# Order-3-7 heptagonal honeycomb

Order-3-7 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,3,7}
Coxeter diagrams
Cells {7,3}
Faces {7}
Edge figure {7}
Vertex figure {3,7}
Dual self-dual
Coxeter group [7,3,7]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-7 heptagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,3,7}.

## Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

 Poincaré disk model Ideal surface

## Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,3,p}:

### Order-3-8 octagonal honeycomb

Order-3-8 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {8,3,8}
{8,(3,4,3)}
Coxeter diagrams
=
Cells {8,3}
Faces {8}
Edge figure {8}
Vertex figure {3,8}
{(3,8,3)}
Dual self-dual
Coxeter group [8,3,8]
[8,((3,4,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-8 octagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {8,3,8}. It has eight octagonal tilings, {8,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.

It has a second construction as a uniform honeycomb, Schläfli symbol {8,(3,4,3)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [8,3,8,1+] = [8,((3,4,3))].

### Order-3-infinite apeirogonal honeycomb

Order-3-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,3,∞}
{∞,(3,∞,3)}
Coxeter diagrams
Cells {∞,3}
Faces {∞}
Edge figure {∞}
Vertex figure {3,∞}
{(3,∞,3)}
Dual self-dual
Coxeter group [∞,3,∞]
[∞,((3,∞,3))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-3-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,3,∞}. It has infinitely many order-3 apeirogonal tiling {∞,3} around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many hexagonal 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,∞,3)}, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.