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Order-6-3 square honeycomb

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Order-6-3 square honeycomb
Type Regular 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]
Properties Regular

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.

Geometry

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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
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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

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Order-6-3 pentagonal honeycomb
Type Regular 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]
Properties Regular

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

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Order-6-3 hexagonal honeycomb
Type Regular 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]
Properties Regular

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

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Order-6-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,6,3}
Coxeter diagram
Cells {∞,6}
Faces Apeirogon {∞}
Vertex figure {6,3}
Dual {3,6,∞}
Coxeter group [∞,6,3]
Properties Regular

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

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References

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  • 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)
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