Order-5-4 square honeycomb

Order-4-5 square honeycomb
Type	Regular honeycomb
Schläfli symbol	{4,5,4}
Coxeter diagrams	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
Cells	{4,5} File:H2-5-4-primal.svg
Faces	{4}
Edge figure	{4}
Vertex figure	{5,4}
Dual	self-dual
Coxeter group	[4,5,4]
Properties	Regular

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

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.

File:Hyperbolic honeycomb 4-5-4 poincare.png Poincaré disk model

File:H3 454 UHS plane at infinity.png Ideal surface

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs {p,5,p}:

Order-5-5 pentagonal honeycomb

Order-5-5 pentagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{5,5,5}
Coxeter diagrams	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png
Cells	{5,5} File:H2 tiling 255-1.png
Faces	{5}
Edge figure	{5}
Vertex figure	{5,5}
Dual	self-dual
Coxeter group	[5,5,5]
Properties	Regular

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

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

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

File:H3 555 UHS plane at infinity.png Ideal surface

Order-5-6 hexagonal honeycomb

Order-5-6 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,5,6} {6,(5,3,5)}
Coxeter diagrams	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node h0.png = File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel split1-55.pngFile:CDel branch.png
Cells	{6,5} File:H2 tiling 256-1.png
Faces	{6}
Edge figure	{6}
Vertex figure	{5,6} File:H2 tiling 256-4.png {(5,3,5)} File:H2 tiling 355-1.png
Dual	self-dual
Coxeter group	[6,5,6] [6,((5,3,5))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.

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

File:H3 656 UHS plane at infinity.png Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(5,3,5)}, Coxeter diagram, File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel split1-55.pngFile:CDel branch.png, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1⁺] = [6,((5,3,5))].

Order-5-7 heptagonal honeycomb

Order-5-7 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{7,5,7}
Coxeter diagrams	File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 7.pngFile:CDel node.png
Cells	{7,5} File:H2 tiling 257-1.png
Faces	{6}
Edge figure	{6}
Vertex figure	{5,7} File:H2 tiling 257-4.png
Dual	self-dual
Coxeter group	[7,5,7]
Properties	Regular

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

File:H3 757 UHS plane at infinity.png Ideal surface

Order-5-infinite apeirogonal honeycomb

Order-5-infinite apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{∞,5,∞} {∞,(5,∞,5)}
Coxeter diagrams	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node h0.png ↔ File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel split1-55.pngFile:CDel branch.pngFile:CDel labelinfin.png
Cells	{∞,5} File:H2 tiling 25i-1.png
Faces	{∞}
Edge figure	{∞}
Vertex figure	File:H2 tiling 25i-4.png {5,∞} File:H2 tiling 55i-4.png {(5,∞,5)}
Dual	self-dual
Coxeter group	[∞,5,∞] [∞,((5,∞,5))]
Properties	Regular

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

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

File:H3 i5i UHS plane at infinity.png Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, Coxeter diagram, File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel split1-55.pngFile:CDel branch.pngFile:CDel labelinfin.png, with alternating types or colors of cells.

See also

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

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]