# Order-4-5 pentagonal honeycomb

Order-4-5 pentagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {5,4,5} |
| Coxeter diagrams | |
| Cells | {5,4} |
| Faces | {5} |
| Edge figure | {5} |
| Vertex figure | {4,5} |
| Dual | self-dual |
| Coxeter group | [5,4,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,4,5}.

## Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-4 pentagonal tilings existing around each edge and with an order-5 square tiling vertex figure.
| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

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

### Order-4-6 hexagonal honeycomb

Order-4-6 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {6,4,6} {6,(4,3,4)} |
| Coxeter diagrams | = |
| Cells | {6,4} |
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {4,6} {(4,3,4)} |
| Dual | self-dual |
| Coxeter group | [6,4,6] [6,((4,3,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, 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-6 square tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(4,3,4)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,4,6,1$^+$] = [6,((4,3,4))].

### Order-4-infinite apeirogonal honeycomb

Order-4-infinite apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {∞,4,∞} {∞,(4,∞,4)} |
| Coxeter diagrams | ↔ |
| Cells | {∞,4} |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {4,∞} {(4,∞,4)} |
| Dual | self-dual |
| Coxeter group | [∞,4,∞] [∞,((4,∞,4))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} 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 square tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of cells.