# Order-3-7 hexagonal honeycomb

Order-3-7 hexagonal honeycomb :
| Poincaré disk model ||
| Type | Regular honeycomb |
| Schläfli symbol | {6,3,7} |
| Coxeter diagrams | |
| Cells | {6,3} |
| Faces | {6} |
| Edge figure | {7} |
| Vertex figure | {3,7} |
| Dual | {7,3,6} |
| Coxeter group | [6,3,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

## Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.
Ideal surface :
| Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model | Closeup |

## Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.
{6,3,p} honeycombs vte :
| Space | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Paracompact | Paracompact | Paracompact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {6,3,3} | {6,3,4} | {6,3,5} | {6,3,6} | {6,3,7} | {6,3,8} | ... {6,3,∞} |
| Coxeter | | | | | | | |
| Image | | | | | | | |
| Vertex figure {3,p} | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |

### Order-3-8 hexagonal honeycomb

Order-3-8 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {6,3,8} {6,(3,4,3)} |
| Coxeter diagrams | = |
| Cells | {6,3} |
| Faces | {6} |
| Edge figure | {8} |
| Vertex figure | {3,8} {(3,4,3)} |
| Dual | {8,3,6} |
| Coxeter group | [6,3,8] [6,((3,4,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.
* Poincaré disk model
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1$^+$] = [6,((3,4,3))].

### Order-3-infinite hexagonal honeycomb

Order-3-infinite hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {6,3,∞} {6,(3,∞,3)} |
| Coxeter diagrams | ↔ ↔ |
| Cells | {6,3} |
| Faces | {6} |
| Edge figure | {∞} |
| Vertex figure | {3,∞}, {(3,∞,3)} |
| Dual | {∞,3,6} |
| Coxeter group | [6,3,∞] [6,((3,∞,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of hexagonal tiling cells.