Order-3-5 heptagonal honeycomb

Order-3-5 heptagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{7,3,5}
Coxeter diagram
Cells	{7,3}
Faces	Heptagon {7}
Vertex figure	icosahedron {3,5}
Dual	{5,3,7}
Coxeter group	[7,3,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-5 heptagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal 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-3-5 heptagonal honeycomb is {7,3,5}, with five heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.

Poincaré disk model (vertex centered)

Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,3,5} Schläfli symbol, and icosahedral vertex figures.

{p,3,5} polytopes
Space	S3	H3
Form	Finite	Compact		Paracompact	Noncompact
Name	{3,3,5}	{4,3,5}	{5,3,5}	{6,3,5}	{7,3,5}	{8,3,5}	... {∞,3,5}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

Order-3-5 octagonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{8,3,5}
Coxeter diagram
Cells	{8,3}
Faces	Octagon {8}
Vertex figure	icosahedron {3,5}
Dual	{5,3,8}
Coxeter group	[8,3,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-5 octagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an octagonal 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 3-5 heptagonal honeycomb is {8,3,5}, with five octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.

Poincaré disk model (vertex centered)

Order-3-5 apeirogonal honeycomb
Type	Regular honeycomb
Schläfli symbol	{∞,3,5}
Coxeter diagram
Cells	{∞,3}
Faces	Apeirogon {∞}
Vertex figure	icosahedron {3,5}
Dual	{5,3,∞}
Coxeter group	[∞,3,5]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-3-5 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-3 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 order-3-5 apeirogonal honeycomb is {∞,3,5}, with five order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an icosahedron, {3,5}.

Poincaré disk model (vertex centered)

Ideal surface

• 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)

• 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]