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 | SchläfliSymbol{p,q,r} | Coxeter | Celltype{p,q} | Facetype{p} | Edgefigure{r} | Vertexfigure{q,r} | Dual | Coxetergroup |
|---|
| 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] |
| |
| 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 | Simplexvolume | 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. 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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|---|
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)(6.6.6) | Tetrahedron | |
|---|
| 2 | rectified hexagonal (rihexah)t1{6,3,3} or r{6,3,3} | (2)(3.3.3) | - | - | (3)(3.6.3.6) | Triangular prism | |
|---|
| 3 | rectified order-6 tetrahedral (rath)t1{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 (thon){3,3,6} | (∞)(3.3.3) | - | - | - | Triangular tiling | |
|---|
| 5 | truncated hexagonal (thexah) t0,1{6,3,3} or t{6,3,3} | (1)(3.3.3) | - | - | (3)(3.12.12) | Triangular pyramid | |
|---|
| 6 | cantellated hexagonal (srihexah)t0,2{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 (sidpithexah)t0,3{6,3,3} | (1)(3.3.3) | (3)(4.4.3) | (3)(4.4.6) | (1)(6.6.6) | | |
|---|
| 8 | cantellated order-6 tetrahedral (srath)t0,2{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 (tehexah)t1,2{6,3,3} or 2t{6,3,3} | (2)(3.6.6) | - | - | (2)(6.6.6) | | |
|---|
| 10 | truncated order-6 tetrahedral (tath)t0,1{3,3,6} or t{3,3,6} | (6)(3.6.6) | - | - | (1)(3.3.3.3.3.3) | | |
|---|
| 11 | cantitruncated hexagonal (grihexah)t0,1,2{6,3,3} or tr{6,3,3} | (1)(3.6.6) | (1)(4.4.3) | - | (2)(4.6.12) | | |
|---|
| 12 | runcitruncated hexagonal (prath)t0,1,3{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 (prihexah)t0,1,3{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 (grath)t0,1,2{3,3,6} or tr{3,3,6} | (2)(4.6.6) | - | (1)(4.4.6) | (1)(6.6.6) | | |
|---|
| 15 | omnitruncated hexagonal (gidpithexah)t0,1,2,3{6,3,3} | (1)(4.6.6) | (1)(4.4.6) | (1)(4.4.12) | (1)(4.6.12) | | |
|---|
Alternated forms
[6,3,4] family
There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or
| # | Name of honeycombCoxeter diagramSchläfli 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)(6.6.6) | (3.3.3.3) | |
|---|
| 17 | rectified order-4 hexagonal (rishexah)t1{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 (rihach)t1{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 (hachon){4,3,6} | (20)(4.4.4) | - | - | - | (3.3.3.3.3.3) | |
|---|
| 20 | truncated order-4 hexagonal (tishexah)t0,1{6,3,4} or t{6,3,4} | (1)(3.3.3.3) | - | - | (4)(3.12.12) | | |
|---|
| 21 | bitruncated order-6 cubic (chexah)t1,2{6,3,4} or 2t{6,3,4} | (2)(4.6.6) | - | - | (2)(6.6.6) | | |
|---|
| 22 | truncated order-6 cubic (thach)t0,1{4,3,6} or t{4,3,6} | (6)(3.8.8) | - | - | (1)(3.3.3.3.3.3) | | |
|---|
| 23 | cantellated order-4 hexagonal (srishexah)t0,2{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 (srihach)t0,2{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 (sidpichexah)t0,3{6,3,4} | (1)(4.4.4) | (3)(4.4.4) | (3)(4.4.6) | (1)(6.6.6) | | |
|---|
| 26 | cantitruncated order-4 hexagonal (grishexah)t0,1,2{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 (grihach)t0,1,2{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 (prihach)t0,1,3{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 (prishexah)t0,1,3{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 (gidpichexah)t0,1,2,3{6,3,4} | (1)(4.6.8) | (1)(4.4.8) | (1)(4.4.12) | (1)(4.6.12) | | |
|---|
Alternated forms
| # | Name of honeycombCoxeter diagramSchläfli 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} | (3.3.3) | | | | (3.3.3.3.3.3) | (3.6.3.6) | |
| [88] | cantic order-6 cubic (tachach) ↔ h2{4,3,6} | (2)(3.6.6) | - | - | (1)(3.6.3.6) | (2)(6.6.6) | | |
| [89] | runcic order-6 cubic (birachach) ↔ h3{4,3,6} | (1)(3.3.3) | - | - | (1)(6.6.6) | (3)(3.4.6.4) | | |
| [90] | runcicantic order-6 cubic (bitachach) ↔ h2,3{4,3,6} | (1)(3.6.6) | - | - | (1)(3.12.12) | (2)(4.6.12) | | |
| [141] | alternated order-4 hexagonal (ashexah) ↔ ↔ h{6,3,4} | - | - | | (3.3.3.3.3.3) | (3.3.3.3) | (4.6.6) | |
| [142] | cantic order-4 hexagonal (tashexah) ↔ ↔ h1{6,3,4} | (1)(3.4.3.4) | - | | (2)(3.6.3.6) | (2)(4.6.6) | | |
| [143] | runcic order-4 hexagonal (birashexah) ↔ h3{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 (bitashexah) ↔ h2,3{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 (quishexah) ↔ q{6,3,4} | (3) | (1) | - | (1) | (3) | | |
| Nonuniform | bisnub order-6 cubic ↔ 2s{4,3,6} | (3.3.3.3.3) | - | - | (3.3.3.3.3.3) | +(3.3.3) | | |
| Nonuniform | runcic bisnub order-6 cubic | | | | | | | |
| Nonuniform | snub rectified order-6 cubic ↔ sr{4,3,6} | (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 cubicsr3{4,3,6} | | | | | | | |
| Nonuniform | snub rectified order-4 hexagonal ↔ sr{6,3,4} | (3.3.3.3.3.3) | (3.3.3) | - | (3.3.3.3.6) | +(3.3.3) | | |
| Nonuniform | runcisnub rectified order-4 hexagonalsr3{6,3,4} | | | | | | | |
| Nonuniform | omnisnub rectified order-6 cubicht0,1,2,3{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 diagramSchläfli 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)(6)3 | Icosahedron | |
|---|
| 32 | rectified order-5 hexagonal (riphexah)t1{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 (rihed)t1{5,3,6} or r{5,3,6} | (5)(3.5.3.5) | - | - | (2)(3)6 | (6.4.4) | |
|---|
| 34 | order-6 dodecahedral (hedhon){5,3,6} | (5.5.5) | - | - | - | (∞) (3)6 | |
|---|
| 35 | truncated order-5 hexagonal (tiphexah)t0,1{6,3,5} or t{6,3,5} | (1)(3.3.3.3.3) | - | - | (5)3.12.12 | | |
|---|
| 36 | cantellated order-5 hexagonal (sriphexah)t0,2{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 (sidpidohexah)t0,3{6,3,5} | (1)(5.5.5) | - | (6)(6.4.4) | (1)(6)3 | | |
|---|
| 38 | cantellated order-6 dodecahedral (srihed)t0,2{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 (dohexah)t1,2{6,3,5} or 2t{6,3,5} | (2)(5.6.6) | - | - | (2)(6)3 | | |
|---|
| 40 | truncated order-6 dodecahedral (thed)t0,1{5,3,6} or t{5,3,6} | (6)(3.10.10) | - | - | (1)(3)6 | | |
|---|
| 41 | cantitruncated order-5 hexagonal (griphexah)t0,1,2{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 (prihed)t0,1,3{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 (priphaxh)t0,1,3{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 (grihed)t0,1,2{5,3,6} or tr{5,3,6} | (1)(4.6.10) | - | (2)(6.4.4) | (1)(6)3 | | |
|---|
| 45 | omnitruncated order-6 dodecahedral (gidpidohaxh)t0,1,2,3{6,3,5} | (1)(4.6.10) | (1)(10.4.4) | (1)(12.4.4) | (1)4.6.12 | | |
|---|
Alternated forms
| # | Honeycomb nameCoxeter diagramSchläfli 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)(3)6 | (12)(3)5 | (5.6.6) | |
| [146] | cantic order-5 hexagonal (taphexah) ↔ h2{6,3,5} | (1)(3.5.3.5) | - | | (2)(3.6.3.6) | (2)(5.6.6) | | |
| [147] | runcic order-5 hexagonal (biraphexah) ↔ h3{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 (bitaphexah) ↔ h2,3{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 ↔ sr{5,3,6} | (3.3.5.3.5) | - | (3.3.3.3) | (3.3.3.3.3.3) | irr. tet | | |
| Nonuniform | omnisnub order-5 hexagonalht0,1,2,3{6,3,5} | (3.3.5.3.5) | (3.3.3.5) | (3.3.3.6) | (3.3.6.3.6) | irr. tet | | |
[6,3,6] family
There are 9 forms, generated by ring permutations of the Coxeter group: [6,3,6] or
| # | Name of honeycombCoxeter diagramSchläfli symbol | Cells by location and count per vertex | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 |
|---|
| 46 | order-6 hexagonal (hihexah){6,3,6} | - | - | - | (20) (6.6.6) | (3.3.3.3.3.3) | |
|---|
| 47 | rectified order-6 hexagonal (rihihexah)t1{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 (thihexah)t0,1{6,3,6} or t{6,3,6} | (1)(3.3.3.3.3.3) | - | - | (6)(3.12.12) | | |
|---|
| 49 | cantellated order-6 hexagonal (srihihexah)t0,2{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 (spiddihexah)t0,3{6,3,6} | (1)(6.6.6) | (3)(4.4.6) | (3)(4.4.6) | (1)(6.6.6) | | |
|---|
| 51 | cantitruncated order-6 hexagonal (grihihexah)t0,1,2{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 (prihihexah)t0,1,3{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 (gidpiddihexah)t0,1,2,3{6,3,6} | (1)(4.6.12) | (1)(4.4.12) | (1)(4.4.12) | (1)(4.6.12) | | |
|---|
| [1] | bitruncated order-6 hexagonal (hexah) ↔ ↔ t1,2{6,3,6} or 2t{6,3,6} | (2)(6.6.6) | - | - | (2)(6.6.6) | | |
Alternated forms
| # | Name of honeycombCoxeter diagramSchläfli 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) (3.3.3.3.3.3) | - | - | (6)(3.6.3.6) | | (6.4.4) | |
| [54] | triangular (trah)( ↔ ) = h{6,3,6} = {3,6,3} | - | - | - | (3.3.3.3.3.3) | (3.3.3.3.3.3) | {6,3} | |
| [55] | cantic order-6 hexagonal (ritrah) ( ↔ ) = h2{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 ↔ h3{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 ↔ h2,3{6,3,6} | (1)(3.12.12) | (1)(4.4.3) | (2)(4.6.12) | (1)(3.6.3.6) | | | |
| [137] | alternated hexagonal (ahexah)( ↔ ↔ ) = 2s{6,3,6} = h{6,3,3} | (3.3.3.3.6) | - | - | (3.3.3.3.6) | +(3.3.3) | (3.6.6) | |
| Nonuniform | snub rectified order-6 hexagonalsr{6,3,6} | (3.3.3.3.3.3) | (3.3.3.3) | - | (3.3.3.3.6) | +(3.3.3) | | |
| Nonuniform | alternated runcinated order-6 hexagonalht0,3{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 hexagonalht0,1,2,3{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 diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 |
|---|
| 54 | triangular (trah){3,6,3} | - | - | - | (∞){3,6} | {6,3} | |
|---|
| 55 | rectified triangular (ritrah)t1{3,6,3} or r{3,6,3} | (2)(6)3 | - | - | (3)(3.6)2 | (3.4.4) | |
|---|
| 56 | cantellated triangular (sritrah)t0,2{3,6,3} or rr{3,6,3} | (1)(3.6)2 | (2)(4.4.3) | - | (2)(3.6.4.6) | | |
|---|
| 57 | runcinated triangular (spidditrah)t0,3{3,6,3} | (1)(3)6 | (6)(4.4.3) | (6)(4.4.3) | (1)(3)6 | | |
|---|
| 58 | bitruncated triangular (ditrah)t1,2{3,6,3} or 2t{3,6,3} | (2)(3.12.12) | - | - | (2)(3.12.12) | | |
|---|
| 59 | cantitruncated triangular (gritrah)t0,1,2{3,6,3} or tr{3,6,3} | (1)(3.12.12) | (1)(4.4.3) | - | (2)(4.6.12) | | |
|---|
| 60 | runcitruncated triangular (pritrah)t0,1,3{3,6,3} | (1)(3.6.4.6) | (1)(4.4.3) | (2)(4.4.6) | (1)(6)3 | | |
|---|
| 61 | omnitruncated triangular (gipidditrah)t0,1,2,3{3,6,3} | (1)(4.6.12) | (1)(4.4.6) | (1)(4.4.6) | (1)(4.6.12) | | |
|---|
| [1] | truncated triangular (hexah) ↔ ↔ t0,1{3,6,3} or t{3,6,3} = {6,3,3} | (1)(6)3 | - | - | (3)(6)3 | {3,3} | |
Alternated forms
| # | Honeycomb nameCoxeter diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 | Alt |
|---|
| [56] | cantellated triangular (sritrah) = s2{3,6,3} | (1)(3.6)2 | - | - | (2)(3.6.4.6) | (3.4.4) | | |
| [60] | runcitruncated triangular (pritrah) = s2,3{3,6,3} | (1)(6)3 | - | (1)(4.4.3) | (1)(3.6.4.6) | (2)(4.4.6) | | |
| [137] | alternated hexagonal (ahexah)( ↔ ) = ( ↔ )s{3,6,3} | (3)6 | - | - | (3)6 | +(3)3 | (3.6.6) | |
| Scaliform | runcisnub triangular (pristrah)s3{3,6,3} | r{6,3} | - | (3.4.4) | (3)6 | tricup | | |
| Nonuniform | omnisnub triangular tiling honeycomb (snatrah)ht0,1,2,3{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 diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 |
|---|
| 62 | square (squah) = {4,4,3} | - | - | - | (6) | Cube | |
|---|
| 63 | rectified square (risquah) = t1{4,4,3} or r{4,4,3} | (2) | - | - | (3) | Triangular prism | |
|---|
| 64 | rectified order-4 octahedral (rocth)t1{3,4,4} or r{3,4,4} | (4) | - | - | (2) | | |
|---|
| 65 | order-4 octahedral (octh){3,4,4} | (∞) | - | - | - | | |
|---|
| 66 | truncated square (tisquah) = t0,1{4,4,3} or t{4,4,3} | (1) | - | - | (3) | | |
|---|
| 67 | truncated order-4 octahedral (tocth)t0,1{3,4,4} or t{3,4,4} | (4) | - | - | (1) | | |
|---|
| 68 | bitruncated square (osquah)t1,2{4,4,3} or 2t{4,4,3} | (2) | - | - | (2) | | |
|---|
| 69 | cantellated square (srisquah)t0,2{4,4,3} or rr{4,4,3} | (1) | (2) | - | (2) | | |
|---|
| 70 | cantellated order-4 octahedral (srocth)t0,2{3,4,4} or rr{3,4,4} | (2) | - | (2) | (1) | | |
|---|
| 71 | runcinated square (sidposquah)t0,3{4,4,3} | (1) | (3) | (3) | (1) | | |
|---|
| 72 | cantitruncated square (grisquah)t0,1,2{4,4,3} or tr{4,4,3} | (1) | (1) | - | (2) | | |
|---|
| 73 | cantitruncated order-4 octahedral (grocth)t0,1,2{3,4,4} or tr{3,4,4} | (2) | - | (1) | (1) | | |
|---|
| 74 | runcitruncated square (procth)t0,1,3{4,4,3} | (1) | (1) | (2) | (1) | | |
|---|
| 75 | runcitruncated order-4 octahedral (prisquah)t0,1,3{3,4,4} | (1) | (2) | (1) | (1) | | |
|---|
| 76 | omnitruncated square (gidposquah)t0,1,2,3{4,4,3} | (1) | (1) | (1) | (1) | | |
|---|
Alternated forms
| # | Honeycomb nameCoxeter diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 | Alt |
|---|
| [83] | alternated square ↔ h{4,4,3} | - | - | - | (6) | (8) | | |
| [84] | cantic square ↔ h2{4,4,3} | (1) | - | - | (2) | (2) | | |
| [85] | runcic square ↔ h3{4,4,3} | (1) | - | - | (1). | (4) | | |
| [86] | runcicantic square ↔ | (1) | - | - | (1) | (2) | | |
| [153] | alternated rectified square ↔ hr{4,4,3} | | - | - | | {}x{3} | | |
| 157 | | | - | - | | {}x{6} | | |
| Scaliform | snub order-4 octahedral = = s{3,4,4} | | - | - | | {}v{4} | | |
| Scaliform | runcisnub order-4 octahedrals3{3,4,4} | | | | | cup-4 | | |
| 152 | snub square = s{4,4,3} | | - | - | | {3,3} | | |
| Nonuniform | snub rectified order-4 octahedralsr{3,4,4} | | - | | | irr. {3,3} | | |
| Nonuniform | alternated runcitruncated squareht0,1,3{3,4,4} | | | | | irr. {}v{4} | | |
| Nonuniform | omnisnub squareht0,1,2,3{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 diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Symmetry | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 |
|---|
| 77 | order-4 square (sisquah){4,4,4} | - | - | - | | [4,4,4] | Cube | |
|---|
| 78 | truncated order-4 square (tissish)t0,1{4,4,4} or t{4,4,4} | | - | - | | [4,4,4] | | |
|---|
| 79 | bitruncated order-4 square (dish)t1,2{4,4,4} or 2t{4,4,4} | | - | - | | [[4,4,4]] | | |
|---|
| 80 | runcinated order-4 square (spiddish)t0,3{4,4,4} | | | | | [[4,4,4]] | | |
|---|
| 81 | runcitruncated order-4 square (prissish)t0,1,3{4,4,4} | | | | | [4,4,4] | | |
|---|
| 82 | omnitruncated order-4 square (gipiddish)t0,1,2,3{4,4,4} | | | | | [[4,4,4]] | | |
|---|
| [62] | square (squah) ↔ t1{4,4,4} or r{4,4,4} | | - | - | | [4,4,4] | Square tiling | |
| [63] | rectified square (risquah) ↔ t0,2{4,4,4} or rr{4,4,4} | | | - | | [4,4,4] | | |
| [66] | truncated order-4 square (tisquah) ↔ t0,1,2{4,4,4} or tr{4,4,4} | | | - | | [4,4,4] | | |
Alternated constructions
| # | Honeycomb nameCoxeter diagramand Schläfli symbol | Cell counts/vertexand positions in honeycomb | Symmetry | Vertex figure | Picture |
|---|
| 0 | 1 | 2 | 3 | Alt |
|---|
| [62] | Square (squah)( ↔ ↔ ↔ ) = | (4.4.4.4) | - | - | | (4.4.4.4) | [1+,4,4,4]=[4,4,4] | | |
| [63] | rectified square (risquah) = s2{4,4,4} | | | - | | | [4+,4,4] | | |
| [77] | order-4 square (sisquah) ↔ ↔ ↔ | - | - | - | | | [1+,4,4,4]=[4,4,4] | Cube | |
| [78] | truncated order-4 square (tissish) ↔ ↔ ↔ | (4.8.8) | - | (4.8.8) | - | (4.4.4.4) | [1+,4,4,4]=[4,4,4] | | |
| [79] | bitruncated order-4 square (dish) ↔ ↔ ↔ | (4.8.8) | - | - | (4.8.8) | (4.8.8) | [1+,4,4,4]=[4,4,4] | | |
| [81] | runcitruncated order-4 square tiling (prissish) = s2,3{4,4,4} | | | | | | [4,4,4] | | |
| [83] | alternated square( ↔ ) ↔ hr{4,4,4} | | - | - | | | [4,1+,4,4] | (4.3.4.3) | |
| [104] | quarter order-4 square ↔ q{4,4,4} | | | | | | [[1+,4,4,4,1+]]=[[4[4]]] | | |
| 153 | alternated rectified square tiling ↔ ↔ hrr{4,4,4} | | | - | | | [((2+,4,4)),4] | | |
| 154 | alternated runcinated order-4 square tilinght0,3{4,4,4} | | | | | | [[(4,4,4,2+)]] | | |
| Scaliform | snub order-4 square tilings{4,4,4} | | - | - | | | [4+,4,4] | | |
| Nonuniform | runcic snub order-4 square tilings3{4,4,4} | | | | | | [4+,4,4] | | |
| Nonuniform | bisnub order-4 square tiling2s{4,4,4} | | - | - | | | [[4,4+,4]] | | |
| [152] | snub square tiling ↔ sr{4,4,4} | | | - | | | [(4,4)+,4] | | |
| Nonuniform | alternated runcitruncated order-4 square tilinght0,1,3{4,4,4} | | | | | | [((2,4)+,4,4)] | | |
| Nonuniform | omnisnub order-4 square tilinght0,1,2,3{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: , with ↔ .
[(4,4,4,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group: .
Alternated forms
[(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 ↔ .
Alternated forms
[(6,3,3,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group: .
Alternated forms
[(6,3,4,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group:
Alternated forms
[(6,3,5,3)] family
There are 9 forms, generated by ring permutations of the Coxeter group:
Alternated forms
[(6,3,6,3)] family
There are 6 forms, generated by ring permutations of the Coxeter group: .
Alternated forms
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]: ↔ .
Alternated forms
[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]: ↔ .
Alternated forms
[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]: ↔ .
Alternated forms
[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]: ↔ .
Alternated forms
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 ↔ .
[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).
Summary enumerations by family
Linear graphs
Paracompact hyperbolic enumeration
| Group | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs |
|---|
| R ¯ 3 {\displaystyle {\bar {R}}_{3}} [4,4,3] | [4,4,3] | 15 | | | | | | | | | | | | | | [1+,4,1+,4,3+] | (6) | (↔ ) (↔ ) | | |
|---|
| [4,4,3]+ | (1) | |
| N ¯ 3 {\displaystyle {\bar {N}}_{3}} [4,4,4] | [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] | [6,3,3] | 15 | | | | | | | | | | | | | | [1+,6,(3,3)+] | (2) | (↔ ) |
|---|
| [6,3,3]+ | (1) | |
| B V ¯ 3 {\displaystyle {\bar {BV}}_{3}} [6,3,4] | [6,3,4] | 15 | | | | | | | | | | | | | | [1+,6,3+,4,1+] | (6) | (↔ ) (↔ ) | | |
|---|
| [6,3,4]+ | (1) | |
| H V ¯ 3 {\displaystyle {\bar {HV}}_{3}} [6,3,5] | [6,3,5] | 15 | | | | | | | | | | | | | | [1+,6,(3,5)+] | (2) | (↔ ) |
|---|
| [6,3,5]+ | (1) | |
| Y ¯ 3 {\displaystyle {\bar {Y}}_{3}} [3,6,3] | [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] | [6,3,6] | 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 | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs |
|---|
| D V ¯ 3 {\displaystyle {\bar {DV}}_{3}} [6,31,1] | [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}} [3,41,1] | [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}} [41,1,1] | [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 | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs |
|---|
| C R ^ 3 {\displaystyle {\widehat {CR}}_{3}} [(4,4,4,3)] | [(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[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]]]+= [2+[4,4,4]]+ | (1) | |
| A V ^ 3 {\displaystyle {\widehat {AV}}_{3}} [(6,3,3,3)] | [(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)] | [(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)] | [(3,5,3,6)] | 6 | | | | | | | |
| [2+[(3,5,3,6)]] | 3 | | | | [2+[(3,5,3,6)]]+ | (1) | |
| V V ^ 3 {\displaystyle {\widehat {VV}}_{3}} [(3,6)[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 | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs |
|---|
| B R ^ 3 {\displaystyle {\widehat {BR}}_{3}} [(3,3,4,4)] | [(3,3,4,4)] | 4 | | | | | |
|---|
| [1[(4,4,3,3)]]=[3,41,1] ↔ | (7) | | | | | | | | [1[(3,3,4,1+,4)]]+= [3+,41,1]+ | (2) | (= ) |
| [1[(3,3,4,4)]]+= [3,41,1]+ | (1) | |
| D P ¯ 3 {\displaystyle {\bar {DP}}_{3}} [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[ ]]]+=[6,3,4]+ | (1) | |
| P P ¯ 3 {\displaystyle {\bar {PP}}_{3}} [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]]]+= [6,3,3]+ | (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 | Extendedsymmetry | Honeycombs | Chiralextendedsymmetry | Alternation honeycombs |
|---|
| P ¯ 3 {\displaystyle {\bar {P}}_{3}} [3,3[3]] | [3,3[3]] | 4 | | | | | |
| [1[3,3[3]]]=[3,3,6] ↔ | (7) | | | | | | | | [1[3,3[3]]]+= [3,3,6]+ | (1) | |
| B P ¯ 3 {\displaystyle {\bar {BP}}_{3}} [4,3[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}} [5,3[3]] | [5,3[3]] | 4 | | | | | |
| [1[5,3[3]]]=[5,3,6] ↔ | (7) | | | | | | | | [1[5,3[3]]]+= [5,3,6]+ | (1) | |
| V P ¯ 3 {\displaystyle {\bar {VP}}_{3}} [6,3[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]]]+= [3,6,3]+ | (1) | ↔ (= ) |
| [1[6,3[3]]]+= [6,3,6]+ | (1) | |
See also
Notes
- 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, 3rd. ed., Dover Publications, 1973. 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 2002
- C. 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 11,12,13
- N. 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".
References
P. Tumarkin, Hyperbolic Coxeter n-polytopes with n+2 facets (2003) https://arxiv.org/abs/math/0301133.pdf