Order-6-3 square honeycomb

In the geometry of hyperbolic 3-space, the order-6-3 square honeycomb or 4,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a hexagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Order-6-3 square honeycomb
TypeRegular honeycomb
Schläfli symbol{4,6,3}
Coxeter diagram
Cells{4,6}
Faces{4}
Vertex figure{6,3}
Dual{3,6,4}
Coxeter group[4,6,3]
PropertiesRegular

Geometry

The Schläfli symbol of the order-6-3 square honeycomb is {4,6,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.


Poincaré disk model

Ideal surface

It is a part of a series of regular polytopes and honeycombs with {p,6,3} Schläfli symbol, and dodecahedral vertex figures:

Order-6-3 pentagonal honeycomb

Order-6-3 pentagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{5,6,3}
Coxeter diagram
Cells{5,6}
Faces{5}
Vertex figure{6,3}
Dual{3,6,5}
Coxeter group[5,6,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-6-3 pentagonal honeycomb or 5,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 pentagonal 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-6-3 pentagonal honeycomb is {5,6,3}, with three order-6 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.


Poincaré disk model

Ideal surface

Order-6-3 hexagonal honeycomb

Order-6-3 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{6,6,3}
Coxeter diagram
Cells{6,6}
Faces{6}
Vertex figure{6,3}
Dual{3,6,6}
Coxeter group[6,6,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-6-3 hexagonal honeycomb or 6,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 order-6-3 hexagonal honeycomb is {6,6,3}, with three order-5 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.


Poincaré disk model

Ideal surface

Order-6-3 apeirogonal honeycomb

Order-6-3 apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbol{∞,6,3}
Coxeter diagram
Cells{∞,6}
FacesApeirogon {∞}
Vertex figure{6,3}
Dual{3,6,∞}
Coxeter group[∞,6,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-6-3 apeirogonal honeycomb or ∞,6,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-6 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 {∞,6,3}, with three order-6 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is a hexagonal tiling, {6,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.


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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