# Order-4-5 square honeycomb

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

## Images

| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}
{4,4,p} honeycombs vte :
| Space | E$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Affine | Paracompact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {4,4,2} | {4,4,3} | {4,4,4} | {4,4,5} | {4,4,6} | ...{4,4,∞} |
| Coxeter | | | | | | |
| Image | | | | | | |
| Vertex figure | {4,2} | {4,3} | {4,4} | {4,5} | {4,6} | {4,∞} |

### Order-4-6 square honeycomb

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

### Order-4-infinite square honeycomb

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