Order-4-4 pentagonal honeycomb

In the geometry of hyperbolic 3-space, the order-4-4 pentagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Order-4-4 pentagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{5,4,4}
{5,41,1}
Coxeter diagram
Cells{5,4}
Faces{5}
Vertex figure{4,4}
Dual{4,4,5}
Coxeter group[5,4,4]
[5,41,1]
PropertiesRegular

Geometry

The Schläfli symbol of the order-4-4 pentagonal honeycomb is {5,4,4}, with four order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.


Poincaré disk model
Ideal surface

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

{p,4,4} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {2,4,4} {3,4,4} {4,4,4} {5,4,4} {6,4,4} ..{∞,4,4}
Coxeter










 





Image
Cells
{2,4}
{3,4}
{4,4}
{5,4}
{6,4}
{∞,4}

Order-4-4 hexagonal honeycomb
Order-4-4 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{6,4,4}
{6,41,1}
Coxeter diagram
Cells{6,4}
Faces{6}
Vertex figure{4,4}
Dual{4,4,6}
Coxeter group[6,4,4]
[6,41,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-4-4 hexagonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

The Schläfli symbol of the octagonal tiling honeycomb is {6,4,4}, with three octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.


Poincaré disk model
Ideal surface

Order-4-4 apeirogonal honeycomb
Order-4-4 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,4,4}
{∞,41,1}
Coxeter diagram
Cells{∞,4}
Faces{∞}
Vertex figure{4,4}
Dual{4,4,∞}
Coxeter group[∞,4,4]
[∞,41,1]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-4-4 apeirogonal honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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 apeirogonal tiling honeycomb is {∞,4,4}, with three order-4 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a square tiling, {4,4}.


Poincaré disk model
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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