Order-4-5 pentagonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, 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-6 square tiling vertex arrangement.

Poincaré disk model

Ideal surface

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

Order-4-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,4,∞} {∞,(4,∞,4)}
Coxeter diagrams	↔
Cells	{∞,4}
Faces	{∞}
Edge figure	{∞}
Vertex figure	{4,∞} {(4,∞,4)}
Dual	self-dual
Coxeter group	[∞,4,∞] [∞,((4,∞,4))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} 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 square tiling vertex arrangement.

Poincaré disk model

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(4,∞,4)}, Coxeter diagram, , with alternating types or colors of cells.