Order-4 icosahedral honeycomb

Order-5 icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,5}
Coxeter diagrams
Cells	{3,5}
Faces	{3}
Edge figure	{5}
Vertex figure	{5,5}
Dual	{5,5,3}
Coxeter group	[3,5,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-5 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,5}. It has five icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-5 pentagonal tiling vertex arrangement.

Poincaré disk model (Cell centered)

Ideal surface

Order-6 icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,6} {3,(5,∞,5)}
Coxeter diagrams	=
Cells	{3,5}
Faces	{3}
Edge figure	{6}
Vertex figure	{5,6}
Dual	{6,5,3}
Coxeter group	[3,5,6]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,6}. It has six icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-6 pentagonal tiling vertex arrangement.

Poincaré disk model (Cell centered)

Ideal surface

Order-7 icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,7}
Coxeter diagrams
Cells	{3,5}
Faces	{3}
Edge figure	{7}
Vertex figure	{5,7}
Dual	{7,5,3}
Coxeter group	[3,5,7]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-7 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,7}. It has seven icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-7 pentagonal tiling vertex arrangement.

Poincaré disk model (Cell centered)

Ideal surface

Order-8 icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,8}
Coxeter diagrams
Cells	{3,5}
Faces	{3}
Edge figure	{8}
Vertex figure	{5,8}
Dual	{8,5,3}
Coxeter group	[3,5,8]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-8 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,8}. It has eight icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an order-8 pentagonal tiling vertex arrangement.

Poincaré disk model (Cell centered)

Infinite-order icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,∞} {3,(5,∞,5)}
Coxeter diagrams	=
Cells	{3,5}
Faces	{3}
Edge figure	{∞}
Vertex figure	{5,∞} {(5,∞,5)}
Dual	{∞,5,3}
Coxeter group	[∞,5,3] [3,((5,∞,5))]
Properties	Regular

In the geometry of hyperbolic 3-space, the infinite-order icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,∞}. It has infinitely many icosahedra, {3,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many icosahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.

Poincaré disk model (Cell centered)

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(5,∞,5)}, Coxeter diagram, = , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,∞,1⁺] = [3,((5,∞,5))].