# Order-7 dodecahedral honeycomb

Order-7 dodecahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,7} |
| Coxeter diagrams | |
| Cells | {5,3} |
| Faces | {5} |
| Edge figure | {7} |
| Vertex figure | {3,7} |
| Dual | {7,3,5} |
| Coxeter group | [5,3,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 dodecahedral honeycomb is a regular space-filling tessellation (or honeycomb).

## Geometry

With Schläfli symbol {5,3,7}, it has seven dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.
| Poincaré disk model Cell-centered | Poincaré disk model | Ideal surface |

## Related polytopes and honeycombs

It a part of a sequence of regular polytopes and honeycombs with dodecahedral cells, {5,3,p}.
{5,3,p} polytopes :
| Space | S$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Finite | Compact | Compact | Paracompact | Noncompact | Noncompact | Noncompact |
| Name | {5,3,3} | {5,3,4} | {5,3,5} | {5,3,6} | {5,3,7} | {5,3,8} | ... {5,3,∞} |
| Image | | | | | | | |
| Vertex figure | {3,3} | {3,4} | {3,5} | {3,6} | {3,7} | {3,8} | {3,∞} |
It a part of a sequence of honeycombs {5,p,7}.
It a part of a sequence of honeycombs {p,3,7}.
| {3,3,7} | {4,3,7} | {5,3,7} | {6,3,7} | {7,3,7} | {8,3,7} | {∞,3,7} |

### Order-8 dodecahedral honeycomb

Order-8 dodecahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,8} {5,(3,4,3)} |
| Coxeter diagrams | = |
| Cells | {5,3} |
| Faces | {5} |
| Edge figure | {8} |
| Vertex figure | {3,8}, {(3,4,3)} |
| Dual | {8,3,5} |
| Coxeter group | [5,3,8] [5,((3,4,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,8}, it has eight dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.
| Poincaré disk model Cell-centered | Poincaré disk model |
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,4,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.

### Infinite-order dodecahedral honeycomb

Infinite-order dodecahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {5,3,∞} {5,(3,∞,3)} |
| Coxeter diagrams | = |
| Cells | {5,3} |
| Faces | {5} |
| Edge figure | {∞} |
| Vertex figure | {3,∞}, {(3,∞,3)} |
| Dual | {∞,3,5} |
| Coxeter group | [5,3,∞] [5,((3,∞,3))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order dodecahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {5,3,∞}. It has infinitely many dodecahedra {5,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many dodecahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
| Poincaré disk model Cell-centered | Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {5,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of dodecahedral cells.