Order-4-3 pentagonal honeycomb

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Order-4-3 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,4,3}
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {5,4} H2-5-4-dual.svg
Faces {5}
Vertex figure {4,3}
Dual {3,4,5}
Coxeter group [5,4,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.

Geometry[]

The Schläfli symbol of the order-4-3 pentagonal honeycomb is {5,4,3}, with three order-4 pentagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Hyperbolic honeycomb 5-4-3 poincare vc.png
Poincaré disk model
(Vertex centered) 		H3 543 UHS plane at infinity.png
Ideal surface

Related polytopes and honeycombs[]

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

Order-4-3 hexagonal honeycomb[]

Order-4-3 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbol {6,4,3}
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {6,4} Uniform tiling 64-t0.png
Faces {6}
Vertex figure {4,3}
Dual {3,4,6}
Coxeter group [6,4,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-3 hexagonal honeycomb or 6,4,3 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 order-4-3 hexagonal honeycomb is {6,4,3}, with three order-4 hexagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Hyperbolic honeycomb 6-4-3 poincare vc.png
Poincaré disk model
(Vertex centered) 		H3 643 UHS plane at infinity.png
Ideal surface

Order-4-3 heptagonal honeycomb[]

Order-4-3 heptagonal honeycomb
Type Regular honeycomb
Schläfli symbol {7,4,3}
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {7,4} Uniform tiling 74-t0.png
Faces {7}
Vertex figure {4,3}
Dual {3,4,7}
Coxeter group [7,4,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-3 heptagonal honeycomb or 7,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 heptagonal 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-4-3 heptagonal honeycomb is {7,4,3}, with three order-4 heptagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Hyperbolic honeycomb 7-4-3 poincare vc.png
Poincaré disk model
(Vertex centered) 		H3 743 UHS plane at infinity.png
Ideal surface

Order-4-3 octagonal honeycomb[]

Order-4-3 octagonal honeycomb
Type Regular honeycomb
Schläfli symbol {8,4,3}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {8,4} Uniform tiling 84-t0.png
Faces {8}
Vertex figure {4,3}
Dual {3,4,8}
Coxeter group [8,4,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-4-3 octagonal honeycomb or 8,4,3 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-4 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-4-3 octagonal honeycomb is {8,4,3}, with three order-4 octagonal tilings meeting at each edge. The vertex figure of this honeycomb is a cube, {4,3}.

Hyperbolic honeycomb 8-4-3 poincare vc.png
Poincaré disk model
(Vertex centered)

Order-4-3 apeirogonal honeycomb[]

Order-4-3 apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbol {∞,4,3}
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Cells {∞,4} H2 tiling 24i-1.png
Faces Apeirogon {∞}
Vertex figure {4,3}
Dual {3,4,∞}
Coxeter group [∞,4,3]
Properties Regular

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

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.

Hyperbolic honeycomb i-4-3 poincare vc.png
Poincaré disk model
(Vertex centered) 		H3 i43 UHS plane at infinity.png
Ideal surface

See also[]

  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

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) [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)

External links[]

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