# Order-3-6 heptagonal honeycomb

Order-3-6 heptagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {7,3,6} {7,3$^{[3]}$} |
| Coxeter diagram | = |
| Cells | {7,3} |
| Faces | {7} |
| Vertex figure | {3,6} |
| Dual | {6,3,7} |
| Coxeter group | [7,3,6] [7,3$^{[3]}$] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal 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-3-6 heptagonal honeycomb is {7,3,6}, with six heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
| Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,3,6} Schläfli symbol, and triangular tiling vertex figures.
Hyperbolic uniform honeycombs: {p,3,6} and {p,3$^{[3]}$} vte :
| Form | Paracompact | Paracompact | Paracompact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {3,3,6} {3,3$^{[3]}$} | {4,3,6} {4,3$^{[3]}$} | {5,3,6} {5,3$^{[3]}$} | {6,3,6} {6,3$^{[3]}$} | {7,3,6} {7,3$^{[3]}$} | {8,3,6} {8,3$^{[3]}$} | ... {∞,3,6} {∞,3$^{[3]}$} |
| Image | | | | | | | |
| Cells | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} |

### Order-3-6 octagonal honeycomb

Order-3-6 octagonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {8,3,6} {8,3$^{[3]}$} |
| Coxeter diagram | = |
| Cells | {8,3} |
| Faces | Octagon {8} |
| Vertex figure | triangular tiling {3,6} |
| Dual | {6,3,8} |
| Coxeter group | [8,3,6] [8,3$^{[3]}$] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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-3-6 octagonal honeycomb is {8,3,6}, with six octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
It has a quasiregular construction, , which can be seen as alternately colored cells.
* Poincaré disk model

### Order-3-6 apeirogonal honeycomb

Order-3-6 apeirogonal honeycomb :
| Type | Regular honeycomb |
| Schläfli symbol | {∞,3,6} {∞,3$^{[3]}$} |
| Coxeter diagram | = |
| Cells | {∞,3} |
| Faces | Apeirogon {∞} |
| Vertex figure | triangular tiling {3,6} |
| Dual | {6,3,∞} |
| Coxeter group | [∞,3,6] [∞,3$^{[3]}$] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-3-6 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 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 order-3-6 apeirogonal honeycomb is {∞,3,6}, with six order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a triangular tiling, {3,6}.
| Poincaré disk model | Ideal surface |
It has a quasiregular construction, , which can be seen as alternately colored cells.