# Order-4-3 pentagonal honeycomb

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

### Order-4-3 hexagonal honeycomb

Order-4-3 hexagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {6,4,3} |
| Coxeter diagram | |
| Cells | {6,4} |
| Faces | {6} |
| Vertex figure | {4,3} |
| Dual | {3,4,6} |
| Coxeter group | [6,4,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 hexagonal honeycomb or 6,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 hexagonal honeycomb is {6,4,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
| Poincaré disk model (Vertex centered) | Ideal surface |

### Order-4-3 heptagonal honeycomb

Order-4-3 heptagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {7,4,3} |
| Coxeter diagram | |
| Cells | {7,4} |
| Faces | {7} |
| Vertex figure | {4,3} |
| Dual | {3,4,7} |
| Coxeter group | [7,4,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 heptagonal honeycomb or 7,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 heptagonal honeycomb is {7,4,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
| Poincaré disk model (Vertex centered) | Ideal surface |

### Order-4-3 octagonal honeycomb

Order-4-3 octagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {8,4,3} |
| Coxeter diagram | |
| Cells | {8,4} |
| Faces | {8} |
| Vertex figure | {4,3} |
| Dual | {3,4,8} |
| Coxeter group | [8,4,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 octagonal honeycomb or 8,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 octagonal honeycomb is {8,4,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
* Poincaré disk model (Vertex centered)

### Order-4-3 apeirogonal honeycomb

Order-4-3 apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {∞,4,3} |
| Coxeter diagram | |
| Cells | {∞,4} |
| Faces | Apeirogon {∞} |
| Vertex figure | {4,3} |
| Dual | {3,4,∞} |
| Coxeter group | [∞,4,3] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4-3 apeirogonal honeycomb or ∞,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an 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 {∞,4,3}, with three apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.
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 |