Order-6-3 square honeycomb

Order-6-3 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{4,6} File:H2 tiling 246-4.png
Faces	{4}
Vertex figure	{6,3}
Dual	{3,6,4}
Coxeter group	[4,6,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Geometry

The Schläfli symbol of the order-6-3 square honeycomb is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.

File:Hyperbolic honeycomb 4-6-3 poincare.png Poincaré disk model

File:H3 463 UHS plane at infinity.png Ideal surface

Related polytopes and honeycombs

It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:

Order-6-3 pentagonal honeycomb

Order-6-3 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{5,6} File:H2 tiling 256-4.png
Faces	{5}
Vertex figure	{6,3}
Dual	{3,6,5}
Coxeter group	[5,6,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-6-3 pentagonal honeycomb is {5,6,3}, with three order-6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.

File:Hyperbolic honeycomb 5-6-3 poincare.png Poincaré disk model

File:H3 563 UHS plane at infinity.png Ideal surface

Order-6-3 hexagonal honeycomb

Order-6-3 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{6,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{6,6} File:H2 tiling 266-4.png
Faces	{6}
Vertex figure	{6,3}
Dual	{3,6,6}
Coxeter group	[6,6,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the order-6-3 hexagonal honeycomb is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.

File:Hyperbolic honeycomb 6-6-3 poincare.png Poincaré disk model

File:H3 663 UHS plane at infinity.png Ideal surface

Order-6-3 apeirogonal honeycomb

Order-6-3 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{∞,6,3}
Coxeter diagram	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
Cells	{∞,6} File:H2 tiling 26i-1.png
Faces	Apeirogon {∞}
Vertex figure	{6,3}
Dual	{3,6,∞}
Coxeter group	[∞,6,3]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.

File:Hyperbolic honeycomb i-6-3 poincare.png Poincaré disk model

File:H3 i63 UHS plane at infinity.png Ideal surface

See also

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

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, (Chapter 10, Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition (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]