3.3) 		40px
{6,3} 		120px
[55] cantic order-6 hexagonal (ritrah)
( ↔ ) =
h2{6,3,6} = r{3,6,3} 		(1)
40px
(3.6.3.6) 		- (2)
40px
(6.6.6) 		(2)
40px
(3.6.3.6) 		80px 120px
[149] runcic order-6 hexagonal
↔
h3{6,3,6} 		(1)
40px
(6.6.6) 		(1)
40px
(4.4.3) 		(3)
40px
(3.4.6.4) 		(1)
40px
(3.3.3.3.3.3) 		80px
[150] runcicantic order-6 hexagonal
↔
h2,3{6,3,6} 		(1)
40px
(3.12.12) 		(1)
40px
(4.4.3) 		(2)
40px
(4.6.12) 		(1)
40px
(3.6.3.6) 		80px
[137] alternated hexagonal (ahexah)
( ↔ ↔ ) =
2s{6,3,6} = h{6,3,3}
40px
(3.3.3.3.6) 		- -
40px
(3.3.3.3.6) 		40px
+(3.3.3) 		40px
(3.6.6)
Nonuniform snub rectified order-6 hexagonal

sr{6,3,6}
40px
(3.3.3.3.3.3)
40px
(3.3.3.3) 		-
40px
(3.3.3.3.6) 		40px
+(3.3.3)
Nonuniform alternated runcinated order-6 hexagonal

ht0,3{6,3,6}
40px
(3.3.3.3.3.3)
40px
(3.3.3.3)
40px
(3.3.3.3)
40px
(3.3.3.3.3.3) 		40px
+(3.3.3)
Nonuniform omnisnub order-6 hexagonal

ht0,1,2,3{6,3,6}
40px
(3.3.3.3.6)
40px
(3.3.3.6)
40px
(3.3.3.6)
40px
(3.3.3.3.6) 		40px
+(3.3.3)

[3,6,3] family

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

# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Vertex figure Picture
0
 1
 2
 3
54 triangular (trah)

{3,6,3} 		- - - (∞)
40px
{3,6} 		40px
{6,3} 		120px
55 rectified triangular (ritrah)

t1{3,6,3} or r{3,6,3} 		(2)
40px
(6)3 		- - (3)
40px
(3.6)2 		80px
(3.4.4) 		120px
56 cantellated triangular (sritrah)

t0,2{3,6,3} or rr{3,6,3} 		(1)
40px
(3.6)2 		(2)
40px
(4.4.3) 		- (2)
40px
(3.6.4.6) 		80px 120px
57 runcinated triangular (spidditrah)

t0,3{3,6,3} 		(1)
40px
(3)6 		(6)
40px
(4.4.3) 		(6)
40px
(4.4.3) 		(1)
40px
(3)6 		80px 120px
58 bitruncated triangular (ditrah)

t1,2{3,6,3} or 2t{3,6,3} 		(2)
40px
(3.12.12) 		- - (2)
40px
(3.12.12) 		80px 120px
59 cantitruncated triangular (gritrah)

t0,1,2{3,6,3} or tr{3,6,3} 		(1)
40px
(3.12.12) 		(1)
40px
(4.4.3) 		- (2)
40px
(4.6.12) 		80px 120px
60 runcitruncated triangular (pritrah)

t0,1,3{3,6,3} 		(1)
40px
(3.6.4.6) 		(1)
40px
(4.4.3) 		(2)
40px
(4.4.6) 		(1)
40px
(6)3 		80px 120px
61 omnitruncated triangular (gipidditrah)

t0,1,2,3{3,6,3} 		(1)
40px
(4.6.12) 		(1)
40px
(4.4.6) 		(1)
40px
(4.4.6) 		(1)
40px
(4.6.12) 		80px 120px
[1] truncated triangular (hexah)
↔ ↔
t0,1{3,6,3} or t{3,6,3} = {6,3,3} 		(1)
40px
(6)3 		- - (3)
40px
(6)3 		80px
{3,3} 		120px
Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Vertex figure Picture
0
 1
 2
 3
 Alt
[56] cantellated triangular (sritrah)
=
s2{3,6,3} 		(1)
40px
(3.6)2
 - - (2)
40px
(3.6.4.6)
 40px
(3.4.4) 		80px 120px
[60] runcitruncated triangular (pritrah)
=
s2,3{3,6,3} 		(1)
40px
(6)3
 - (1)
40px
(4.4.3)
 (1)
40px
(3.6.4.6)
 (2)
40px
(4.4.6) 		80px 120px
[137] alternated hexagonal (ahexah)
( ↔ ) = ( ↔ )
s{3,6,3} 		40px
(3)6
 - - 40px
(3)6
 40px
+(3)3 		40px
(3.6.6)
Scaliform runcisnub triangular (pristrah)

s3{3,6,3} 		40px
r{6,3}
 - 40px
(3.4.4)
 40px
(3)6
 40px
tricup
Nonuniform omnisnub triangular tiling honeycomb (snatrah)

ht0,1,2,3{3,6,3} 		40px
(3.3.3.3.6)
 40px
(3)4
 40px
(3)4
 40px
(3.3.3.3.6)
 40px
+(3)3

[4,4,3] family

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

# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Vertex figure Picture
0
 1
 2
 3
62 square (squah)
=
{4,4,3} 		- - - (6)

40px 		80px
Cube		 120px
63 rectified square (risquah)
=
t1{4,4,3} or r{4,4,3} 		(2)

40px 		- - (3)

40px 		80px

Triangular prism		 120px
64 rectified order-4 octahedral (rocth)

t1{3,4,4} or r{3,4,4} 		(4)

40px 		- - (2)

40px 		80px 120px
65 order-4 octahedral (octh)

{3,4,4} 		(∞)

40px 		- - - 40px 120px
66 truncated square (tisquah)
=
t0,1{4,4,3} or t{4,4,3} 		(1)

40px 		- - (3)

40px 		80px 120px
67 truncated order-4 octahedral (tocth)

t0,1{3,4,4} or t{3,4,4} 		(4)

40px 		- - (1)

40px		 80px 120px
68 bitruncated square (osquah)

t1,2{4,4,3} or 2t{4,4,3} 		(2)

40px 		- - (2)

40px 		80px 120px
69 cantellated square (srisquah)

t0,2{4,4,3} or rr{4,4,3} 		(1)

40px 		(2)

40px 		- (2)

40px 		80px 120px
70 cantellated order-4 octahedral (srocth)

t0,2{3,4,4} or rr{3,4,4} 		(2)

40px 		- (2)

40px 		(1)

40px 		80px 120px
71 runcinated square (sidposquah)

t0,3{4,4,3} 		(1)

40px 		(3)

40px 		(3)

40px 		(1)

40px 		80px 120px
72 cantitruncated square (grisquah)

t0,1,2{4,4,3} or tr{4,4,3} 		(1)

40px 		(1)

40px 		- (2)

40px 		80px 120px
73 cantitruncated order-4 octahedral (grocth)

t0,1,2{3,4,4} or tr{3,4,4} 		(2)

40px 		- (1)

40px 		(1)

40px 		80px 120px
74 runcitruncated square (procth)

t0,1,3{4,4,3} 		(1)

40px 		(1)

40px 		(2)

40px 		(1)

40px 		80px 120px
75 runcitruncated order-4 octahedral (prisquah)

t0,1,3{3,4,4} 		(1)

40px 		(2)

40px 		(1)

40px 		(1)

40px 		80px 120px
76 omnitruncated square (gidposquah)

t0,1,2,3{4,4,3} 		(1)

40px 		(1)

40px 		(1)

40px 		(1)

40px 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Vertex figure Picture
0
 1
 2
 3
 Alt
[83] alternated square
↔
h{4,4,3}		 - - - (6)

40px		 (8)

40px		 40px
[84] cantic square
↔
h2{4,4,3} 		(1)

40px 		- - (2)

40px 		(2)

40px 		80px
[85] runcic square
↔
h3{4,4,3} 		(1)

40px 		- - (1)

40px. 		(4)

40px 		80px
[86] runcicantic square
↔ 		(1)

40px 		- - (1)

40px 		(2)

40px 		80px
[153] alternated rectified square
↔
hr{4,4,3}
40px		 - -
40px		 {}x{3}
157
40px		 - -
40px		 {}x{6}
Scaliform snub order-4 octahedral
= =
s{3,4,4}
40px		 - -
40px		 {}v{4}
Scaliform runcisnub order-4 octahedral

s3{3,4,4}
40px
40px
40px
40px		 cup-4
152 snub square
=
s{4,4,3}
40px		 - -
40px		 {3,3} 80px
Nonuniform snub rectified order-4 octahedral

sr{3,4,4}
40px		 -
40px
40px		 irr. {3,3}
Nonuniform alternated runcitruncated square

ht0,1,3{3,4,4}
40px
40px
40px
40px		 irr. {}v{4}
Nonuniform omnisnub square

ht0,1,2,3{4,4,3}
40px
40px
40px
40px		 irr. {3,3}

[4,4,4] family

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

# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Symmetry Vertex figure Picture
0
 1
 2
 3
77 order-4 square (sisquah)

{4,4,4}		 - - -
40px		 [4,4,4]
40px
Cube		 120px
78 truncated order-4 square (tissish)

t0,1{4,4,4} or t{4,4,4}
40px		 - -
40px		 [4,4,4] 80px 120px
79 bitruncated order-4 square (dish)

t1,2{4,4,4} or 2t{4,4,4}
40px		 - -
40px		 4,4,4 80px 120px
80 runcinated order-4 square (spiddish)

t0,3{4,4,4}
40px
40px
40px
40px		 4,4,4 80px 120px
81 runcitruncated order-4 square (prissish)

t0,1,3{4,4,4}
40px
40px
40px
40px		 [4,4,4] 80px 120px
82 omnitruncated order-4 square (gipiddish)

t0,1,2,3{4,4,4}
40px
40px
40px
40px		 4,4,4 80px 120px
[62] square (squah)
↔
t1{4,4,4} or r{4,4,4}
40px		 - -
40px		 [4,4,4] 40px
Square tiling		 120px
[63] rectified square (risquah)
↔
t0,2{4,4,4} or rr{4,4,4}
40px
40px		 -
40px		 [4,4,4] 80px 120px
[66] truncated order-4 square (tisquah)
↔
t0,1,2{4,4,4} or tr{4,4,4}
40px
40px		 -
40px		 [4,4,4] 80px 120px
Alternated constructions
# Honeycomb name
Coxeter diagram
and Schläfli symbol 		Cell counts/vertex
and positions in honeycomb 		Symmetry Vertex figure Picture
0
 1
 2
 3
 Alt
[62] Square (squah)
( ↔ ↔ ↔ ) = 		40px
(4.4.4.4) 		- - 40px
(4.4.4.4) 		[1+,4,4,4]
=[4,4,4] 		80px 120px
[63] rectified square (risquah)
=
s2{4,4,4}
40px
40px		 -
40px		 [4+,4,4] 80px 120px
[77] order-4 square (sisquah)
↔ ↔ ↔ 		- - -
40px
40px		 [1+,4,4,4]
=[4,4,4]
40px
Cube		 120px
[78] truncated order-4 square (tissish)
↔ ↔ ↔ 		40px
(4.8.8) 		- 40px
(4.8.8) 		- 40px
(4.4.4.4) 		[1+,4,4,4]
=[4,4,4] 		80px 120px
[79] bitruncated order-4 square (dish)
↔ ↔ ↔ 		40px
(4.8.8) 		- - 40px
(4.8.8) 		40px
(4.8.8) 		[1+,4,4,4]
=[4,4,4] 		80px 120px
[81] runcitruncated order-4 square tiling (prissish)
=
s2,3{4,4,4}
40px
40px
40px
40px		 [4,4,4] 80px 120px
[83] alternated square
( ↔ ) ↔
hr{4,4,4}
40px		 - -
40px		 40px [4,1+,4,4] 40px
(4.3.4.3)
[104] quarter order-4 square
↔
q{4,4,4} 		1+,4,4,4,1+
=4[4]		 80px
153 alternated rectified square tiling
↔
↔
hrr{4,4,4}
40px
40px		 -
40px		 [((2+,4,4)),4]
154 alternated runcinated order-4 square tiling

ht0,3{4,4,4}
40px
40px
40px
40px		 (4,4,4,2+) 80px
Scaliform snub order-4 square tiling

s{4,4,4}
40px		 - -
40px		 [4+,4,4]
Nonuniform runcic snub order-4 square tiling

s3{4,4,4}		 [4+,4,4]
Nonuniform bisnub order-4 square tiling

2s{4,4,4}
40px		 - -
40px		 4,4+,4 80px
[152] snub square tiling
↔
sr{4,4,4}
40px
40px		 -
40px		 [(4,4)+,4] 80px
Nonuniform alternated runcitruncated order-4 square tiling

ht0,1,3{4,4,4}
40px
40px
40px
40px		 [((2,4)+,4,4)]
Nonuniform omnisnub order-4 square tiling

ht0,1,2,3{4,4,4}
40px
40px
40px
40px		 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:

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 0'
 3
83 alternated square
↔ 		- - 40px
(4.4.4) 		40px
(4.4.4.4) 		40px
(4.3.4.3)
84 cantic square
↔ 		40px
(3.4.3.4) 		- 40px
(3.8.8) 		40px
(4.8.8) 		80px
85 runcic square
↔ 		40px
(4.4.4.4) 		- 40px
(3.4.4.4) 		40px
(4.4.4.4) 		80px
86 runcicantic square
↔ 		40px
(4.6.6) 		- 40px
(3.4.4.4) 		40px
(4.8.8) 		80px
[63] rectified square (risquah)
↔ 		40px
(4.4.4) 		- 40px
(4.4.4) 		40px
(4.4.4.4) 		80px 120px
[64] rectified order-4 octahedral (rocth)
↔ 		40px
(3.4.3.4) 		- 40px
(3.4.3.4) 		40px
(4.4.4.4) 		80px 120px
[65] order-4 octahedral (octh)
↔ 		40px
(4.4.4.4) 		- 40px
(4.4.4.4) 		- 40px 120px
[67] truncated order-4 octahedral (tocth)
↔ 		40px
(4.6.6) 		- 40px
(4.6.6) 		40px
(4.4.4.4) 		80px 120px
[68] bitruncated square (osquah)
↔ 		40px
(3.8.8) 		- 40px
(3.8.8) 		40px
(4.8.8) 		80px 120px
[70] cantellated order-4 octahedral (srocth)
↔ 		40px
(3.4.4.4) 		40px
(4.4.4) 		40px
(3.4.4.4) 		40px

(4.4.4.4) 		80px 120px
[73] cantitruncated order-4 octahedral (grocth)
↔ 		40px
(4.6.8) 		40px
(4.4.4) 		40px
(4.6.8) 		40px
(4.8.8) 		80px 120px
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 0'
 3
 Alt
Scaliform snub order-4 octahedral
= =
s{3,41,1}		 - - irr. {}v{4}
Nonuniform snub rectified order-4 octahedral
↔
sr{3,41,1} 		40px
(3.3.3.3.4) 		40px
(3.3.3) 		40px
(3.3.3.3.4) 		40px
(3.3.4.3.4) 		40px
+(3.3.3)

[4,41,1] 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 Vertex figure Picture
0
 1
 0'
 3
[62] Square (squah)
( ↔ ) = 		40px
(4.4.4.4) 		- 40px
(4.4.4.4) 		40px
(4.4.4.4) 		40px 120px
[62] Square (squah)
( ↔ ) = 		40px
(4.4.4.4) 		- 40px
(4.4.4.4) 		40px
(4.4.4.4) 		40px 120px
[63] rectified square (risquah)
( ↔ ) = 		40px
(4.4.4.4) 		40px
(4.4.4) 		40px
(4.4.4.4) 		40px
(4.4.4.4) 		80px 120px
[66] truncated square (tisquah)
( ↔ ) = 		40px
(4.8.8) 		40px
(4.4.4) 		40px
(4.8.8) 		40px
(4.8.8) 		80px 120px
[77] order-4 square (sisquah)
↔ 		40px
(4.4.4.4) 		- 40px
(4.4.4.4) 		- 40px 120px
[78] truncated order-4 square (tissish)
↔ 		40px
(4.8.8) 		- 40px
(4.8.8) 		40px
(4.4.4.4) 		80px 120px
[79] bitruncated order-4 square (dish)
↔ 		40px
(4.8.8) 		- 40px
(4.8.8) 		40px
(4.8.8) 		80px 120px
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 0'
 3
 Alt
[77] order-4 square (sisquah)
( ↔ ↔ ) = 		- -
40px
Cube 		40px 120px
[78] truncated order-4 square (tissish)
( ↔ ) = ( ↔ ) 		80px 120px
[83] Alternated square
↔ 		-
40px
40px
Scaliform Snub order-4 square
 -
Nonuniform -
Nonuniform -
[153] ( ↔ )
= ( ↔ )
Nonuniform Snub square
↔ ↔
40px
(3.3.4.3.4)
40px
(3.3.3)
40px
(3.3.4.3.4)
40px
(3.3.4.3.4) 		40px
+(3.3.3)

[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 .

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 0'
 3
87 alternated order-6 cubic (ahach)
↔ 		- - (∞)
40px
(3.3.3.3.3) 		(∞)
40px
(3.3.3)
40px
(3.6.3.6)
88 cantic order-6 cubic (tachach)
↔ 		(1)
40px
(3.6.3.6) 		- (2)
40px
(6.6.6) 		(2)
40px
(3.6.6) 		80px
89 runcic order-6 cubic (birachach)
↔ 		(1)
40px
(6.6.6) 		- (3)
40px
(3.4.6.4) 		(1)
40px
(3.3.3) 		80px
90 runcicantic order-6 cubic (bitachach)
↔ 		(1)
40px
(3.12.12) 		- (2)
40px
(4.6.12) 		(1)
40px
(3.6.6) 		80px
[16] order-4 hexagonal (shexah)
↔ 		(4)
40px
(6.6.6) 		- (4)
40px
(6.6.6) 		- 80px
(3.3.3.3) 		120px
[17] rectified order-4 hexagonal (rishexah)
↔ 		(2)
40px
(3.6.3.6) 		- (2)
40px
(3.6.3.6) 		(2)
40px
(3.3.3.3) 		80px 120px
[18] rectified order-6 cubic (rihach)
↔ 		(1)
40px
(3.3.3.3.3) 		- (1)
40px
(3.3.3.3.3) 		(6)
40px
(3.4.3.4) 		80px 120px
[20] truncated order-4 hexagonal (tishexah)
↔ 		(2)
40px
(3.12.12) 		- (2)
40px
(3.12.12) 		(1)
40px
(3.3.3.3) 		80px 120px
[21] bitruncated order-6 cubic (chexah)
↔ 		(1)
40px
(6.6.6) 		- (1)
40px
(6.6.6) 		(2)
40px
(4.6.6) 		80px 120px
[24] cantellated order-6 cubic (srihach)
↔ 		(1)
40px
(3.4.6.4) 		(2)
40px
(4.4.4) 		(1)
40px
(3.4.6.4) 		(1)
40px
(3.4.3.4) 		80px 120px
[27] cantitruncated order-6 cubic (grihach)
↔ 		(1)
40px
(4.6.12) 		(1)
40px
(4.4.4) 		(1)
40px
(4.6.12) 		(1)
40px
(4.6.6) 		80px 120px
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 0'
 3
 Alt
[141] alternated order-4 hexagonal (ashexah)
↔ ↔ ↔ 		40px
(4.6.6)
Nonuniform bisnub order-4 hexagonal
↔ 		80px
Nonuniform snub rectified order-4 hexagonal
↔ 		40px
(3.3.3.3.6) 		40px
(3.3.3) 		40px
(3.3.3.3.6) 		40px
(3.3.3.3.3) 		40px
+(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 location Vertex figure Picture
0
 1
 2
 3
91 tetrahedral-square
 - (6)

40px
(444) 		(8)

40px
(333) 		(12)

40px
(3434)
40px
(3444)
92 cyclotruncated square-tetrahedral
40px
(444)
40px
(488)
40px
(333)
40px
(388) 		80px
93 cyclotruncated tetrahedral-square
 (1)

40px
(3333) 		(1)

40px
(444) 		(4)

40px
(366) 		(4)

40px
(466) 		80px
94 truncated tetrahedral-square
 (1)

40px
(3444) 		(1)

40px
(488) 		(1)

40px
(366) 		(2)

40px
(468) 		80px
[64] ( ↔ ) =
rectified order-4 octahedral (rocth)
40px
(3434)
40px
(4444)
40px
(3434)
40px
(3434) 		80px 120px
[65] ( ↔ ) =
order-4 octahedral (octh)
40px
(3333) 		-
40px
(3333)
40px
(3333) 		40px 120px
[67] ( ↔ ) =
truncated order-4 octahedral (tocth)
40px
(466)
40px
(4444)
40px
(3434)
40px
(466) 		80px 120px
[83] alternated square
( ↔ ) =
40px
(444)
40px
(4444) 		-
40px
(444) 		40px
(4.3.4.3)
[84] cantic square
( ↔ ) =
40px
(388)
40px
(488)
40px
(3434)
40px
(388) 		80px
[85] runcic square
( ↔ ) =
40px
(3444)
40px
(3434)
40px
(3333)
40px
(3444) 		80px
[86] runcicantic square
( ↔ ) =
40px
(468)
40px
(488)
40px
(466)
40px
(468) 		80px
# Honeycomb name
Coxeter diagram 		Cells by location Vertex figure Picture
0
 1
 2
 3
 Alt
Scaliform snub order-4 octahedral
= = 		- - irr. {}v{4}
Nonuniform
155 alternated tetrahedral-square
↔ 		r{4,3}

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

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
95 cubic-square
 (8)
40px
(4.4.4)
 - (6)
40px
(4.4.4.4)
 (12)
40px
(4.4.4.4)
 40px
(3.4.4.4)
96 octahedral-square
 40px
(3.4.3.4)
 40px
(3.3.3.3)
 - 40px
(4.4.4.4)
 40px
(4.4.4.4)
97 cyclotruncated cubic-square
 (4)
40px
(3.8.8)
 (1)
40px
(3.3.3.3)
 (1)
40px
(4.4.4.4)
 (4)
40px
(4.8.8)
 80px
98 cyclotruncated square-cubic
 (1)
40px
(4.4.4)
 (1)
40px
(4.4.4)
 (3)
40px
(4.8.8)
 (3)
40px
(4.8.8)
 80px
99 cyclotruncated octahedral-square
 (4)
40px
(4.6.6)
 (4)
40px
(4.6.6)
 (1)
40px
(4.4.4.4)
 (1)
40px
(4.4.4.4)
 80px
100 rectified cubic-square
 (1)
40px
(3.4.3.4)
 (2)
40px
(3.4.4.4)
 (1)
40px
(4.4.4.4)
 (2)
40px
(4.4.4.4)
 80px
101 truncated cubic-square
 (1)
40px
(4.8.8)
 (1)
40px
(3.4.4.4)
 (2)
40px
(4.8.8)
 (1)
40px
(4.8.8)
 80px
102 truncated octahedral-square
 (2)
40px
(4.6.8
 (1)
40px
(4.6.6)
 (1)
40px
(4.4.4.4)
 (1)
40px
(4.8.8)
 80px
103 omnitruncated octahedral-square
 (1)
40px
(4.6.8)
 (1)
40px
(4.6.8)
 (1)
40px
(4.8.8)
 (1)
40px
(4.8.8)
 80px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
 Alt
156 alternated cubic-square
↔ 		- 40px
 40px
 40px
 40px
(3.4.4.4)
Nonuniform snub octahedral-square
 40px
 40px
 40px
 40px
Nonuniform cyclosnub square-cubic
 40px
 40px
 40px
 40px
Nonuniform cyclosnub octahedral-square
 40px
 40px
 40px
 40px
Nonuniform omnisnub cubic-square
 40px
(3.3.3.3.4)
 40px
(3.3.3.3.4)
 40px
(3.3.4.3.4)
 40px
(3.3.4.3.4)
 40px
+(3.3.3)

[(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 ↔ .

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
104 quarter order-4 square
↔ 		40px
(4.8.8)
 40px
(4.4.4.4)
 40px
(4.4.4.4)
 40px
(4.8.8)
 80px
[62] square (squah)
↔ ↔ 		40px
(4.4.4.4)
 40px
(4.4.4.4)
 40px
(4.4.4.4)
 40px
(4.4.4.4)
 80px 120px
[77] order-4 square (sisquah)
( ↔ ) = 		40px
(4.4.4.4)
 - 40px
(4.4.4.4)
 40px
(4.4.4.4)
 40px
(4.4.4.4) 		120px
[78] truncated order-4 square (tissish)
( ↔ ) = 		40px
(4.8.8)
 40px
(4.4.4.4)
 40px
(4.8.8)
 40px
(4.8.8)
 80px 120px
[79] bitruncated order-4 square (dish)
↔ 		40px
(4.8.8)
 40px
(4.8.8)
 40px
(4.8.8)
 40px
(4.8.8)
 80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
 Alt
[83] alternated square
( ↔ ↔ ) = 		(6)
40px
(4.4.4.4)
 (6)
40px
(4.4.4.4)
 (6)
40px
(4.4.4.4)
 (6)
40px
(4.4.4.4)
 (8)
40px
(4.4.4) 		40px
(4.3.4.3)
[77] alternated order-4 square (sisquah)
↔
-

158 cantic order-4 square
↔



Nonuniform cyclosnub square



Nonuniform snub order-4 square



Nonuniform bisnub order-4 square
↔ 		40px
(3.3.4.3.4)
 40px
(3.3.4.3.4)
 40px
(3.3.4.3.4)
 40px
(3.3.4.3.4)
 40px
+(3.3.3) 		80px

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

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
105 tetrahedral-hexagonal
 (4)
40px
(3.3.3) 		- (4)
40px
(6.6.6) 		(6)
40px
(3.6.3.6) 		40px
(3.4.3.4)
106 tetrahedral-triangular
40px
(3.3.3.3)
40px
(3.3.3) 		-
40px
(3.3.3.3.3.3) 		40px
(3.4.6.4)
107 cyclotruncated tetrahedral-hexagonal
 (3)
40px
(3.6.6) 		(1)
40px
(3.3.3) 		(1)
40px
(6.6.6) 		(3)
40px
(6.6.6) 		80px
108 cyclotruncated hexagonal-tetrahedral
 (1)
40px
(3.3.3) 		(1)
40px
(3.3.3) 		(4)
40px
(3.12.12) 		(4)
40px
(3.12.12) 		80px
109 cyclotruncated tetrahedral-triangular
 (6)
40px
(3.6.6) 		(6)
40px
(3.6.6) 		(1)
40px
(3.3.3.3.3.3) 		(1)
40px
(3.3.3.3.3.3) 		80px
110 rectified tetrahedral-hexagonal
 (1)
40px
(3.3.3.3) 		(2)
40px
(3.4.3.4) 		(1)
40px
(3.6.3.6) 		(2)
40px
(3.4.6.4) 		80px
111 truncated tetrahedral-hexagonal
 (1)
40px
(3.6.6) 		(1)
40px
(3.4.3.4) 		(1)
40px
(3.12.12) 		(2)
40px
(4.6.12) 		80px
112 truncated tetrahedral-triangular
 (2)
40px
(4.6.6) 		(1)
40px
(3.6.6) 		(1)
40px
(3.4.6.4) 		(1)
40px
(6.6.6) 		80px
113 omnitruncated tetrahedral-hexagonal
 (1)
40px
(4.6.6) 		(1)
40px
(4.6.6) 		(1)
40px
(4.6.12) 		(1)
40px
(4.6.12) 		80px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
 Alt
Nonuniform omnisnub tetrahedral-hexagonal
 40px
(3.3.3.3.3) 		40px
(3.3.3.3.3) 		40px
(3.3.3.3.6) 		40px
(3.3.3.3.6) 		40px
+(3.3.3) 		80px

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

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
114 octahedral-hexagonal
 (6)
40px
(3.3.3.3)
 - (8)
40px
(6.6.6)
 (12)
40px
(3.6.3.6)
 80px
115 cubic-triangular
 (∞)
40px
(3.4.3.4)
 (∞)
40px
(4.4.4)
 - (∞)
40px
(3.3.3.3.3.3)
 40px
(3.4.6.4)
116 cyclotruncated octahedral-hexagonal
 (3)
40px
(4.6.6)
 (1)
40px
(4.4.4)
 (1)
40px
(6.6.6)
 (3)
40px
(6.6.6)
 80px
117 cyclotruncated hexagonal-octahedral
 (1)
40px
(3.3.3.3)
 (1)
40px
(3.3.3.3)
 (4)
40px
(3.12.12)
 (4)
40px
(3.12.12)
 80px
118 cyclotruncated cubic-triangular
 (6)
40px
(3.8.8)
 (6)
40px
(3.8.8)
 (1)
40px
(3.3.3.3.3.3)
 (1)
40px
(3.3.3.3.3.3)
 80px
119 rectified octahedral-hexagonal
 (1)
40px
(3.4.3.4)
 (2)
40px
(3.4.4.4)
 (1)
40px
(3.6.3.6)
 (2)
40px
(3.4.6.4)
 80px
120 truncated octahedral-hexagonal
 (1)
40px
(4.6.6)
 (1)
40px
(3.4.4.4)
 (1)
40px
(3.12.12)
 (2)
40px
(4.6.12)
 80px
121 truncated cubic-triangular
 (2)
40px
(4.6.8)
 (1)
40px
(3.8.8)
 (1)
40px
(3.4.6.4)
 (1)
40px
(6.6.6)
 80px
122 omnitruncated octahedral-hexagonal
 (1)
40px
(4.6.8)
 (1)
40px
(4.6.8)
 (1)
40px
(4.6.12)
 (1)
40px
(4.6.12)
 80px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure
0
 1
 2
 3
 Alt
Nonuniform cyclosnub octahedral-hexagonal
 40px
(3.3.3.3.3)
 40px
(3.3.3)
 40px
(3.3.3.3.3.3)
 40px
(3.3.3.3.3.3)
 40px
irr. {3,4} 		80px
Nonuniform omnisnub octahedral-hexagonal
 40px
(3.3.3.3.4)
 40px
(3.3.3.3.4)
 40px
(3.3.3.3.6)
 40px
(3.3.3.3.6)
 40px
irr. {3,3} 		80px

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

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
123 icosahedral-hexagonal
 (6)
50px
(3.3.3.3.3) 		- (8)
50px
(6.6.6) 		(12)
50px
(3.6.3.6) 		50px
3.4.5.4
124 dodecahedral-triangular
 (30)
50px
(3.5.3.5) 		(20)
50px
(5.5.5) 		- (12)
50px
(3.3.3.3.3.3) 		50px
(3.4.6.4)
125 cyclotruncated icosahedral-hexagonal
 (3)
50px
(5.6.6) 		(1)
50px
(5.5.5) 		(1)
50px
(6.6.6) 		(3)
50px
(6.6.6) 		80px
126 cyclotruncated hexagonal-icosahedral
 (1)
50px
(3.3.3.3.3) 		(1)
50px
(3.3.3.3.3) 		(5)
50px
(3.12.12) 		(5)
50px
(3.12.12) 		80px
127 cyclotruncated dodecahedral-triangular
 (6)
50px
(3.10.10) 		(6)
50px
(3.10.10) 		(1)
50px
(3.3.3.3.3.3) 		(1)
50px
(3.3.3.3.3.3) 		80px
128 rectified icosahedral-hexagonal
 (1)
50px
(3.5.3.5) 		(2)
50px
(3.4.5.4) 		(1)
50px
(3.6.3.6) 		(2)
50px
(3.4.6.4) 		80px
129 truncated icosahedral-hexagonal
 (1)
50px
(5.6.6) 		(1)
50px
(3.5.5.5) 		(1)
50px
(3.12.12) 		(2)
50px
(4.6.12) 		80px
130 truncated dodecahedral-triangular
 (2)
50px
(4.6.10) 		(1)
50px
(3.10.10) 		(1)
50px
(3.4.6.4) 		(1)
50px
(6.6.6) 		80px
131 omnitruncated icosahedral-hexagonal
 (1)
50px
(4.6.10) 		(1)
50px
(4.6.10) 		(1)
50px
(4.6.12) 		(1)
50px
(4.6.12) 		80px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
 Alt
Nonuniform omnisnub icosahedral-hexagonal
 50px
(3.3.3.3.5) 		50px
(3.3.3.3.5) 		50px
(3.3.3.3.6) 		50px
(3.3.3.3.6) 		50px
+(3.3.3) 		80px

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

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
132 hexagonal-triangular
 40px
(3.3.3.3.3.3) 		- 40px
(6.6.6) 		40px
(3.6.3.6) 		40px
(3.4.6.4)
133 cyclotruncated hexagonal-triangular
 (1)
40px
(3.3.3.3.3.3) 		(1)
40px
(3.3.3.3.3.3) 		(3)
40px
(3.12.12) 		(3)
40px
(3.12.12) 		80px
134 cyclotruncated triangular-hexagonal
 (1)
40px
(3.6.3.6) 		(2)
40px
(3.4.6.4) 		(1)
40px
(3.6.3.6) 		(2)
40px
(3.4.6.4) 		80px
135 rectified hexagonal-triangular
 (1)
40px
(6.6.6) 		(1)
40px
(3.4.6.4) 		(1)
40px
(3.12.12) 		(2)
40px
(4.6.12) 		80px
136 truncated hexagonal-triangular
 (1)
40px
(4.6.12) 		(1)
40px
(4.6.12) 		(1)
40px
(4.6.12) 		(1)
40px
(4.6.12) 		80px
[16] order-4 hexagonal tiling (shexah)

= 		(3)
40px
(6.6.6) 		(1)
40px
(6.6.6) 		(1)
40px
(6.6.6) 		(3)
40px
(6.6.6) 		80px
(3.3.3.3) 		120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
 Alt
[141] alternated order-4 hexagonal (ashexah)
↔ ↔ ↔ 		40px
(3.3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
+(3.3.3.3) 		40px
(4.6.6)
Nonuniform cyclocantisnub hexagonal-triangular
Nonuniform cycloruncicantisnub hexagonal-triangular
Nonuniform snub rectified hexagonal-triangular
 40px
(3.3.3.3.6) 		40px
(3.3.3.3.6) 		40px
(3.3.3.3.6) 		40px
(3.3.3.3.6) 		40px
+(3.3.3) 		80px

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 . 7 are half symmetry forms of [3,3,6]: ↔ .

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
137 alternated hexagonal (ahexah)
( ↔ ) = 		- - 40px
(3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(3.6.6)
138 cantic hexagonal (tahexah)
↔ 		(1)
40px
(3.3.3.3) 		- (2)
40px
(3.6.6) 		(2)
40px
(3.6.3.6) 		80px
139 runcic hexagonal (birahexah)
↔ 		(1)
40px
(4.4.4) 		(1)
40px
(4.4.3) 		(3)
40px
(3.4.3.4) 		(1)
40px
(3.3.3.3.3.3) 		80px
140 runcicantic hexagonal (bitahexah)
↔ 		(1)
40px
(3.10.10) 		(1)
40px
(4.4.3) 		(2)
40px
(4.6.6) 		(1)
40px
(3.6.3.6) 		80px
[2] rectified hexagonal (rihexah)
↔ 		(1)
40px
(3.3.3) 		- (1)
40px
(3.3.3) 		(6)
40px
(3.6.3.6) 		80px
Triangular prism 		120px
[3] rectified order-6 tetrahedral (rath)
↔ 		(2)
40px
(3.3.3.3) 		- (2)
40px
(3.3.3.3) 		(2)
40px
(3.3.3.3.3.3) 		80px
Hexagonal prism 		120px
[4] order-6 tetrahedral (thon)
↔ 		(4)
40px
(4.4.4) 		- (4)
40px
(4.4.4) 		- 60px 120px
[8] cantellated order-6 tetrahedral (srath)
↔ 		(1)
40px
(3.3.3.3) 		(2)
40px
(4.4.6) 		(1)
40px
(3.3.3.3) 		(1)
40px
(3.6.3.6) 		80px 120px
[9] bitruncated order-6 tetrahedral (tehexah)
↔ 		(1)
40px
(3.6.6) 		- (1)
40px
(3.6.6) 		(2)
40px
(6.6.6) 		80px 120px
[10] truncated order-6 tetrahedral (tath)
↔ 		(2)
40px
(3.10.10) 		- (2)
40px
(3.10.10) 		(1)
40px
(3.6.3.6) 		80px 120px
[14] cantitruncated order-6 tetrahedral (grath)
↔ 		(1)
40px
(4.6.6) 		(1)
40px
(4.4.6) 		(1)
40px
(4.6.6) 		(1)
40px
(6.6.6) 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure
0
 1
 0'
 3
 Alt
Nonuniform snub rectified order-6 tetrahedral
↔ 		40px
(3.3.3.3.3) 		40px
(3.3.3.3) 		40px
(3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
+(3.3.3) 		80px

[4,3[3]] family

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
141 alternated order-4 hexagonal (ashexah)
↔ 		- - 40px
(3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(4.6.6)
142 cantic order-4 hexagonal (tashexah)
↔ ↔ 		(1)
40px
(3.4.3.4) 		- (2)
40px
(4.6.6) 		(2)
40px
(3.6.3.6) 		80px
143 runcic order-4 hexagonal (birashexah)
↔ 		(1)
40px
(4.4.4) 		(1)
40px
(4.4.3) 		(3)
40px
(3.4.4.4) 		(1)
40px
(3.3.3.3.3.3) 		80px
144 runcicantic order-4 hexagonal (bitashexah)
↔ 		(1)
40px
(3.8.8) 		(1)
40px
(4.4.3) 		(2)
40px
(4.6.8) 		(1)
40px
(3.6.3.6) 		80px
[16] order-4 hexagonal (shexah)
↔ 		(4)
40px
(4.4.4) 		- (4)
40px
(4.4.4) 		- 80px 120px
[17] rectified order-4 hexagonal (rishexah)
↔ 		(1)
40px
(3.3.3.3) 		- (1)
40px
(3.3.3.3) 		(6)
40px
(3.6.3.6) 		80px 120px
[18] rectified order-6 cubic (rihach)
↔ 		(2)
40px
(3.4.3.4) 		- (2)
40px
(3.4.3.4) 		(2)
40px
(3.3.3.3.3.3) 		80px 120px
[21] bitruncated order-4 hexagonal (chexah)
↔ 		(1)
40px
(4.6.6) 		- (1)
40px
(4.6.6) 		(2)
40px
(6.6.6) 		80px 120px
[22] truncated order-6 cubic (thach)
↔ 		(2)
40px
(3.8.8) 		- (2)
40px
(3.8.8) 		(1)
40px
(3.6.3.6) 		80px 120px
[23] cantellated order-4 hexagonal (srishexah)
↔ 		(1)
40px
(3.4.4.4) 		(2)
40px
(4.4.6) 		(1)
40px
(3.4.4.4) 		(1)
40px
(3.6.3.6) 		80px 120px
[26] cantitruncated order-4 hexagonal (grishexah)
↔ 		(1)
40px
(4.6.8) 		(1)
40px
(4.4.6) 		(1)
40px
(4.6.8) 		(1)
40px
(6.6.6) 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure
0
 1
 0'
 3
 Alt
Nonuniform snub rectified order-4 hexagonal
↔ 		40px
(3.3.3.3.4) 		40px
(3.3.3.3) 		40px
(3.3.3.3.4) 		40px
(3.3.3.3.3.3) 		40px
+(3.3.3)

[5,3[3]] family

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
145 alternated order-5 hexagonal (aphexah)
↔ 		- - 40px
(3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(3.6.3.6)
146 cantic order-5 hexagonal (taphexah)
↔ 		(1)
40px
(3.5.3.5) 		- (2)
40px
(5.6.6) 		(2)
40px
(3.6.3.6) 		80px
147 runcic order-5 hexagonal (biraphexah)
↔ 		(1)
40px
(5.5.5) 		(1)
40px
(4.4.3) 		(3)
40px
(3.4.5.4) 		(1)
40px
(3.3.3.3.3.3) 		80px
148 runcicantic order-5 hexagonal (bitaphexah)
↔ 		(1)
40px
(3.10.10) 		(1)
40px
(4.4.3) 		(2)
40px
(4.6.10) 		(1)
40px
(3.6.3.6) 		80px
[32] rectified order-5 hexagonal (riphexah)
↔ 		(1)
40px
(3.3.3.3.3) 		- (1)
40px
(3.3.3.3.3) 		(6)
40px
(3.6.3.6) 		80px 120px
[33] rectified order-6 dodecahedral (rihed)
↔ 		(2)
40px
(3.5.3.5) 		- (2)
40px
(3.5.3.5) 		(2)
40px
(3.3.3.3.3.3) 		80px 120px
[34] Order-5 hexagonal (hedhon)
↔ 		(4)
40px
(5.5.5) 		- (4)
40px
(5.5.5) 		- 80px 120px
[40] truncated order-6 dodecahedral (thed)
↔ 		(2)
40px
(3.10.10) 		- (2)
40px
(3.10.10) 		(1)
40px
(3.6.3.6) 		80px 120px
[36] cantellated order-5 hexagonal (sriphexah)
↔ 		(1)
40px
(3.4.5.4) 		(2)
40px
(6.4.4) 		(1)
40px
(3.4.5.4) 		(1)
40px
(3.6.3.6) 		80px 120px
[39] bitruncated order-5 hexagonal (dohexah)
↔ 		(1)
40px
(5.6.6) 		- (1)
40px
(5.6.6) 		(2)
40px
(6.6.6) 		80px 120px
[41] cantitruncated order-5 hexagonal (griphexah)
↔ 		(1)
40px
(4.6.10) 		(1)
40px
(6.4.4) 		(1)
40px
(4.6.10) 		(1)
40px
(6.6.6) 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
 Alt
Nonuniform snub rectified order-5 hexagonal
↔ 		40px
(3.3.3.3.5) 		40px
(3.3.3) 		40px
(3.3.3.3.5) 		40px
(3.3.3.3.3.3) 		40px
+(3.3.3)

[6,3[3]] family

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
149 runcic order-6 hexagonal
↔ 		(1)
40px
(6.6.6) 		(1)
40px
(4.4.3) 		(3)
40px
(3.4.6.4) 		(1)
40px
(3.3.3.3.3.3) 		80px
150 runcicantic order-6 hexagonal
↔ 		(1)
40px
(3.12.12) 		(1)
40px
(4.4.3) 		(2)
40px
(4.6.12) 		(1)
40px
(3.6.3.6) 		80px
[1] hexagonal (hexah)
↔ ↔ ↔ 		(1)
40px
(6.6.6) 		- (1)
40px
(6.6.6) 		(2)
40px
(6.6.6) 		80px 120px
[46] order-6 hexagonal (hihexah)
↔ 		(4)
40px
(6.6.6) 		- (4)
40px
(6.6.6) 		- 40px 120px
[47] rectified order-6 hexagonal (rihihexah)
↔ 		(2)
40px
(3.6.3.6) 		- (2)
40px
(3.6.3.6) 		(2)
40px
(3.3.3.3.3.3) 		80px 120px
[47] rectified order-6 hexagonal (rihihexah)
↔ 		(1)
40px
(3.3.3.3.3.3) 		- (1)
40px
(3.3.3.3.3.3) 		(6)
40px
(3.6.3.6) 		80px 120px
[48] truncated order-6 hexagonal (thihexah)
↔ 		(2)
40px
(3.12.12) 		- (2)
40px
(3.12.12) 		(1)
40px
(3.3.3.3.3.3) 		80px 120px
[49] cantellated order-6 hexagonal (srihihexah)
↔ 		(1)
40px
(3.4.6.4) 		(2)
40px
(6.4.4) 		(1)
40px
(3.4.6.4) 		(1)
40px
(3.6.3.6) 		80px 120px
[51] cantitruncated order-6 hexagonal (grihihexah)
↔ 		(1)
40px
(4.6.12) 		(1)
40px
(6.4.4) 		(1)
40px
(4.6.12) 		(1)
40px
(6.6.6) 		80px 120px
[54] triangular tiling honeycomb (trah)
( ↔ ) = 		- - 40px
(3.3.3.3.3.3) 		40px
(3.3.3.3.3.3) 		40px
(6.6.6) 		120px
[55] cantic order-6 hexagonal (ritrah)
( ↔ ) = 		(1)
40px
(3.6.3.6) 		- (2)
40px
(6.6.6) 		(2)
40px
(3.6.3.6) 		80px 120px
Alternated forms
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		vertex figure Picture
0
 1
 0'
 3
 Alt
[54] triangular tiling honeycomb (trah)
( ↔ ↔ ) = 		40px
 - 40px
 - 40px 40px
(6.6.6) 		120px
[137] alternated hexagonal (ahexah)
( ↔ ) = ( ↔ ) 		40px
 - 40px
 40px
 40px
+(3.6.6) 		40px
(3.6.6)
[47] rectified order-6 hexagonal (rihihexah)
↔ ↔ ↔ 		40px
(3.6.3.6) 		- 40px
(3.6.3.6) 		40px
(3.3.3.3.3.3) 		80px 120px
[55] cantic order-6 hexagonal (ritrah)
( ↔ ) = ( ↔ ) = 		(1)
40px
(3.6.3.6) 		- (2)
40px
(6.6.6) 		(2)
40px
(3.6.3.6) 		80px 120px
Nonuniform snub rectified order-6 hexagonal
↔
40px
(3.3.3.3.6)
40px
(3.3.3.3)
40px
(3.3.3.3.6)
40px
(3.3.3.3.3.3) 		40px
+(3.3.3)

Multicyclic graphs

[3[ ]×[ ]] family

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

# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
151 Quarter order-4 hexagonal (quishexah)
↔
40px
40px
40px
40px 		80px
[17] rectified order-4 hexagonal (rishexah)
↔ ↔ ↔
40px
40px
40px
40px 		60px
(4.4.4) 		120px
[18] rectified order-6 cubic (rihach)
↔ ↔ ↔
40px
40px
40px
40px 		60px
(6.4.4) 		120px
[21] bitruncated order-6 cubic (chexah)
↔ ↔ ↔
40px
40px
40px
40px 		60px 120px
[87] alternated order-6 cubic (ahach)
↔ ↔ 		-
40px
40px
40px 		40px
(3.6.3.6)
[88] cantic order-6 cubic (tachach)
↔ ↔
40px
40px
40px
40px 		60px
[141] alternated order-4 hexagonal (ashexah)
↔ ↔
40px 		-
40px
40px 		40px
(4.6.6)
[142] cantic order-4 hexagonal (tashexah)
↔ ↔
40px
40px
40px
40px 		80px
# Honeycomb name
Coxeter diagram 		Cells by location
(and count around each vertex) 		Vertex figure Picture
0
 1
 2
 3
 Alt
Nonuniform bisnub order-6 cubic
↔ 		40px
 40px
 40px
 40px
 40px
irr. {3,3} 		80px

[3[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).

# Name
Coxeter diagram 		0 1 2 3 vertex figure Picture
[1] hexagonal (hexah)
↔ 		40px
 40px
 40px
 40px
 60px
{3,3} 		120px
[47] rectified order-6 hexagonal (rihihexah)
↔ 		40px
 40px
 40px
 40px
 60px
t{2,3} 		120px
[54] triangular tiling honeycomb (trah)
( ↔ ) = 		40px
 - 40px
 40px
 40px
t{3[3]} 		120px
[55] rectified triangular (ritrah)
↔ 		40px
 40px
 40px
 40px
 60px
t{2,3} 		120px
# Name
Coxeter diagram 		0 1 2 3 Alt vertex figure Picture
[137] alternated hexagonal (ahexah)
( ↔ ) = 		40px

s{3[3]} 		40px

s{3[3]} 		40px

s{3[3]} 		40px

s{3[3]} 		40px

{3,3} 		40px
(4.6.6)

Summary enumerations by family

Linear graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry 		Honeycombs Chiral
extended
symmetry 		Alternation honeycombs
R¯3
[4,4,3]
 [4,4,3]
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Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". 		[1+,4,1+,4,3+] (6) Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[4,4,3]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
N¯3
[4,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[4,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1+,4,1+,4,1+,4,1+] (3) Lua error: Internal error: The interpreter has terminated with signal "24".
[4,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24". 		(3) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1+,4,1+,4,1+,4,1+] (3) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[4,4,4]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(4,4+,4,2+)]] (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[4,4,4]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
V¯3
[6,3,3]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[6,3,3]
Lua error: Internal error: The interpreter has terminated with signal "24".		 15 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". 		[1+,6,(3,3)+] (2) Lua error: Internal error: The interpreter has terminated with signal "24". (↔ Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[6,3,3]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
BV¯3
[6,3,4]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[6,3,4]
Lua error: Internal error: The interpreter has terminated with signal "24".		 15 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". 		[1+,6,3+,4,1+] (6) Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[6,3,4]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
HV¯3
[6,3,5]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[6,3,5]
Lua error: Internal error: The interpreter has terminated with signal "24".		 15 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". 		[1+,6,(3,5)+] (2) Lua error: Internal error: The interpreter has terminated with signal "24". (↔ Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[6,3,5]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
Y¯3
[3,6,3]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[3,6,3]
Lua error: Internal error: The interpreter has terminated with signal "24".		 5 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[3,6,3]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [2+[3+,6,3+]] (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[3,6,3]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[3,6,3]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
Z¯3
[6,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[6,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24".		 6 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". 		[1+,6,3+,6,1+] (2) Lua error: Internal error: The interpreter has terminated with signal "24". (↔ Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[6,3,6]]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(6,3+,6,2+)]] (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[6,3,6]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 2 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[6,3,6]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".

Tridental graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry 		Honeycombs Chiral
extended
symmetry 		Alternation honeycombs
DV¯3
[6,31,1]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [6,31,1] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[1[6,31,1]]=[6,3,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[1+,6,31,1]]+ (2) Lua error: Internal error: The interpreter has terminated with signal "24". (↔ Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[1[6,31,1]]+=[6,3,4]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
O¯3
[3,41,1]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [3,41,1] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [3+,41,1]+ (2) Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24".
[1[3,41,1]]=[3,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[3+,41,1]]+ (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[1[3,41,1]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
M¯3
[41,1,1]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [41,1,1] 0 (none)
[1[41,1,1]]=[4,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (4) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[1+,4,1+,41,1]]+=[(4,1+,4,1+,4,2+)] (4) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[3[41,1,1]]=[4,4,3]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (3) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [3[1+,41,1,1]]+=[1+,4,1+,4,3+] (2) Lua error: Internal error: The interpreter has terminated with signal "24". (↔ Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[3[41,1,1]]+=[4,4,3]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".

Cyclic graphs

Paracompact hyperbolic enumeration
Group Extended
symmetry 		Honeycombs Chiral
extended
symmetry 		Alternation honeycombs
CR^3
[(4,4,4,3)]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(4,4,4,3)] 6 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [(4,1+,4,1+,4,3+)] (2) Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(4,4,4,3)]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(4,4+,4,3+)]] (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(4,4,4,3)]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
RR^3
[4[4]]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[4[4]] (none)
[2+[4[4]]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 1 Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(4+,4)[2]]] (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[1[4[4]]]=[4,41,1]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (2) Lua error: Internal error: The interpreter has terminated with signal "24". Lua error: Internal error: The interpreter has terminated with signal "24". [(1+,4)[4]] (2) Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24".
[2[4[4]]]=[4,4,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(1+,4,4)[2]]] (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[(2+,4)[4[4]]]=[2+[4,4,4]]
Lua error: Internal error: The interpreter has terminated with signal "24". = Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [(2+,4)[4[4]]]+
= [2+[4,4,4]]+ 		(1) Lua error: Internal error: The interpreter has terminated with signal "24".
AV^3
[(6,3,3,3)]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(6,3,3,3)] 6 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(6,3,3,3)]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(6,3,3,3)]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
BV^3
[(3,4,3,6)]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(3,4,3,6)] 6 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [(3+,4,3+,6)] (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(3,4,3,6)]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(3,4,3,6)]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
HV^3
[(3,5,3,6)]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(3,5,3,6)] 6 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(3,5,3,6)]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 3 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(3,5,3,6)]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
VV^3
[(3,6)[2]]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(3,6)[2]] 2 Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(3,6)[2]]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 1 Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(3,6)[2]]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 1 Lua error: Internal error: The interpreter has terminated with signal "24".
[2+[(3,6)[2]]]
Lua error: Internal error: The interpreter has terminated with signal "24". = Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [2+[(3+,6)[2]]] (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[(2,2)+[(3,6)[2]]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 1 Lua error: Internal error: The interpreter has terminated with signal "24". [(2,2)+[(3,6)[2]]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
Paracompact hyperbolic enumeration
Group Extended
symmetry 		Honeycombs Chiral
extended
symmetry 		Alternation honeycombs
BR^3
[(3,3,4,4)]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[(3,3,4,4)] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[1[(4,4,3,3)]]=[3,41,1]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[(3,3,4,1+,4)]]+
= [3+,41,1]+		 (2) Lua error: Internal error: The interpreter has terminated with signal "24". (= Lua error: Internal error: The interpreter has terminated with signal "24".)
Lua error: Internal error: The interpreter has terminated with signal "24".
[1[(3,3,4,4)]]+
= [3,41,1]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".
DP¯3
[3[ ]x[ ]]
Lua error: Internal error: The interpreter has terminated with signal "24". 		[3[ ]x[ ]] 1 Lua error: Internal error: The interpreter has terminated with signal "24".
[1[3[ ]x[ ]]]=[6,31,1]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[1[3[ ]x[ ]]]=[4,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[2[3[ ]x[ ]]]=[6,3,4]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (3) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [2[3[ ]x[ ]]]+
=[6,3,4]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".
PP¯3
[3[3,3]]
Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24". 		[3[3,3]] 0 (none)
[1[3[3,3]]]=[6,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 0 (none)
[3[3[3,3]]]=[3,6,3]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (2) Lua error: Internal error: The interpreter has terminated with signal "24".
[2[3[3,3]]]=[6,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[(3,3)[3[3,3]]]=[6,3,3]
Lua error: Internal error: The interpreter has terminated with signal "24". = Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [(3,3)[3[3,3]]]+
= [6,3,3]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".

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
symmetry 		Honeycombs Chiral
extended
symmetry 		Alternation honeycombs
P¯3
[3,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [3,3[3]] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[1[3,3[3]]]=[3,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[3,3[3]]]+
= [3,3,6]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".
BP¯3
[4,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [4,3[3]] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[1[4,3[3]]]=[4,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1+,4,(3[3])+] (2) Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".
Lua error: Internal error: The interpreter has terminated with signal "24".
[4,3[3]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
HP¯3
[5,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [5,3[3]] 4 Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24".
[1[5,3[3]]]=[5,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (7) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [1[5,3[3]]]+
= [5,3,6]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".
VP¯3
[6,3[3]]
Lua error: Internal error: The interpreter has terminated with signal "24".		 [6,3[3]] 2 Lua error: Internal error: The interpreter has terminated with signal "24".
[6,3[3]] = (2) (Lua error: Internal error: The interpreter has terminated with signal "24". = Lua error: Internal error: The interpreter has terminated with signal "24".)
[(3,3)[1+,6,3[3]]]=[6,3,3]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". [(3,3)[1+,6,3[3]]]+ (1) Lua error: Internal error: The interpreter has terminated with signal "24".
[1[6,3[3]]]=[6,3,6]
Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24".		 (6) Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". | Lua error: Internal error: The interpreter has terminated with signal "24". [3[1+,6,3[3]]]+
= [3,6,3]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24". ↔ Lua error: Internal error: The interpreter has terminated with signal "24". (= Lua error: Internal error: The interpreter has terminated with signal "24". )
[1[6,3[3]]]+
= [6,3,6]+		 (1) Lua error: Internal error: The interpreter has terminated with signal "24".

See also

Notes

  1. ↑ P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003)

Lua error: Internal error: The interpreter has terminated with signal "24".

References


Lua error: Internal error: The interpreter has terminated with signal "24".


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