# Order-3-7 heptagonal honeycomb

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

## Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.
| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,3,p}:
{p,3,p} regular honeycombs :
| Space | S$^3$ | Euclidean E$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Finite | Affine | Compact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {3,3,3} | {4,3,4} | {5,3,5} | {6,3,6} | {7,3,7} | {8,3,8} | ...{∞,3,∞} |
| Image | | | | | | | |
| Cells | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |
| Vertex figure | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |

### Order-3-8 octagonal honeycomb

Order-3-8 octagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {8,3,8} {8,(3,4,3)} |
| Coxeter diagrams | = |
| Cells | {8,3} |
| Faces | {8} |
| Edge figure | {8} |
| Vertex figure | {3,8} {(3,8,3)} |
| Dual | self-dual |
| Coxeter group | [8,3,8] [8,((3,4,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {8,3,8}. It has eight octagonal tilings, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal 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 {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1$^+$] = [8,((3,4,3))].

### Order-3-infinite apeirogonal honeycomb

Order-3-infinite apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {∞,3,∞} {∞,(3,∞,3)} |
| Coxeter diagrams | ↔ |
| Cells | {∞,3} |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {3,∞} {(3,∞,3)} |
| Dual | self-dual |
| Coxeter group | [∞,3,∞] [∞,((3,∞,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,3,∞}. It has infinitely many order-3 apeirogonal tiling {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal 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 {∞,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.