# Order-4 icosahedral honeycomb

Order-4 icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,4} |
| Coxeter diagrams | |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {4} |
| Vertex figure | {5,4} |
| Dual | {4,5,3} |
| Coxeter group | [3,5,4] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.

## Geometry

It has four icosahedra {3,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-4 pentagonal tiling vertex arrangement.
| Poincaré disk model (Cell centered) | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,5$^{1,1}$}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,4,1$^+$] = [3,5$^{1,1}$].

## Related polytopes and honeycombs

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

### Order-5 icosahedral honeycomb

Order-5 icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,5} |
| Coxeter diagrams | |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {5} |
| Vertex figure | {5,5} |
| Dual | {5,5,3} |
| Coxeter group | [3,5,5] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.
| Poincaré disk model (Cell centered) | Ideal surface |

### Order-6 icosahedral honeycomb

Order-6 icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,6} {3,(5,∞,5)} |
| Coxeter diagrams | = |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {5,6} |
| Dual | {6,5,3} |
| Coxeter group | [3,5,6] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.
| Poincaré disk model (Cell centered) | Ideal surface |

### Order-7 icosahedral honeycomb

Order-7 icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,7} |
| Coxeter diagrams | |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {7} |
| Vertex figure | {5,7} |
| Dual | {7,5,3} |
| Coxeter group | [3,5,7] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.
| Poincaré disk model (Cell centered) | Ideal surface |

### Order-8 icosahedral honeycomb

Order-8 icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,8} |
| Coxeter diagrams | |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {8} |
| Vertex figure | {5,8} |
| Dual | {8,5,3} |
| Coxeter group | [3,5,8] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.
* Poincaré disk model (Cell centered)

### Infinite-order icosahedral honeycomb

Infinite-order icosahedral honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,5,∞} {3,(5,∞,5)} |
| Coxeter diagrams | = |
| Cells | {3,5} |
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {5,∞} {(5,∞,5)} |
| Dual | {∞,5,3} |
| Coxeter group | [∞,5,3] [3,((5,∞,5))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.
| Poincaré disk model (Cell centered) | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, = , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1$^+$] = [3,((5,∞,5))].