Order-6-4 triangular honeycomb

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

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

It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.

Poincaré disk model

Ideal surface

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

It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}

{3,6,p} polytopes
Space	H3
Form	Paracompact	Noncompact
Name	{3,6,3}	{3,6,4}	{3,6,5}	{3,6,6}	... {3,6,∞}
Image
Vertex figure	{6,3}	{6,4}	{6,5}	{6,6}	{6,∞}

Order-6-5 triangular honeycomb
Type	Regular honeycomb
Schläfli symbol	{3,6,5}
Coxeter diagram
Cells	{3,6}
Faces	{3}
Edge figure	{5}
Vertex figure	{6,5}
Dual	{5,6,3}
Coxeter group	[3,6,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,5}. It has five triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement.

Poincaré disk model

Ideal surface

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

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

Poincaré disk model

Ideal surface

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

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

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

• 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)

• Spherical Video: {3,6,∞} honeycomb with parabolic Möbius transform YouTube, Roice Nelson
• 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]