# Order-6-3 square honeycomb

Order-6-3 square honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {4,6,3} |
| Coxeter diagram | |
| Cells | {4,6} |
| Faces | {4} |
| Vertex figure | {6,3} |
| Dual | {3,6,4} |
| Coxeter group | [4,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

## Geometry

The Schläfli symbol of the order-6-3 square honeycomb is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:

### Order-6-3 pentagonal honeycomb

Order-6-3 pentagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {5,6,3} |
| Coxeter diagram | |
| Cells | {5,6} |
| Faces | {5} |
| Vertex figure | {6,3} |
| Dual | {3,6,5} |
| Coxeter group | [5,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-6-3 pentagonal honeycomb is {5,6,3}, with three order-6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
| Poincaré disk model | Ideal surface |

### Order-6-3 hexagonal honeycomb

Order-6-3 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {6,6,3} |
| Coxeter diagram | |
| Cells | {6,6} |
| Faces | {6} |
| Vertex figure | {6,3} |
| Dual | {3,6,6} |
| Coxeter group | [6,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the order-6-3 hexagonal honeycomb is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
| Poincaré disk model | Ideal surface |

### Order-6-3 apeirogonal honeycomb

Order-6-3 apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {∞,6,3} |
| Coxeter diagram | |
| Cells | {∞,6} |
| Faces | Apeirogon {∞} |
| Vertex figure | {6,3} |
| Dual | {3,6,∞} |
| Coxeter group | [∞,6,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
| Poincaré disk model | Ideal surface |