# Order-6-4 triangular honeycomb

Order-6-4 triangular honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,4} |
| Coxeter diagrams | = |
| Cells | {3,6} |
| Faces | {3} |
| Edge figure | {4} |
| Vertex figure | {6,4} r{6,6} |
| Dual | {4,6,3} |
| Coxeter group | [3,6,4] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,4}.

## Geometry

It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,6$^{1,1}$}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1$^+$] = [3,6$^{1,1}$].

## Related polytopes and honeycombs

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

### Order-6-5 triangular honeycomb

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

### Order-6-6 triangular honeycomb

Order-6-6 triangular honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,6} {3,(6,3,6)} |
| Coxeter diagrams | = |
| Cells | {3,6} |
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {6,6} {(6,3,6)} |
| Dual | {6,6,3} |
| Coxeter group | [3,6,6] [3,((6,3,6))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,6}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-6 triangular tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,3,6)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,6,1$^+$] = [3,((6,3,6))].

### Order-6-infinite triangular honeycomb

Order-6-infinite triangular honeycomb :
| Type | Regular honeycomb |
| Schläfli symbols | {3,6,∞} {3,(6,∞,6)} |
| Coxeter diagrams | = |
| Cells | {3,6} |
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {6,∞} {(6,∞,6)} |
| Dual | {∞,6,3} |
| Coxeter group | [∞,6,3] [3,((6,∞,6))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,∞}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.
| Poincaré disk model | Ideal surface |
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,∞,6)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1$^+$] = [3,((6,∞,6))].