Order-4 icosahedral honeycomb

Order-4 icosahedral honeycomb
Type	Regular honeycomb
Schläfli symbols	{3,5,4}
Coxeter diagrams
Cells	{3,5}
Faces	{3}
Edge figure	{4}
Vertex figure	{5,4}
Dual	{4,5,3}
Coxeter group	[3,5,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4 icosahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,5,4}.

Geometry

It has four 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-4 pentagonal tiling vertex arrangement.

Poincaré disk model (Cell centered)	Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,5^1,1}, Coxeter diagram, , with alternating types or colors of icosahedral cells. In Coxeter notation the half symmetry is [3,5,4,1⁺] = [3,5^1,1].

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with icosahedral cells: {3,5,p}

{3,5,p} polytopes
Space	H³
Form	Compact	Noncompact
Name	{3,5,3}	{3,5,4}	{3,5,5}	{3,5,6}	{3,5,7}	{3,5,8}	... {3,5,∞}
Image
Vertex figure	{5,3}	{5,4}	{5,5}	{5,6}	{5,7}	{5,8}	{5,∞}

Order-5 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

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

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

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

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))].

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I, II)
George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)

External links

John Baez, Visual insights: {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)
Danny Calegari, Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]