Order-3-5 heptagonal honeycomb

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.

Order-3-5 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,5}
Coxeter diagram
Cells{7,3}
FacesHeptagon {7}
Vertex figureicosahedron {3,5}
Dual{5,3,7}
Coxeter group[7,3,5]
PropertiesRegular

Geometry

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
Order-3-5 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,5}
Coxeter diagram
Cells{8,3}
FacesOctagon {8}
Vertex figureicosahedron {3,5}
Dual{5,3,8}
Coxeter group[8,3,5]
PropertiesRegular

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
Order-3-5 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,5}
Coxeter diagram
Cells{∞,3}
FacesApeirogon {∞}
Vertex figureicosahedron {3,5}
Dual{5,3,∞}
Coxeter group[∞,3,5]
PropertiesRegular

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

See also

References
    • 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)
    • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)
    • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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