# Order-6-4 square honeycomb

Order-4-6 square honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {4,6,4} |
| Coxeter diagrams | |
| Cells | {4,6} |
| Faces | {4} |
| Edge figure | {4} |
| Vertex figure | {6,4} |
| Dual | self-dual |
| Coxeter group | [4,6,4] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.

## Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.
| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,6,p}:

### Order-6-5 hexagonal honeycomb

Order-6-5 pentagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {5,6,5} |
| Coxeter diagrams | |
| Cells | {5,6} |
| Faces | {5} |
| Edge figure | {5} |
| Vertex figure | {6,5} |
| Dual | self-dual |
| Coxeter group | [5,6,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.
All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.
| Poincaré disk model | Ideal surface |

### Order-6-6 hexagonal honeycomb

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

### Order-6-infinite apeirogonal honeycomb

Order-6-infinite apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {∞,6,∞} {∞,(6,∞,6)} |
| Coxeter diagrams | ↔ |
| Cells | {∞,6} |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {6,∞} {(6,∞,6)} |
| Dual | self-dual |
| Coxeter group | [∞,6,∞] [∞,((6,∞,6))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal 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 {∞,(6,∞,6)}, Coxeter diagram, , with alternating types or colors of cells.