# Order-5 octahedral honeycomb

Order-5 octahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,5} |
| Coxeter diagrams | |
| Cells | {3,4} |
| Faces | {3} |
| Edge figure | {5} |
| Vertex figure | {4,5} |
| Dual | {5,4,3} |
| Coxeter group | [3,4,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.

## Images

| Poincaré disk model (cell centered) | Ideal surface |

## Related polytopes and honeycombs

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

### Order-6 octahedral honeycomb

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

### Order-7 octahedral honeycomb

Order-7 octahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,7} |
| Coxeter diagrams | |
| Cells | {3,4} |
| Faces | {3} |
| Edge figure | {7} |
| Vertex figure | {4,7} |
| Dual | {7,4,3} |
| Coxeter group | [3,4,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,7}. It has seven octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-7 square tiling vertex arrangement.
| Poincaré disk model (cell centered) | Ideal surface |

### Order-8 octahedral honeycomb

Order-8 octahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,4,8} |
| Coxeter diagrams | |
| Cells | {3,4} |
| Faces | {3} |
| Edge figure | {8} |
| Vertex figure | {4,8} |
| Dual | {8,4,3} |
| Coxeter group | [3,4,8] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,8}. It has eight octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-8 square tiling vertex arrangement.
* Poincaré disk model (cell centered)

### Infinite-order octahedral honeycomb

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