Order-7 tetrahedral honeycomb

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

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

Poincaré disk model (cell-centered)

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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

Infinite-order tetrahedral honeycomb
Type	Hyperbolic regular honeycomb
Schläfli symbols	{3,3,∞} {3,(3,∞,3)}
Coxeter diagrams	=
Cells	{3,3}
Faces	{3}
Edge figure	{∞}
Vertex figure	{3,∞} {(3,∞,3)}
Dual	{∞,3,3}
Coxeter group	[∞,3,3] [3,((3,∞,3))]
Properties	Regular

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

Poincaré disk model (cell-centered)

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

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