# Order-5-4 square honeycomb

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

## Geometry

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

## Related polytopes and honeycombs

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

### Order-5-5 pentagonal honeycomb

Order-5-5 pentagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {5,5,5} |
| Coxeter diagrams | |
| Cells | {5,5} |
| Faces | {5} |
| Edge figure | {5} |
| Vertex figure | {5,5} |
| Dual | self-dual |
| Coxeter group | [5,5,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-5 pentagonal honeycomb (or 5,5,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,5,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-5 pentagonal tilings existing around each edge and with an order-5 pentagonal tiling vertex figure.
| Poincaré disk model | Ideal surface |

### Order-5-6 hexagonal honeycomb

Order-5-6 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {6,5,6} {6,(5,3,5)} |
| Coxeter diagrams | = |
| Cells | {6,5} |
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {5,6} {(5,3,5)} |
| Dual | self-dual |
| Coxeter group | [6,5,6] [6,((5,3,5))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1$^+$] = [6,((5,3,5))].

### Order-5-7 heptagonal honeycomb

Order-5-7 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {7,5,7} |
| Coxeter diagrams | |
| Cells | {7,5} |
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {5,7} |
| Dual | self-dual |
| Coxeter group | [7,5,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
* Ideal surface

### Order-5-infinite apeirogonal honeycomb

Order-5-infinite apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {∞,5,∞} {∞,(5,∞,5)} |
| Coxeter diagrams | ↔ |
| Cells | {∞,5} |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {5,∞} {(5,∞,5)} |
| Dual | self-dual |
| Coxeter group | [∞,5,∞] [∞,((5,∞,5))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, , with alternating types or colors of cells.