Order-3-7 hexagonal honeycomb

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

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

Related polytopes and honeycombs
It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.

Order-3-8 hexagonal honeycomb
In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-8 triangular tiling vertex arrangement.

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram,, with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].

Order-3-infinite hexagonal honeycomb
In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} 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.

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram,, with alternating types or colors of hexagonal tiling cells.