Order-3-7 hexagonal honeycomb

Order-3-8 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,3,8} {6,(3,4,3)}
Coxeter diagrams	=
Cells	{6,3}
Faces	{6}
Edge figure	{8}
Vertex figure	{3,8} {(3,4,3)}
Dual	{8,3,6}
Coxeter group	[6,3,8] [6,((3,4,3))]
Properties	Regular

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.

Poincaré disk model

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
Type	Regular honeycomb
Schläfli symbols	{6,3,∞} {6,(3,∞,3)}
Coxeter diagrams	↔ ↔
Cells	{6,3}
Faces	{6}
Edge figure	{∞}
Vertex figure	{3,∞}, {(3,∞,3)}
Dual	{∞,3,6}
Coxeter group	[6,3,∞] [6,((3,∞,3))]
Properties	Regular

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.

Poincaré disk model

Ideal surface

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.