Order-3-7 hexagonal honeycomb

Order-3-7 hexagonal honeycomb
Poincaré disk model
Type	Regular honeycomb
Schläfli symbol	{6,3,7}
Coxeter diagrams
Cells	{6,3}
Faces	{6}
Edge figure	{7}
Vertex figure	{3,7}
Dual	{7,3,6}
Coxeter group	[6,3,7]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

Ideal surface

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model

Closeup

It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.

{6,3,p} honeycombs v t e
Space	H3
Form	Paracompact				Noncompact
Name	{6,3,3}	{6,3,4}	{6,3,5}	{6,3,6}	{6,3,7}	{6,3,8}	... {6,3,∞}
Coxeter
Image
Vertex figure {3,p}	{3,3}	{3,4}	{3,5}	{3,6}	{3,7}	{3,8}	{3,∞}

Order-3-8 hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,3,8} {6,(3,4,3)}
Coxeter diagrams	=
Cells	{6,3}
Faces	{6}
Edge figure	{8}
Vertex figure	{3,8} {(3,4,3)}
Dual	{8,3,6}
Coxeter group	[6,3,8] [6,((3,4,3))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, 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-8 triangular tiling vertex arrangement.

Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1⁺] = [6,((3,4,3))].

Order-3-infinite hexagonal honeycomb
Type	Regular honeycomb
Schläfli symbols	{6,3,∞} {6,(3,∞,3)}
Coxeter diagrams	↔ ↔
Cells	{6,3}
Faces	{6}
Edge figure	{∞}
Vertex figure	{3,∞}, {(3,∞,3)}
Dual	{∞,3,6}
Coxeter group	[6,3,∞] [6,((3,∞,3))]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement.

Poincaré disk model

Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of hexagonal tiling 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]