Order-6-4 square honeycomb

Order-4-6 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,6,4}
Coxeter diagrams
Cells	{4,6}
Faces	{4}
Edge figure	{4}
Vertex figure	{6,4}
Dual	self-dual
Coxeter group	[4,6,4]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.

Poincaré disk model

Ideal surface

It a part of a sequence of regular polychora and honeycombs {p,6,p}:

Order-6-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,6,5}
Coxeter diagrams
Cells	{5,6}
Faces	{5}
Edge figure	{5}
Vertex figure	{6,5}
Dual	self-dual
Coxeter group	[5,6,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.

Poincaré disk model

Ideal surface

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

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

Poincaré disk model

Ideal surface

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

Order-6-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,6,∞} {∞,(6,∞,6)}
Coxeter diagrams	↔
Cells	{∞,6}
Faces	{∞}
Edge figure	{∞}
Vertex figure	{6,∞} {(6,∞,6)}
Dual	self-dual
Coxeter group	[∞,6,∞] [∞,((6,∞,6))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal 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 {∞,(6,∞,6)}, Coxeter diagram, , with alternating types or colors of cells.

• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes
• Infinite-order dodecahedral honeycomb

• 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]