# Paracompact uniform honeycombs

Example paracompact regular honeycombs :
| {3,3,6} | {6,3,3} | {4,3,6} | {6,3,4} |
| {5,3,6} | {6,3,5} | {6,3,6} | {3,6,3} |
| {4,4,3} | {3,4,4} | {4,4,4} |
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of paracompact uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter diagrams for each family. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity, similar to the hyperbolic uniform tilings in 2-dimensions.

## Regular paracompact honeycombs

Of the uniform paracompact H$^3$ 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} | 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} | [6,3,3] |
| 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} | [4,4,3] |
| 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} | [6,3,4] |
| 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} | [6,3,5] |
| Order-6 hexagonal tiling honeycomb | {6,3,6} | {6,3} | {6} | {6} | {3,6} | Self-dual | [6,3,6] |

## Coxeter groups of paracompact uniform honeycombs

* These graphs show subgroup relations of paracompact hyperbolic Coxeter groups. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry. |
This is a complete enumeration of the 151 unique Wythoffian paracompact uniform honeycombs generated from tetrahedral fundamental domains (rank 4 paracompact coxeter groups). The honeycombs are indexed here for cross-referencing duplicate forms, with brackets around the nonprimary constructions.
The alternations are listed, but are either repeats or don't generate uniform solutions. Single-hole alternations represent a mirror removal operation. If an end-node is removed, another simplex (tetrahedral) family is generated. If a hole has two branches, a Vinberg polytope is generated, although only Vinberg polytope with mirror symmetry are related to the simplex groups, and their uniform honeycombs have not been systematically explored. These nonsimplectic (pyramidal) Coxeter groups are not enumerated on this page, except as special cases of half groups of the tetrahedral ones. 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 |
| [3,4$^{1,1}$] | 0.1526609324 | [3$^+$,4$^{1+,1+}$] | 4 |
| [3,6,3] | 0.1691569344 | [3$^+$,6,3$^+$] | 8 |
| [6,3,5] | 0.1715016613 | [1$^+$,6,(3,5)$^+$] = [5,3$^{[3]}$]$^+$ | 15 |
| [6,3$^{1,1}$] | 0.2114461680 | [1$^+$,6,(3$^{1,1}$)$^+$] = [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 |
| [4$^{1,1,1}$] | 0.4579827971 | [1$^+$,4$^{1+,1+,1+}$] | 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 H$^3$ 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 |
| H$^3$ | 5 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

## Linear graphs
### [6,3,3] family

| # | Honeycomb name Coxeter diagram: Schläfli symbol | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram: Schläfli symbol | 1 | 2 | 3 | 4 | Vertex figure | Picture |
| 1 | hexagonal (hexah) {6,3,3} | - | - | - | (4) (6.6.6) | Tetrahedron | |
| 2 | rectified hexagonal (rihexah) t$_1${6,3,3} or r{6,3,3} | (2) (3.3.3) | - | - | (3) (3.6.3.6) | Triangular prism | |
| 3 | rectified order-6 tetrahedral (rath) t$_1${3,3,6} or r{3,3,6} | (6) (3.3.3.3) | - | - | (2) (3.3.3.3.3.3) | Hexagonal prism | |
| 4 | order-6 tetrahedral (thon) {3,3,6} | (∞) (3.3.3) | - | - | - | Triangular tiling | |
| 5 | truncated hexagonal (thexah) t$_{0,1}${6,3,3} or t{6,3,3} | (1) (3.3.3) | - | - | (3) (3.12.12) | Triangular pyramid | |
| 6 | cantellated hexagonal (srihexah) t$_{0,2}${6,3,3} or rr{6,3,3} | (1) 3.3.3.3 | (2) (4.4.3) | - | (2) (3.4.6.4) | | |
| 7 | runcinated hexagonal (sidpithexah) t$_{0,3}${6,3,3} | (1) (3.3.3) | (3) (4.4.3) | (3) (4.4.6) | (1) (6.6.6) | | |
| 8 | cantellated order-6 tetrahedral (srath) t$_{0,2}${3,3,6} or rr{3,3,6} | (1) (3.4.3.4) | - | (2) (4.4.6) | (2) (3.6.3.6) | | |
| 9 | bitruncated hexagonal (tehexah) t$_{1,2}${6,3,3} or 2t{6,3,3} | (2) (3.6.6) | - | - | (2) (6.6.6) | | |
| 10 | truncated order-6 tetrahedral (tath) t$_{0,1}${3,3,6} or t{3,3,6} | (6) (3.6.6) | - | - | (1) (3.3.3.3.3.3) | | |
| 11 | cantitruncated hexagonal (grihexah) t$_{0,1,2}${6,3,3} or tr{6,3,3} | (1) (3.6.6) | (1) (4.4.3) | - | (2) (4.6.12) | | |
| 12 | runcitruncated hexagonal (prath) t$_{0,1,3}${6,3,3} | (1) (3.4.3.4) | (2) (4.4.3) | (1) (4.4.12) | (1) (3.12.12) | | |
| 13 | runcitruncated order-6 tetrahedral (prihexah) t$_{0,1,3}${3,3,6} | (1) (3.6.6) | (1) (4.4.6) | (2) (4.4.6) | (1) (3.4.6.4) | | |
| 14 | cantitruncated order-6 tetrahedral (grath) t$_{0,1,2}${3,3,6} or tr{3,3,6} | (2) (4.6.6) | - | (1) (4.4.6) | (1) (6.6.6) | | |
| 15 | omnitruncated hexagonal (gidpithexah) t$_{0,1,2,3}${6,3,3} | (1) (4.6.6) | (1) (4.4.6) | (1) (4.4.12) | (1) (4.6.12) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram: Schläfli symbol | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram: Schläfli symbol | 1 | 2 | 3 | 4 | Alt | Vertex figure | Picture |
| [137] | alternated hexagonal (ahexah) ( ↔ ) = | - | - | | (4) (3.3.3.3.3.3) | (4) (3.3.3) | (3.6.6) | |
| [138] | cantic hexagonal (tahexah) ↔ | (1) (3.3.3.3) | - | | (2) (3.6.3.6) | (2) (3.6.6) | | |
| [139] | runcic hexagonal (birahexah) ↔ | (1) (4.4.4) | (1) (4.4.3) | | (1) (3.3.3.3.3.3) | (3) (3.4.3.4) | | |
| [140] | runcicantic hexagonal (bitahexah) ↔ | (1) (3.6.6) | (1) (4.4.3) | | (1) (3.6.3.6) | (2) (4.6.6) | | |
| Nonuniform | snub rectified order-6 tetrahedral ↔ sr{3,3,6} | | | | | Irr. (3.3.3) | | |
| Nonuniform | cantic snub order-6 tetrahedral sr$_3${3,3,6} | | | | | | | |
| Nonuniform | omnisnub order-6 tetrahedral ht$_{0,1,2,3}${6,3,3} | | | | | Irr. (3.3.3) | | |

### [6,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group: [6,3,4] or
| # | Name of honeycomb Coxeter diagram Schläfli symbol | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Vertex figure | Picture |
| # | Name of honeycomb Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 16 | (Regular) order-4 hexagonal (shexah) {6,3,4} | - | - | - | (8) (6.6.6) | (3.3.3.3) | |
| 17 | rectified order-4 hexagonal (rishexah) t$_1${6,3,4} or r{6,3,4} | (2) (3.3.3.3) | - | - | (4) (3.6.3.6) | (4.4.4) | |
| 18 | rectified order-6 cubic (rihach) t$_1${4,3,6} or r{4,3,6} | (6) (3.4.3.4) | - | - | (2) (3.3.3.3.3.3) | (6.4.4) | |
| 19 | order-6 cubic (hachon) {4,3,6} | (20) (4.4.4) | - | - | - | (3.3.3.3.3.3) | |
| 20 | truncated order-4 hexagonal (tishexah) t$_{0,1}${6,3,4} or t{6,3,4} | (1) (3.3.3.3) | - | - | (4) (3.12.12) | | |
| 21 | bitruncated order-6 cubic (chexah) t$_{1,2}${6,3,4} or 2t{6,3,4} | (2) (4.6.6) | - | - | (2) (6.6.6) | | |
| 22 | truncated order-6 cubic (thach) t$_{0,1}${4,3,6} or t{4,3,6} | (6) (3.8.8) | - | - | (1) (3.3.3.3.3.3) | | |
| 23 | cantellated order-4 hexagonal (srishexah) t$_{0,2}${6,3,4} or rr{6,3,4} | (1) (3.4.3.4) | (2) (4.4.4) | - | (2) (3.4.6.4) | | |
| 24 | cantellated order-6 cubic (srihach) t$_{0,2}${4,3,6} or rr{4,3,6} | (2) (3.4.4.4) | - | (2) (4.4.6) | (1) (3.6.3.6) | | |
| 25 | runcinated order-6 cubic (sidpichexah) t$_{0,3}${6,3,4} | (1) (4.4.4) | (3) (4.4.4) | (3) (4.4.6) | (1) (6.6.6) | | |
| 26 | cantitruncated order-4 hexagonal (grishexah) t$_{0,1,2}${6,3,4} or tr{6,3,4} | (1) (4.6.6) | (1) (4.4.4) | - | (2) (4.6.12) | | |
| 27 | cantitruncated order-6 cubic (grihach) t$_{0,1,2}${4,3,6} or tr{4,3,6} | (2) (4.6.8) | - | (1) (4.4.6) | (1) (6.6.6) | | |
| 28 | runcitruncated order-4 hexagonal (prihach) t$_{0,1,3}${6,3,4} | (1) (3.4.4.4) | (1) (4.4.4) | (2) (4.4.12) | (1) (3.12.12) | | |
| 29 | runcitruncated order-6 cubic (prishexah) t$_{0,1,3}${4,3,6} | (1) (3.8.8) | (2) (4.4.8) | (1) (4.4.6) | (1) (3.4.6.4) | | |
| 30 | omnitruncated order-6 cubic (gidpichexah) t$_{0,1,2,3}${6,3,4} | (1) (4.6.8) | (1) (4.4.8) | (1) (4.4.12) | (1) (4.6.12) | | |
Alternated forms :
| # | Name of honeycomb Coxeter diagram Schläfli symbol | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Vertex figure | Picture |
| # | Name of honeycomb Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [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) ↔ h$_2${4,3,6} | (2) (3.6.6) | - | - | (1) (3.6.3.6) | (2) (6.6.6) | | |
| [89] | runcic order-6 cubic (birachach) ↔ h$_3${4,3,6} | (1) (3.3.3) | - | - | (1) (6.6.6) | (3) (3.4.6.4) | | |
| [90] | runcicantic order-6 cubic (bitachach) ↔ h$_{2,3}${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) ↔ ↔ h$_1${6,3,4} | (1) (3.4.3.4) | - | | (2) (3.6.3.6) | (2) (4.6.6) | | |
| [143] | runcic order-4 hexagonal (birashexah) ↔ h$_3${6,3,4} | (1) (4.4.4) | (1) (4.4.3) | | (1) (3.3.3.3.3.3) | (3) (3.4.4.4) | | |
| [144] | runcicantic order-4 hexagonal (bitashexah) ↔ h$_{2,3}${6,3,4} | (1) (3.8.8) | (1) (4.4.3) | | (1) (3.6.3.6) | (2) (4.6.8) | | |
| [151] | quarter order-4 hexagonal (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 cubic sr$_3${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 hexagonal sr$_3${6,3,4} | | | | | | | |
| Nonuniform | omnisnub rectified order-6 cubic ht$_{0,1,2,3}${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 name Coxeter diagram Schläfli symbol | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 31 | order-5 hexagonal (phexah) {6,3,5} | - | - | - | (20) (6)$^3$ | Icosahedron | |
| 32 | rectified order-5 hexagonal (riphexah) t$_1${6,3,5} or r{6,3,5} | (2) (3.3.3.3.3) | - | - | (5) (3.6)$^2$ | (5.4.4) | |
| 33 | rectified order-6 dodecahedral (rihed) t$_1${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) t$_{0,1}${6,3,5} or t{6,3,5} | (1) (3.3.3.3.3) | - | - | (5) 3.12.12 | | |
| 36 | cantellated order-5 hexagonal (sriphexah) t$_{0,2}${6,3,5} or rr{6,3,5} | (1) (3.5.3.5) | (2) (5.4.4) | - | (2) 3.4.6.4 | | |
| 37 | runcinated order-6 dodecahedral (sidpidohexah) t$_{0,3}${6,3,5} | (1) (5.5.5) | - | (6) (6.4.4) | (1) (6)$^3$ | | |
| 38 | cantellated order-6 dodecahedral (srihed) t$_{0,2}${5,3,6} or rr{5,3,6} | (2) (4.3.4.5) | - | (2) (6.4.4) | (1) (3.6)$^2$ | | |
| 39 | bitruncated order-6 dodecahedral (dohexah) t$_{1,2}${6,3,5} or 2t{6,3,5} | (2) (5.6.6) | - | - | (2) (6)$^3$ | | |
| 40 | truncated order-6 dodecahedral (thed) t$_{0,1}${5,3,6} or t{5,3,6} | (6) (3.10.10) | - | - | (1) (3)$^6$ | | |
| 41 | cantitruncated order-5 hexagonal (griphexah) t$_{0,1,2}${6,3,5} or tr{6,3,5} | (1) (5.6.6) | (1) (5.4.4) | - | (2) 4.6.10 | | |
| 42 | runcitruncated order-5 hexagonal (prihed) t$_{0,1,3}${6,3,5} | (1) (4.3.4.5) | (1) (5.4.4) | (2) (12.4.4) | (1) 3.12.12 | | |
| 43 | runcitruncated order-6 dodecahedral (priphaxh) t$_{0,1,3}${5,3,6} | (1) (3.10.10) | (1) (10.4.4) | (2) (6.4.4) | (1) 3.4.6.4 | | |
| 44 | cantitruncated order-6 dodecahedral (grihed) t$_{0,1,2}${5,3,6} or tr{5,3,6} | (1) (4.6.10) | - | (2) (6.4.4) | (1) (6)$^3$ | | |
| 45 | omnitruncated order-6 dodecahedral (gidpidohaxh) t$_{0,1,2,3}${6,3,5} | (1) (4.6.10) | (1) (10.4.4) | (1) (12.4.4) | (1) 4.6.12 | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram Schläfli symbol | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [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) ↔ h$_2${6,3,5} | (1) (3.5.3.5) | - | | (2) (3.6.3.6) | (2) (5.6.6) | | |
| [147] | runcic order-5 hexagonal (biraphexah) ↔ h$_3${6,3,5} | (1) (5.5.5) | (1) (4.4.3) | | (1) (3.3.3.3.3.3) | (3) (3.4.5.4) | | |
| [148] | runcicantic order-5 hexagonal (bitaphexah) ↔ h$_{2,3}${6,3,5} | (1) (3.10.10) | (1) (4.4.3) | | (1) (3.6.3.6) | (2) (4.6.10) | | |
| Nonuniform | snub rectified order-6 dodecahedral ↔ 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 hexagonal ht$_{0,1,2,3}${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 honeycomb Coxeter diagram Schläfli symbol | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Vertex figure | Picture |
| # | Name of honeycomb Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 46 | order-6 hexagonal (hihexah) {6,3,6} | - | - | - | (20) (6.6.6) | (3.3.3.3.3.3) | |
| 47 | rectified order-6 hexagonal (rihihexah) t$_1${6,3,6} or r{6,3,6} | (2) (3.3.3.3.3.3) | - | - | (6) (3.6.3.6) | (6.4.4) | |
| 48 | truncated order-6 hexagonal (thihexah) t$_{0,1}${6,3,6} or t{6,3,6} | (1) (3.3.3.3.3.3) | - | - | (6) (3.12.12) | | |
| 49 | cantellated order-6 hexagonal (srihihexah) t$_{0,2}${6,3,6} or rr{6,3,6} | (1) (3.6.3.6) | (2) (4.4.6) | - | (2) (3.6.4.6) | | |
| 50 | Runcinated order-6 hexagonal (spiddihexah) t$_{0,3}${6,3,6} | (1) (6.6.6) | (3) (4.4.6) | (3) (4.4.6) | (1) (6.6.6) | | |
| 51 | cantitruncated order-6 hexagonal (grihihexah) t$_{0,1,2}${6,3,6} or tr{6,3,6} | (1) (6.6.6) | (1) (4.4.6) | - | (2) (4.6.12) | | |
| 52 | runcitruncated order-6 hexagonal (prihihexah) t$_{0,1,3}${6,3,6} | (1) (3.6.4.6) | (1) (4.4.6) | (2) (4.4.12) | (1) (3.12.12) | | |
| 53 | omnitruncated order-6 hexagonal (gidpiddihexah) t$_{0,1,2,3}${6,3,6} | (1) (4.6.12) | (1) (4.4.12) | (1) (4.4.12) | (1) (4.6.12) | | |
| [1] | bitruncated order-6 hexagonal (hexah) ↔ ↔ t$_{1,2}${6,3,6} or 2t{6,3,6} | (2) (6.6.6) | - | - | (2) (6.6.6) | | |
Alternated forms :
| # | Name of honeycomb Coxeter diagram Schläfli symbol | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Cells by location and count per vertex | Vertex figure | Picture |
| # | Name of honeycomb Coxeter diagram Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [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) ( ↔ ) = h$_2${6,3,6} = r{3,6,3} | (1) (3.6.3.6) | - | (2) (6.6.6) | (2) (3.6.3.6) | | | |
| [149] | runcic order-6 hexagonal ↔ h$_3${6,3,6} | (1) (6.6.6) | (1) (4.4.3) | (3) (3.4.6.4) | (1) (3.3.3.3.3.3) | | | |
| [150] | runcicantic order-6 hexagonal ↔ h$_{2,3}${6,3,6} | (1) (3.12.12) | (1) (4.4.3) | (2) (4.6.12) | (1) (3.6.3.6) | | | |
| [137] | alternated hexagonal (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 hexagonal sr{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 hexagonal ht$_{0,3}${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 hexagonal ht$_{0,1,2,3}${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 name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 54 | triangular (trah) {3,6,3} | - | - | - | (∞) {3,6} | {6,3} | |
| 55 | rectified triangular (ritrah) t$_1${3,6,3} or r{3,6,3} | (2) (6)$^3$ | - | - | (3) (3.6)$^2$ | (3.4.4) | |
| 56 | cantellated triangular (sritrah) t$_{0,2}${3,6,3} or rr{3,6,3} | (1) (3.6)$^2$ | (2) (4.4.3) | - | (2) (3.6.4.6) | | |
| 57 | runcinated triangular (spidditrah) t$_{0,3}${3,6,3} | (1) (3)$^6$ | (6) (4.4.3) | (6) (4.4.3) | (1) (3)$^6$ | | |
| 58 | bitruncated triangular (ditrah) t$_{1,2}${3,6,3} or 2t{3,6,3} | (2) (3.12.12) | - | - | (2) (3.12.12) | | |
| 59 | cantitruncated triangular (gritrah) t$_{0,1,2}${3,6,3} or tr{3,6,3} | (1) (3.12.12) | (1) (4.4.3) | - | (2) (4.6.12) | | |
| 60 | runcitruncated triangular (pritrah) t$_{0,1,3}${3,6,3} | (1) (3.6.4.6) | (1) (4.4.3) | (2) (4.4.6) | (1) (6)$^3$ | | |
| 61 | omnitruncated triangular (gipidditrah) t$_{0,1,2,3}${3,6,3} | (1) (4.6.12) | (1) (4.4.6) | (1) (4.4.6) | (1) (4.6.12) | | |
| [1] | truncated triangular (hexah) ↔ ↔ t$_{0,1}${3,6,3} or t{3,6,3} = {6,3,3} | (1) (6)$^3$ | - | - | (3) (6)$^3$ | {3,3} | |
Alternated forms :
| # | Honeycomb name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [56] | cantellated triangular (sritrah) = s$_2${3,6,3} | (1) (3.6)$^2$ | - | - | (2) (3.6.4.6) | (3.4.4) | | |
| [60] | runcitruncated triangular (pritrah) = s$_{2,3}${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) s$_3${3,6,3} | r{6,3} | - | (3.4.4) | (3)$^6$ | tricup | | |
| Nonuniform | omnisnub triangular tiling honeycomb (snatrah) ht$_{0,1,2,3}${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 name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 62 | square (squah) = {4,4,3} | - | - | - | (6) | Cube | |
| 63 | rectified square (risquah) = t$_1${4,4,3} or r{4,4,3} | (2) | - | - | (3) | Triangular prism | |
| 64 | rectified order-4 octahedral (rocth) t$_1${3,4,4} or r{3,4,4} | (4) | - | - | (2) | | |
| 65 | order-4 octahedral (octh) {3,4,4} | (∞) | - | - | - | | |
| 66 | truncated square (tisquah) = t$_{0,1}${4,4,3} or t{4,4,3} | (1) | - | - | (3) | | |
| 67 | truncated order-4 octahedral (tocth) t$_{0,1}${3,4,4} or t{3,4,4} | (4) | - | - | (1) | | |
| 68 | bitruncated square (osquah) t$_{1,2}${4,4,3} or 2t{4,4,3} | (2) | - | - | (2) | | |
| 69 | cantellated square (srisquah) t$_{0,2}${4,4,3} or rr{4,4,3} | (1) | (2) | - | (2) | | |
| 70 | cantellated order-4 octahedral (srocth) t$_{0,2}${3,4,4} or rr{3,4,4} | (2) | - | (2) | (1) | | |
| 71 | runcinated square (sidposquah) t$_{0,3}${4,4,3} | (1) | (3) | (3) | (1) | | |
| 72 | cantitruncated square (grisquah) t$_{0,1,2}${4,4,3} or tr{4,4,3} | (1) | (1) | - | (2) | | |
| 73 | cantitruncated order-4 octahedral (grocth) t$_{0,1,2}${3,4,4} or tr{3,4,4} | (2) | - | (1) | (1) | | |
| 74 | runcitruncated square (procth) t$_{0,1,3}${4,4,3} | (1) | (1) | (2) | (1) | | |
| 75 | runcitruncated order-4 octahedral (prisquah) t$_{0,1,3}${3,4,4} | (1) | (2) | (1) | (1) | | |
| 76 | omnitruncated square (gidposquah) t$_{0,1,2,3}${4,4,3} | (1) | (1) | (1) | (1) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [83] | alternated square ↔ h{4,4,3} | - | - | - | (6) | (8) | | |
| [84] | cantic square ↔ h$_2${4,4,3} | (1) | - | - | (2) | (2) | | |
| [85] | runcic square ↔ h$_3${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 octahedral s$_3${3,4,4} | | | | | cup-4 | | |
| 152 | snub square = s{4,4,3} | | - | - | | {3,3} | | |
| Nonuniform | snub rectified order-4 octahedral sr{3,4,4} | | - | | | irr. {3,3} | | |
| Nonuniform | alternated runcitruncated square ht$_{0,1,3}${3,4,4} | | | | | irr.v{4} | | |
| Nonuniform | omnisnub square ht$_{0,1,2,3}${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 name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Symmetry | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Symmetry | Vertex figure | Picture |
| 77 | order-4 square (sisquah) {4,4,4} | - | - | - | | [4,4,4] | Cube | |
| 78 | truncated order-4 square (tissish) t$_{0,1}${4,4,4} or t{4,4,4} | | - | - | | [4,4,4] | | |
| 79 | bitruncated order-4 square (dish) t$_{1,2}${4,4,4} or 2t{4,4,4} | | - | - | | [[4,4,4]] | | |
| 80 | runcinated order-4 square (spiddish) t$_{0,3}${4,4,4} | | | | | [[4,4,4]] | | |
| 81 | runcitruncated order-4 square (prissish) t$_{0,1,3}${4,4,4} | | | | | [4,4,4] | | |
| 82 | omnitruncated order-4 square (gipiddish) t$_{0,1,2,3}${4,4,4} | | | | | [[4,4,4]] | | |
| [62] | square (squah) ↔ t$_1${4,4,4} or r{4,4,4} | | - | - | | [4,4,4] | Square tiling | |
| [63] | rectified square (risquah) ↔ t$_{0,2}${4,4,4} or rr{4,4,4} | | | - | | [4,4,4] | | |
| [66] | truncated order-4 square (tisquah) ↔ t$_{0,1,2}${4,4,4} or tr{4,4,4} | | | - | | [4,4,4] | | |
Alternated constructions :
| # | Honeycomb name Coxeter diagram and Schläfli symbol | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Cell counts/vertex and positions in honeycomb | Symmetry | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram and Schläfli symbol | 0 | 1 | 2 | 3 | Alt | Symmetry | Vertex figure | Picture |
| [62] | Square (squah) ( ↔ ↔ ↔ ) = | (4.4.4.4) | - | - | | (4.4.4.4) | [1$^+$,4,4,4] =[4,4,4] | | |
| [63] | rectified square (risquah) = s$_2${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) = s$_{2,3}${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 tiling ht$_{0,3}${4,4,4} | | | | | | [[(4,4,4,2$^+$)]] | | |
| Scaliform | snub order-4 square tiling s{4,4,4} | | - | - | | | [4$^+$,4,4] | | |
| Nonuniform | runcic snub order-4 square tiling s$_3${4,4,4} | | | | | | [4$^+$,4,4] | | |
| Nonuniform | bisnub order-4 square tiling 2s{4,4,4} | | - | - | | | [[4,4$^+$,4]] | | |
| [152] | snub square tiling ↔ sr{4,4,4} | | | - | | | [(4,4)$^+$,4] | | |
| Nonuniform | alternated runcitruncated order-4 square tiling ht$_{0,1,3}${4,4,4} | | | | | | [((2,4)$^+$,4,4)] | | |
| Nonuniform | omnisnub order-4 square tiling ht$_{0,1,2,3}${4,4,4} | | | | | | [[4,4,4]]$^+$ | | |

## Tridental graphs
### [3,4$^{1,1}$] 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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Vertex figure | Picture |
| 83 | alternated square ↔ | - | - | (4.4.4) | (4.4.4.4) | (4.3.4.3) | |
| 84 | cantic square ↔ | (3.4.3.4) | - | (3.8.8) | (4.8.8) | | |
| 85 | runcic square ↔ | (4.4.4.4) | - | (3.4.4.4) | (4.4.4.4) | | |
| 86 | runcicantic square ↔ | (4.6.6) | - | (3.4.4.4) | (4.8.8) | | |
| [63] | rectified square (risquah) ↔ | (4.4.4) | - | (4.4.4) | (4.4.4.4) | | |
| [64] | rectified order-4 octahedral (rocth) ↔ | (3.4.3.4) | - | (3.4.3.4) | (4.4.4.4) | | |
| [65] | order-4 octahedral (octh) ↔ | (4.4.4.4) | - | (4.4.4.4) | - | | |
| [67] | truncated order-4 octahedral (tocth) ↔ | (4.6.6) | - | (4.6.6) | (4.4.4.4) | | |
| [68] | bitruncated square (osquah) ↔ | (3.8.8) | - | (3.8.8) | (4.8.8) | | |
| [70] | cantellated order-4 octahedral (srocth) ↔ | (3.4.4.4) | (4.4.4) | (3.4.4.4) | (4.4.4.4) | | |
| [73] | cantitruncated order-4 octahedral (grocth) ↔ | (4.6.8) | (4.4.4) | (4.6.8) | (4.8.8) | | |
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Alt | Vertex figure | Picture |
| Scaliform | snub order-4 octahedral = = s{3,4$^{1,1}$} | | - | - | | irr.v{4} | | |
| Nonuniform | snub rectified order-4 octahedral ↔ sr{3,4$^{1,1}$} | (3.3.3.3.4) | (3.3.3) | (3.3.3.3.4) | (3.3.4.3.4) | +(3.3.3) | | |

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

### [6,3$^{1,1}$] family

There are 11 forms (and only 4 not shared with [6,3,4] family), generated by ring permutations of the Coxeter group: [6,3$^{1,1}$] or .
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Vertex figure | Picture |
| 87 | alternated order-6 cubic (ahach) ↔ | - | - | (∞) (3.3.3.3.3) | (∞) (3.3.3) | (3.6.3.6) | |
| 88 | cantic order-6 cubic (tachach) ↔ | (1) (3.6.3.6) | - | (2) (6.6.6) | (2) (3.6.6) | | |
| 89 | runcic order-6 cubic (birachach) ↔ | (1) (6.6.6) | - | (3) (3.4.6.4) | (1) (3.3.3) | | |
| 90 | runcicantic order-6 cubic (bitachach) ↔ | (1) (3.12.12) | - | (2) (4.6.12) | (1) (3.6.6) | | |
| [16] | order-4 hexagonal (shexah) ↔ | (4) (6.6.6) | - | (4) (6.6.6) | - | (3.3.3.3) | |
| [17] | rectified order-4 hexagonal (rishexah) ↔ | (2) (3.6.3.6) | - | (2) (3.6.3.6) | (2) (3.3.3.3) | | |
| [18] | rectified order-6 cubic (rihach) ↔ | (1) (3.3.3.3.3) | - | (1) (3.3.3.3.3) | (6) (3.4.3.4) | | |
| [20] | truncated order-4 hexagonal (tishexah) ↔ | (2) (3.12.12) | - | (2) (3.12.12) | (1) (3.3.3.3) | | |
| [21] | bitruncated order-6 cubic (chexah) ↔ | (1) (6.6.6) | - | (1) (6.6.6) | (2) (4.6.6) | | |
| [24] | cantellated order-6 cubic (srihach) ↔ | (1) (3.4.6.4) | (2) (4.4.4) | (1) (3.4.6.4) | (1) (3.4.3.4) | | |
| [27] | cantitruncated order-6 cubic (grihach) ↔ | (1) (4.6.12) | (1) (4.4.4) | (1) (4.6.12) | (1) (4.6.6) | | |
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Alt | Vertex figure | Picture |
| [141] | alternated order-4 hexagonal (ashexah) ↔ ↔ ↔ | | | | | | (4.6.6) | |
| Nonuniform | bisnub order-4 hexagonal ↔ | | | | | | | |
| Nonuniform | snub rectified order-4 hexagonal ↔ | (3.3.3.3.6) | (3.3.3) | (3.3.3.3.6) | (3.3.3.3.3) | +(3.3.3) | | |

## Cyclic graphs
### [(4,4,3,3)] family

There are 11 forms, 4 unique to this family, generated by ring permutations of the Coxeter group: , with ↔ .
| # | Honeycomb name Coxeter diagram | Cells by location | Cells by location | Cells by location | Cells by location | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 91 | tetrahedral-square | - | (6) (444) | (8) (333) | (12) (3434) | (3444) | |
| 92 | cyclotruncated square-tetrahedral | (444) | (488) | (333) | (388) | | |
| 93 | cyclotruncated tetrahedral-square | (1) (3333) | (1) (444) | (4) (366) | (4) (466) | | |
| 94 | truncated tetrahedral-square | (1) (3444) | (1) (488) | (1) (366) | (2) (468) | | |
| [64] | ( ↔ ) = rectified order-4 octahedral (rocth) | (3434) | (4444) | (3434) | (3434) | | |
| [65] | ( ↔ ) = order-4 octahedral (octh) | (3333) | - | (3333) | (3333) | | |
| [67] | ( ↔ ) = truncated order-4 octahedral (tocth) | (466) | (4444) | (3434) | (466) | | |
| [83] | alternated square ( ↔ ) = | (444) | (4444) | - | (444) | (4.3.4.3) | |
| [84] | cantic square ( ↔ ) = | (388) | (488) | (3434) | (388) | | |
| [85] | runcic square ( ↔ ) = | (3444) | (3434) | (3333) | (3444) | | |
| [86] | runcicantic square ( ↔ ) = | (468) | (488) | (466) | (468) | | |
| # | Honeycomb name Coxeter diagram | Cells by location | Cells by location | Cells by location | Cells by location | Cells by location | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| 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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 95 | cubic-square | (8) (4.4.4) | - | (6) (4.4.4.4) | (12) (4.4.4.4) | (3.4.4.4) | |
| 96 | octahedral-square | (3.4.3.4) | (3.3.3.3) | - | (4.4.4.4) | (4.4.4.4) | |
| 97 | cyclotruncated cubic-square | (4) (3.8.8) | (1) (3.3.3.3) | (1) (4.4.4.4) | (4) (4.8.8) | | |
| 98 | cyclotruncated square-cubic | (1) (4.4.4) | (1) (4.4.4) | (3) (4.8.8) | (3) (4.8.8) | | |
| 99 | cyclotruncated octahedral-square | (4) (4.6.6) | (4) (4.6.6) | (1) (4.4.4.4) | (1) (4.4.4.4) | | |
| 100 | rectified cubic-square | (1) (3.4.3.4) | (2) (3.4.4.4) | (1) (4.4.4.4) | (2) (4.4.4.4) | | |
| 101 | truncated cubic-square | (1) (4.8.8) | (1) (3.4.4.4) | (2) (4.8.8) | (1) (4.8.8) | | |
| 102 | truncated octahedral-square | (2) (4.6.8 | (1) (4.6.6) | (1) (4.4.4.4) | (1) (4.8.8) | | |
| 103 | omnitruncated octahedral-square | (1) (4.6.8) | (1) (4.6.8) | (1) (4.8.8) | (1) (4.8.8) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure |
| 156 | alternated cubic-square ↔ | - | | | | (3.4.4.4) | |
| Nonuniform | snub octahedral-square | | | | | | |
| Nonuniform | cyclosnub square-cubic | | | | | | |
| Nonuniform | cyclosnub octahedral-square | | | | | | |
| Nonuniform | omnisnub cubic-square | (3.3.3.3.4) | (3.3.3.3.4) | (3.3.4.3.4) | (3.3.4.3.4) | +(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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 104 | quarter order-4 square ↔ | (4.8.8) | (4.4.4.4) | (4.4.4.4) | (4.8.8) | | |
| [62] | square (squah) ↔ ↔ | (4.4.4.4) | (4.4.4.4) | (4.4.4.4) | (4.4.4.4) | | |
| [77] | order-4 square (sisquah) ( ↔ ) = | (4.4.4.4) | - | (4.4.4.4) | (4.4.4.4) | (4.4.4.4) | |
| [78] | truncated order-4 square (tissish) ( ↔ ) = | (4.8.8) | (4.4.4.4) | (4.8.8) | (4.8.8) | | |
| [79] | bitruncated order-4 square (dish) ↔ | (4.8.8) | (4.8.8) | (4.8.8) | (4.8.8) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure |
| [83] | alternated square ( ↔ ↔ ) = | (6) (4.4.4.4) | (6) (4.4.4.4) | (6) (4.4.4.4) | (6) (4.4.4.4) | (8) (4.4.4) | (4.3.4.3) |
| [77] | alternated order-4 square (sisquah) ↔ | | - | | | | |
| 158 | cantic order-4 square ↔ | | | | | | |
| Nonuniform | cyclosnub square | | | | | | |
| Nonuniform | snub order-4 square | | | | | | |
| Nonuniform | bisnub order-4 square ↔ | (3.3.4.3.4) | (3.3.4.3.4) | (3.3.4.3.4) | (3.3.4.3.4) | +(3.3.3) | |

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

There are 9 forms, generated by ring permutations of the Coxeter group: .
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure |
| 105 | tetrahedral-hexagonal | (4) (3.3.3) | - | (4) (6.6.6) | (6) (3.6.3.6) | (3.4.3.4) |
| 106 | tetrahedral-triangular | (3.3.3.3) | (3.3.3) | - | (3.3.3.3.3.3) | (3.4.6.4) |
| 107 | cyclotruncated tetrahedral-hexagonal | (3) (3.6.6) | (1) (3.3.3) | (1) (6.6.6) | (3) (6.6.6) | |
| 108 | cyclotruncated hexagonal-tetrahedral | (1) (3.3.3) | (1) (3.3.3) | (4) (3.12.12) | (4) (3.12.12) | |
| 109 | cyclotruncated tetrahedral-triangular | (6) (3.6.6) | (6) (3.6.6) | (1) (3.3.3.3.3.3) | (1) (3.3.3.3.3.3) | |
| 110 | rectified tetrahedral-hexagonal | (1) (3.3.3.3) | (2) (3.4.3.4) | (1) (3.6.3.6) | (2) (3.4.6.4) | |
| 111 | truncated tetrahedral-hexagonal | (1) (3.6.6) | (1) (3.4.3.4) | (1) (3.12.12) | (2) (4.6.12) | |
| 112 | truncated tetrahedral-triangular | (2) (4.6.6) | (1) (3.6.6) | (1) (3.4.6.4) | (1) (6.6.6) | |
| 113 | omnitruncated tetrahedral-hexagonal | (1) (4.6.6) | (1) (4.6.6) | (1) (4.6.12) | (1) (4.6.12) | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure |
| Nonuniform | omnisnub tetrahedral-hexagonal | (3.3.3.3.3) | (3.3.3.3.3) | (3.3.3.3.6) | (3.3.3.3.6) | +(3.3.3) | |

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

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

### [(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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 132 | hexagonal-triangular | (3.3.3.3.3.3) | - | (6.6.6) | (3.6.3.6) | (3.4.6.4) | |
| 133 | cyclotruncated hexagonal-triangular | (1) (3.3.3.3.3.3) | (1) (3.3.3.3.3.3) | (3) (3.12.12) | (3) (3.12.12) | | |
| 134 | cyclotruncated triangular-hexagonal | (1) (3.6.3.6) | (2) (3.4.6.4) | (1) (3.6.3.6) | (2) (3.4.6.4) | | |
| 135 | rectified hexagonal-triangular | (1) (6.6.6) | (1) (3.4.6.4) | (1) (3.12.12) | (2) (4.6.12) | | |
| 136 | truncated hexagonal-triangular | (1) (4.6.12) | (1) (4.6.12) | (1) (4.6.12) | (1) (4.6.12) | | |
| [16] | order-4 hexagonal tiling (shexah) = | (3) (6.6.6) | (1) (6.6.6) | (1) (6.6.6) | (3) (6.6.6) | (3.3.3.3) | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| [141] | alternated order-4 hexagonal (ashexah) ↔ ↔ ↔ | (3.3.3.3.3.3) | (3.3.3.3.3.3) | (3.3.3.3.3.3) | (3.3.3.3.3.3) | +(3.3.3.3) | (4.6.6) | |
| Nonuniform | cyclocantisnub hexagonal-triangular | | | | | | | |
| Nonuniform | cycloruncicantisnub hexagonal-triangular | | | | | | | |
| Nonuniform | snub rectified hexagonal-triangular | (3.3.3.3.6) | (3.3.3.3.6) | (3.3.3.3.6) | (3.3.3.3.6) | +(3.3.3) | | |

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

### [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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | vertex figure | Picture |
| 141 | alternated order-4 hexagonal (ashexah) ↔ | - | - | (3.3.3.3) | (3.3.3.3.3.3) | (4.6.6) | |
| 142 | cantic order-4 hexagonal (tashexah) ↔ ↔ | (1) (3.4.3.4) | - | (2) (4.6.6) | (2) (3.6.3.6) | | |
| 143 | runcic order-4 hexagonal (birashexah) ↔ | (1) (4.4.4) | (1) (4.4.3) | (3) (3.4.4.4) | (1) (3.3.3.3.3.3) | | |
| 144 | runcicantic order-4 hexagonal (bitashexah) ↔ | (1) (3.8.8) | (1) (4.4.3) | (2) (4.6.8) | (1) (3.6.3.6) | | |
| [16] | order-4 hexagonal (shexah) ↔ | (4) (4.4.4) | - | (4) (4.4.4) | - | | |
| [17] | rectified order-4 hexagonal (rishexah) ↔ | (1) (3.3.3.3) | - | (1) (3.3.3.3) | (6) (3.6.3.6) | | |
| [18] | rectified order-6 cubic (rihach) ↔ | (2) (3.4.3.4) | - | (2) (3.4.3.4) | (2) (3.3.3.3.3.3) | | |
| [21] | bitruncated order-4 hexagonal (chexah) ↔ | (1) (4.6.6) | - | (1) (4.6.6) | (2) (6.6.6) | | |
| [22] | truncated order-6 cubic (thach) ↔ | (2) (3.8.8) | - | (2) (3.8.8) | (1) (3.6.3.6) | | |
| [23] | cantellated order-4 hexagonal (srishexah) ↔ | (1) (3.4.4.4) | (2) (4.4.6) | (1) (3.4.4.4) | (1) (3.6.3.6) | | |
| [26] | cantitruncated order-4 hexagonal (grishexah) ↔ | (1) (4.6.8) | (1) (4.4.6) | (1) (4.6.8) | (1) (6.6.6) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Alt | vertex figure |
| Nonuniform | snub rectified order-4 hexagonal ↔ | (3.3.3.3.4) | (3.3.3.3) | (3.3.3.3.4) | (3.3.3.3.3.3) | +(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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | vertex figure | Picture |
| 145 | alternated order-5 hexagonal (aphexah) ↔ | - | - | (3.3.3.3.3) | (3.3.3.3.3.3) | (3.6.3.6) | |
| 146 | cantic order-5 hexagonal (taphexah) ↔ | (1) (3.5.3.5) | - | (2) (5.6.6) | (2) (3.6.3.6) | | |
| 147 | runcic order-5 hexagonal (biraphexah) ↔ | (1) (5.5.5) | (1) (4.4.3) | (3) (3.4.5.4) | (1) (3.3.3.3.3.3) | | |
| 148 | runcicantic order-5 hexagonal (bitaphexah) ↔ | (1) (3.10.10) | (1) (4.4.3) | (2) (4.6.10) | (1) (3.6.3.6) | | |
| [32] | rectified order-5 hexagonal (riphexah) ↔ | (1) (3.3.3.3.3) | - | (1) (3.3.3.3.3) | (6) (3.6.3.6) | | |
| [33] | rectified order-6 dodecahedral (rihed) ↔ | (2) (3.5.3.5) | - | (2) (3.5.3.5) | (2) (3.3.3.3.3.3) | | |
| [34] | Order-5 hexagonal (hedhon) ↔ | (4) (5.5.5) | - | (4) (5.5.5) | - | | |
| [40] | truncated order-6 dodecahedral (thed) ↔ | (2) (3.10.10) | - | (2) (3.10.10) | (1) (3.6.3.6) | | |
| [36] | cantellated order-5 hexagonal (sriphexah) ↔ | (1) (3.4.5.4) | (2) (6.4.4) | (1) (3.4.5.4) | (1) (3.6.3.6) | | |
| [39] | bitruncated order-5 hexagonal (dohexah) ↔ | (1) (5.6.6) | - | (1) (5.6.6) | (2) (6.6.6) | | |
| [41] | cantitruncated order-5 hexagonal (griphexah) ↔ | (1) (4.6.10) | (1) (6.4.4) | (1) (4.6.10) | (1) (6.6.6) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Alt | vertex figure | Picture |
| Nonuniform | snub rectified order-5 hexagonal ↔ | (3.3.3.3.5) | (3.3.3) | (3.3.3.3.5) | (3.3.3.3.3.3) | +(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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | vertex figure | Picture |
| 149 | runcic order-6 hexagonal ↔ | (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 ↔ | (1) (3.12.12) | (1) (4.4.3) | (2) (4.6.12) | (1) (3.6.3.6) | | |
| [1] | hexagonal (hexah) ↔ ↔ ↔ | (1) (6.6.6) | - | (1) (6.6.6) | (2) (6.6.6) | | |
| [46] | order-6 hexagonal (hihexah) ↔ | (4) (6.6.6) | - | (4) (6.6.6) | - | | |
| [47] | rectified order-6 hexagonal (rihihexah) ↔ | (2) (3.6.3.6) | - | (2) (3.6.3.6) | (2) (3.3.3.3.3.3) | | |
| [47] | rectified order-6 hexagonal (rihihexah) ↔ | (1) (3.3.3.3.3.3) | - | (1) (3.3.3.3.3.3) | (6) (3.6.3.6) | | |
| [48] | truncated order-6 hexagonal (thihexah) ↔ | (2) (3.12.12) | - | (2) (3.12.12) | (1) (3.3.3.3.3.3) | | |
| [49] | cantellated order-6 hexagonal (srihihexah) ↔ | (1) (3.4.6.4) | (2) (6.4.4) | (1) (3.4.6.4) | (1) (3.6.3.6) | | |
| [51] | cantitruncated order-6 hexagonal (grihihexah) ↔ | (1) (4.6.12) | (1) (6.4.4) | (1) (4.6.12) | (1) (6.6.6) | | |
| [54] | triangular tiling honeycomb (trah) ( ↔ ) = | - | - | (3.3.3.3.3.3) | (3.3.3.3.3.3) | (6.6.6) | |
| [55] | cantic order-6 hexagonal (ritrah) ( ↔ ) = | (1) (3.6.3.6) | - | (2) (6.6.6) | (2) (3.6.3.6) | | |
Alternated forms :
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 0' | 3 | Alt | vertex figure | Picture |
| [54] | triangular tiling honeycomb (trah) ( ↔ ↔ ) = | | - | | - | | (6.6.6) | |
| [137] | alternated hexagonal (ahexah) ( ↔ ) = ( ↔ ) | | - | | | +(3.6.6) | (3.6.6) | |
| [47] | rectified order-6 hexagonal (rihihexah) ↔ ↔ ↔ | (3.6.3.6) | - | (3.6.3.6) | (3.3.3.3.3.3) | | | |
| [55] | cantic order-6 hexagonal (ritrah) ( ↔ ) = ( ↔ ) = | (1) (3.6.3.6) | - | (2) (6.6.6) | (2) (3.6.3.6) | | | |
| Nonuniform | snub rectified order-6 hexagonal ↔ | (3.3.3.3.6) | (3.3.3.3) | (3.3.3.3.6) | (3.3.3.3.3.3) | +(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) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Vertex figure | Picture |
| 151 | Quarter order-4 hexagonal (quishexah) ↔ | | | | | | |
| [17] | rectified order-4 hexagonal (rishexah) ↔ ↔ ↔ | | | | | (4.4.4) | |
| [18] | rectified order-6 cubic (rihach) ↔ ↔ ↔ | | | | | (6.4.4) | |
| [21] | bitruncated order-6 cubic (chexah) ↔ ↔ ↔ | | | | | | |
| [87] | alternated order-6 cubic (ahach) ↔ ↔ | - | | | | (3.6.3.6) | |
| [88] | cantic order-6 cubic (tachach) ↔ ↔ | | | | | | |
| [141] | alternated order-4 hexagonal (ashexah) ↔ ↔ | | - | | | (4.6.6) | |
| [142] | cantic order-4 hexagonal (tashexah) ↔ ↔ | | | | | | |
| # | Honeycomb name Coxeter diagram | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Cells by location (and count around each vertex) | Vertex figure | Picture |
| # | Honeycomb name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | Vertex figure | Picture |
| Nonuniform | bisnub order-6 cubic ↔ | | | | | irr. {3,3} | | |

### [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 | 1 | vertex figure |
| [1] | hexagonal (hexah) ↔ | | {3,3} |
| [47] | rectified order-6 hexagonal (rihihexah) ↔ | | t{2,3} |
| [54] | triangular tiling honeycomb (trah) ( ↔ ) = | - | t{3$^{[3]}$} |
| [55] | rectified triangular (ritrah) ↔ | | t{2,3} |
| # | Name Coxeter diagram | 0 | 1 | 2 | 3 | Alt | vertex figure | Picture |
| [137] | alternated hexagonal (ahexah) ( ↔ ) = | s{3$^{[3]}$} | s{3$^{[3]}$} | s{3$^{[3]}$} | s{3$^{[3]}$} | {3,3} | (4.6.6) |

## Summary enumerations by family
### Linear graphs

Paracompact hyperbolic enumeration :
| Group | Extended symmetry | Honeycombs | Honeycombs | Chiral extended symmetry | Alternation honeycombs | Alternation honeycombs |
| ${\bar {R}}_{3}$ [4,4,3] | [4,4,3] | 15 | | | | | | | | | | | | | | [1$^+$,4,1$^+$,4,3$^+$] | (6) | | |
| ${\bar {R}}_{3}$ [4,4,3] | [4,4,3] | 15 | | | | | | | | | | | | | | [4,4,3]$^+$ | (1) | |
| ${\bar {N}}_{3}$ [4,4,4] | [4,4,4] | 3 | | | [1$^+$,4,1$^+$,4,1$^+$,4,1$^+$] | (3) | |
| ${\bar {N}}_{3}$ [4,4,4] | [4,4,4] ↔ | (3) | | | [1$^+$,4,1$^+$,4,1$^+$,4,1$^+$] | (3) | |
| ${\bar {N}}_{3}$ [4,4,4] | [2$^+$[4,4,4]] | 3 | | | | [2$^+$[(4,4$^+$,4,2$^+$)]] | (2) | |
| ${\bar {N}}_{3}$ [4,4,4] | [2$^+$[4,4,4]] | 3 | | | | [2$^+$[4,4,4]]$^+$ | (1) | |
| ${\bar {V}}_{3}$ [6,3,3] | [6,3,3] | 15 | | | | | | | | | | | | | | [1$^+$,6,(3,3)$^+$] | (2) | (↔ ) |
| ${\bar {V}}_{3}$ [6,3,3] | [6,3,3] | 15 | | | | | | | | | | | | | | [6,3,3]$^+$ | (1) | |
| ${\bar {BV}}_{3}$ [6,3,4] | [6,3,4] | 15 | | | | | | | | | | | | | | [1$^+$,6,3$^+$,4,1$^+$] | (6) | | |
| ${\bar {BV}}_{3}$ [6,3,4] | [6,3,4] | 15 | | | | | | | | | | | | | | [6,3,4]$^+$ | (1) | |
| ${\bar {HV}}_{3}$ [6,3,5] | [6,3,5] | 15 | | | | | | | | | | | | | | [1$^+$,6,(3,5)$^+$] | (2) | (↔ ) |
| ${\bar {HV}}_{3}$ [6,3,5] | [6,3,5] | 15 | | | | | | | | | | | | | | [6,3,5]$^+$ | (1) | |
| ${\bar {Y}}_{3}$ [3,6,3] | [3,6,3] | 5 | | | | | | | |
| ${\bar {Y}}_{3}$ [3,6,3] | [3,6,3] ↔ | (1) | | [2$^+$[3$^+$,6,3$^+$]] | (1) | |
| ${\bar {Y}}_{3}$ [3,6,3] | [2$^+$[3,6,3]] | 3 | | | [2$^+$[3,6,3]]$^+$ | (1) | |
| ${\bar {Z}}_{3}$ [6,3,6] | [6,3,6] | 6 | | | | | [1$^+$,6,3$^+$,6,1$^+$] | (2) | (↔ ) |
| ${\bar {Z}}_{3}$ [6,3,6] | [2$^+$[6,3,6]] ↔ | (1) | | [2$^+$[(6,3$^+$,6,2$^+$)]] | (2) | |
| ${\bar {Z}}_{3}$ [6,3,6] | [2$^+$[6,3,6]] | 2 | | | [2$^+$[(6,3$^+$,6,2$^+$)]] | (2) | |
| ${\bar {Z}}_{3}$ [6,3,6] | [2$^+$[6,3,6]] | 2 | | | [2$^+$[6,3,6]]$^+$ | (1) | |

### Tridental graphs

Paracompact hyperbolic enumeration :
| Group | Extended symmetry | Honeycombs | Honeycombs | Chiral extended symmetry | Alternation honeycombs | Alternation honeycombs |
| ${\bar {DV}}_{3}$ [6,3$^{1,1}$] | [6,3$^{1,1}$] | 4 | | | | | | |
| ${\bar {DV}}_{3}$ [6,3$^{1,1}$] | [1[6,3$^{1,1}$]]=[6,3,4] ↔ | (7) | | | | | | | | [1[1$^+$,6,3$^{1,1}$]]$^+$ | (2) | (↔ ) |
| ${\bar {DV}}_{3}$ [6,3$^{1,1}$] | [1[6,3$^{1,1}$]]=[6,3,4] ↔ | (7) | | | | | | | | [1[6,3$^{1,1}$]]$^+$=[6,3,4]$^+$ | (1) | |
| ${\bar {O}}_{3}$ [3,4$^{1,1}$] | [3,4$^{1,1}$] | 4 | | | | [3$^+$,4$^{1,1}$]$^+$ | (2) | ↔ |
| ${\bar {O}}_{3}$ [3,4$^{1,1}$] | [1[3,4$^{1,1}$]]=[3,4,4] ↔ | (7) | | | | | | | | [1[3$^+$,4$^{1,1}$]]$^+$ | (2) | |
| ${\bar {O}}_{3}$ [3,4$^{1,1}$] | [1[3,4$^{1,1}$]]=[3,4,4] ↔ | (7) | | | | | | | | [1[3,4$^{1,1}$]]$^+$ | (1) | |
| ${\bar {M}}_{3}$ [4$^{1,1,1}$] | [4$^{1,1,1}$] | 0 | (none) |
| ${\bar {M}}_{3}$ [4$^{1,1,1}$] | [1[4$^{1,1,1}$]]=[4,4,4] ↔ | (4) | | | | [1[1$^+$,4,1$^+$,4$^{1,1}$]]$^+$=[(4,1$^+$,4,1$^+$,4,2$^+$)] | (4) | | |
| ${\bar {M}}_{3}$ [4$^{1,1,1}$] | [3[4$^{1,1,1}$]]=[4,4,3] ↔ | (3) | | | | [3[1$^+$,4$^{1,1,1}$]]$^+$=[1$^+$,4,1$^+$,4,3$^+$] | (2) | (↔ ) |
| ${\bar {M}}_{3}$ [4$^{1,1,1}$] | [3[4$^{1,1,1}$]]=[4,4,3] ↔ | (3) | | | | [3[4$^{1,1,1}$]]$^+$=[4,4,3]$^+$ | (1) | |

### Cyclic graphs

Paracompact hyperbolic enumeration :
| Group | Extended symmetry | Honeycombs | Honeycombs | Chiral extended symmetry | Alternation honeycombs | Alternation honeycombs |
| ${\widehat {CR}}_{3}$ [(4,4,4,3)] | [(4,4,4,3)] | 6 | | | | | | [(4,1$^+$,4,1$^+$,4,3$^+$)] | (2) | ↔ |
| ${\widehat {CR}}_{3}$ [(4,4,4,3)] | [2$^+$[(4,4,4,3)]] | 3 | | | | [2$^+$[(4,4$^+$,4,3$^+$)]] | (2) | |
| ${\widehat {CR}}_{3}$ [(4,4,4,3)] | [2$^+$[(4,4,4,3)]] | 3 | | | | [2$^+$[(4,4,4,3)]]$^+$ | (1) | |
| ${\widehat {RR}}_{3}$ [4$^{[4]}$] | [4$^{[4]}$] | (none) | (none) | (none) | (none) | (none) |
| ${\widehat {RR}}_{3}$ [4$^{[4]}$] | [2$^+$[4$^{[4]}$]] | 1 | | [2$^+$[(4$^+$,4)$^{[2]}$]] | (1) | |
| ${\widehat {RR}}_{3}$ [4$^{[4]}$] | [1[4$^{[4]}$]]=[4,4$^{1,1}$] ↔ | (2) | | [(1$^+$,4)$^{[4]}$] | (2) | ↔ |
| ${\widehat {RR}}_{3}$ [4$^{[4]}$] | [2[4$^{[4]}$]]=[4,4,4] ↔ | (1) | | [2$^+$[(1$^+$,4,4)$^{[2]}$]] | (1) | |
| ${\widehat {RR}}_{3}$ [4$^{[4]}$] | [(2$^+$,4)[4$^{[4]}$]]=[2$^+$[4,4,4]] = | (1) | | [(2$^+$,4)[4$^{[4]}$]]$^+$ = [2$^+$[4,4,4]]$^+$ | (1) | |
| ${\widehat {AV}}_{3}$ [(6,3,3,3)] | [(6,3,3,3)] | 6 | | | | | | | | |
| ${\widehat {AV}}_{3}$ [(6,3,3,3)] | [2$^+$[(6,3,3,3)]] | 3 | | | [2$^+$[(6,3,3,3)]]$^+$ | (1) | |
| ${\widehat {BV}}_{3}$ [(3,4,3,6)] | [(3,4,3,6)] | 6 | | | | | | [(3$^+$,4,3$^+$,6)] | (1) | |
| ${\widehat {BV}}_{3}$ [(3,4,3,6)] | [2$^+$[(3,4,3,6)]] | 3 | | | [2$^+$[(3,4,3,6)]]$^+$ | (1) | |
| ${\widehat {HV}}_{3}$ [(3,5,3,6)] | [(3,5,3,6)] | 6 | | | | | | | | |
| ${\widehat {HV}}_{3}$ [(3,5,3,6)] | [2$^+$[(3,5,3,6)]] | 3 | | | [2$^+$[(3,5,3,6)]]$^+$ | (1) | |
| ${\widehat {VV}}_{3}$ [(3,6)$^{[2]}$] | [(3,6)$^{[2]}$] | 2 | | | | |
| ${\widehat {VV}}_{3}$ [(3,6)$^{[2]}$] | [2$^+$[(3,6)$^{[2]}$]] | 1 | | | | |
| ${\widehat {VV}}_{3}$ [(3,6)$^{[2]}$] | [2$^+$[(3,6)$^{[2]}$]] | 1 | | | | |
| ${\widehat {VV}}_{3}$ [(3,6)$^{[2]}$] | [2$^+$[(3,6)$^{[2]}$]] = | (1) | | [2$^+$[(3$^+$,6)$^{[2]}$]] | (1) | |
| ${\widehat {VV}}_{3}$ [(3,6)$^{[2]}$] | [(2,2)$^+$[(3,6)$^{[2]}$]] | 1 | | [(2,2)$^+$[(3,6)$^{[2]}$]]$^+$ | (1) | |
Paracompact hyperbolic enumeration :
| Group | Extended symmetry | Honeycombs | Honeycombs | Chiral extended symmetry | Alternation honeycombs | Alternation honeycombs |
| ${\widehat {BR}}_{3}$ [(3,3,4,4)] | [(3,3,4,4)] | 4 | | | | | | |
| ${\widehat {BR}}_{3}$ [(3,3,4,4)] | [1[(4,4,3,3)]]=[3,4$^{1,1}$] ↔ | (7) | | | | | | | | [1[(3,3,4,1$^+$,4)]]$^+$ = [3$^+$,4$^{1,1}$]$^+$ | (2) | (= ) |
| ${\widehat {BR}}_{3}$ [(3,3,4,4)] | [1[(4,4,3,3)]]=[3,4$^{1,1}$] ↔ | (7) | | | | | | | | [1[(3,3,4,4)]]$^+$ = [3,4$^{1,1}$]$^+$ | (1) | |
| ${\bar {DP}}_{3}$ [3$^{x}$] | [3$^{x}$] | 1 | | | | |
| ${\bar {DP}}_{3}$ [3$^{x}$] | [1[3$^{x}$]]=[6,3$^{1,1}$] ↔ | (2) | | | | |
| ${\bar {DP}}_{3}$ [3$^{x}$] | [1[3$^{x}$]]=[4,3$^{[3]}$] ↔ | (2) | | | | |
| ${\bar {DP}}_{3}$ [3$^{x}$] | [2[3$^{x}$]]=[6,3,4] ↔ | (3) | | | [2[3$^{x}$]]$^+$ =[6,3,4]$^+$ | (1) | |
| ${\bar {PP}}_{3}$ [3$^{[3,3]}$] | [3$^{[3,3]}$] | 0 | (none) | (none) | (none) | (none) |
| ${\bar {PP}}_{3}$ [3$^{[3,3]}$] | [1[3$^{[3,3]}$]]=[6,3$^{[3]}$] ↔ | 0 | (none) | (none) | (none) | (none) |
| ${\bar {PP}}_{3}$ [3$^{[3,3]}$] | [3[3$^{[3,3]}$]]=[3,6,3] ↔ | (2) | | | | |
| ${\bar {PP}}_{3}$ [3$^{[3,3]}$] | [2[3$^{[3,3]}$]]=[6,3,6] ↔ | (1) | | | | |
| ${\bar {PP}}_{3}$ [3$^{[3,3]}$] | [(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 | Extended symmetry | Honeycombs | Honeycombs | Chiral extended symmetry | Alternation honeycombs | Alternation honeycombs |
| ${\bar {P}}_{3}$ [3,3$^{[3]}$] | [3,3$^{[3]}$] | 4 | | | | | | |
| ${\bar {P}}_{3}$ [3,3$^{[3]}$] | [1[3,3$^{[3]}$]]=[3,3,6] ↔ | (7) | | | | | | | [1[3,3$^{[3]}$]]$^+$ = [3,3,6]$^+$ | (1) | |
| ${\bar {BP}}_{3}$ [4,3$^{[3]}$] | [4,3$^{[3]}$] | 4 | | | | | | |
| ${\bar {BP}}_{3}$ [4,3$^{[3]}$] | [1[4,3$^{[3]}$]]=[4,3,6] ↔ | (7) | | | | | | | | [1$^+$,4,(3$^{[3]}$)$^+$] | (2) | ↔ |
| ${\bar {BP}}_{3}$ [4,3$^{[3]}$] | [1[4,3$^{[3]}$]]=[4,3,6] ↔ | (7) | | | | | | | | [4,3$^{[3]}$]$^+$ | (1) | |
| ${\bar {HP}}_{3}$ [5,3$^{[3]}$] | [5,3$^{[3]}$] | 4 | | | | | | |
| ${\bar {HP}}_{3}$ [5,3$^{[3]}$] | [1[5,3$^{[3]}$]]=[5,3,6] ↔ | (7) | | | | | | | [1[5,3$^{[3]}$]]$^+$ = [5,3,6]$^+$ | (1) | |
| ${\bar {VP}}_{3}$ [6,3$^{[3]}$] | [6,3$^{[3]}$] | 2 | | | | |
| ${\bar {VP}}_{3}$ [6,3$^{[3]}$] | [6,3$^{[3]}$] = | (2) | ( = ) | | | |
| ${\bar {VP}}_{3}$ [6,3$^{[3]}$] | [(3,3)[1$^+$,6,3$^{[3]}$]]=[6,3,3] ↔ ↔ | (1) | | [(3,3)[1$^+$,6,3$^{[3]}$]]$^+$ | (1) | |
| ${\bar {VP}}_{3}$ [6,3$^{[3]}$] | [1[6,3$^{[3]}$]]=[6,3,6] ↔ | (6) | | | | | | | [3[1$^+$,6,3$^{[3]}$]]$^+$ = [3,6,3]$^+$ | (1) | ↔ (= ) |
| ${\bar {VP}}_{3}$ [6,3$^{[3]}$] | [1[6,3$^{[3]}$]]=[6,3,6] ↔ | (6) | | | | | | | [1[6,3$^{[3]}$]]$^+$ = [6,3,6]$^+$ | (1) | |