Order-3-4 heptagonal honeycomb

In the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 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-3-4 heptagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{7,3,4}
Coxeter diagram
=
Cells{7,3}
Facesheptagon {7}
Vertex figureoctahedron {3,4}
Dual{4,3,7}
Coxeter group[7,3,4]
PropertiesRegular

Geometry

The Schläfli symbol of the order-3-4 heptagonal honeycomb is {7,3,4}, with four heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.


Poincaré disk model
(vertex centered)
One hyperideal cell limits to a circle on the ideal surface
Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,3,4} Schläfli symbol, and octahedral vertex figures:

{p,3,4} regular honeycombs
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,4}

 {4,3,4}



 {5,3,4}

 {6,3,4}



 {7,3,4}

 {8,3,4}



 ... {∞,3,4}



Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}

Order-3-4 octagonal honeycomb
Order-3-4 octagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{8,3,4}
Coxeter diagram
=

Cells{8,3}
Facesoctagon {8}
Vertex figureoctahedron {3,4}
Dual{4,3,8}
Coxeter group[8,3,4]
[8,31,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-4 octagonal honeycomb or 8,3,4 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-3-4 octagonal honeycomb is {8,3,4}, with four octagonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.


Poincaré disk model
(vertex centered)

Order-3-4 apeirogonal honeycomb
Order-3-4 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,3,4}
Coxeter diagram
=

Cells{∞,3}
Facesapeirogon {∞}
Vertex figureoctahedron {3,4}
Dual{4,3,∞}
Coxeter group[∞,3,4]
[∞,31,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-4 apeirogonal honeycomb or ∞,3,4 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-3-4 apeirogonal honeycomb is {∞,3,4}, with four order-3 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octahedron, {3,4}.


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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