# Order-5-3 square honeycomb

Order-5-3 square honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {4,5,3} |
| Coxeter diagram | |
| Cells | {4,5} |
| Faces | {4} |
| Vertex figure | {5,3} |
| Dual | {3,5,4} |
| Coxeter group | [4,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 square honeycomb or 4,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal 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-5-3 square honeycomb is {4,5,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
| Poincaré disk model (Vertex centered) | Ideal surface |

## Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,5,3} Schläfli symbol, and dodecahedral vertex figures:

### Order-5-3 pentagonal honeycomb

Order-5-3 pentagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {5,5,3} |
| Coxeter diagram | |
| Cells | {5,5} |
| Faces | {5} |
| Vertex figure | {5,3} |
| Dual | {3,5,5} |
| Coxeter group | [5,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 pentagonal honeycomb or 5,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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-5-3 pentagonal honeycomb is {5,5,3}, with three order-5 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
| Poincaré disk model (Vertex centered) | Ideal surface |

### Order-5-3 hexagonal honeycomb

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

### Order-5-3 heptagonal honeycomb

Order-5-3 heptagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {7,5,3} |
| Coxeter diagram | |
| Cells | {7,5} |
| Faces | {7} |
| Vertex figure | {5,3} |
| Dual | {3,5,7} |
| Coxeter group | [7,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 heptagonal honeycomb or 7,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 heptagonal 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-5-3 heptagonal honeycomb is {7,5,3}, with three order-5 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
| Poincaré disk model (Vertex centered) | Ideal surface |

### Order-5-3 octagonal honeycomb

Order-5-3 octagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {8,5,3} |
| Coxeter diagram | |
| Cells | {8,5} |
| Faces | {8} |
| Vertex figure | {5,3} |
| Dual | {3,5,8} |
| Coxeter group | [8,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 octagonal honeycomb or 8,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 octagonal 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-5-3 octagonal honeycomb is {8,5,3}, with three order-5 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
* Poincaré disk model (Vertex centered)

### Order-5-3 apeirogonal honeycomb

Order-5-3 apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {∞,5,3} |
| Coxeter diagram | |
| Cells | {∞,5} |
| Faces | Apeirogon {∞} |
| Vertex figure | {5,3} |
| Dual | {3,5,∞} |
| Coxeter group | [∞,5,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5-3 apeirogonal honeycomb or ∞,5,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-5 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 {∞,5,3}, with three order-5 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a dodecahedron, {5,3}.
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 (Vertex centered) | Ideal surface |