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

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

Poincaré disk model

Ideal surface

It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}

{4,4,p} honeycombs
Space	E3	H3
Form	Affine	Paracompact		Noncompact
Name	{4,4,2}	{4,4,3}	{4,4,4}	{4,4,5}	{4,4,6}	...{4,4,∞}
Coxeter
Image
Vertex figure	{4,2}	{4,3}	{4,4}	{4,5}	{4,6}	{4,∞}

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

In the geometry of hyperbolic 3-space, the order-4-6 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,6}. It has six square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling 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 {4,(4,3,4)}, Coxeter diagram, , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1⁺] = [4,((4,3,4))].

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

In the geometry of hyperbolic 3-space, the order-4-infinite square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,∞}. It has infinitely many square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling 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,∞,4)}, Coxeter diagram, = , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1⁺] = [4,((4,∞,4))].

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

• 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)

• 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]