# 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 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}.

It a part of a sequence of honeycombs {5,p,7}.

It a part of a sequence of honeycombs {p,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.