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Order-6-4 square honeycomb

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Order-4-6 square honeycomb
Type Regular honeycomb
Schläfli symbol {4,6,4}
Coxeter diagrams
Cells {4,6}
Faces {4}
Edge figure {4}
Vertex figure {6,4}
Dual self-dual
Coxeter group [4,6,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-4 square honeycomb (or 4,6,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,6,4}.

Geometry

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All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-6 square tilings existing around each edge and with an order-4 hexagonal tiling vertex figure.


Poincaré disk model
Ideal surface
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It a part of a sequence of regular polychora and honeycombs {p,6,p}:

Order-6-5 hexagonal honeycomb

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Order-6-5 pentagonal honeycomb
Type Regular honeycomb
Schläfli symbol {5,6,5}
Coxeter diagrams
Cells {5,6}
Faces {5}
Edge figure {5}
Vertex figure {6,5}
Dual self-dual
Coxeter group [5,6,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-5 pentagonal honeycomb (or 5,6,5 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,6,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five order-6 pentagonal tilings existing around each edge and with an order-5 hexagonal tiling vertex figure.


Poincaré disk model
Ideal surface

Order-6-6 hexagonal honeycomb

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Order-5-6 hexagonal honeycomb
Type Regular honeycomb
Schläfli symbols {6,6,6}
{6,(6,3,6)}
Coxeter diagrams
=
Cells {6,6}
Faces {6}
Edge figure {6}
Vertex figure {6,6}
{(6,3,6)}
Dual self-dual
Coxeter group [6,5,6]
[6,((6,3,6))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-6 hexagonal honeycomb (or 6,6,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,6,6}. It has six order-6 hexagonal tilings, {6,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 hexagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(6,3,6)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,6,6,1+] = [6,((6,3,6))].

Order-6-infinite apeirogonal honeycomb

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Order-6-infinite apeirogonal honeycomb
Type Regular honeycomb
Schläfli symbols {∞,6,∞}
{∞,(6,∞,6)}
Coxeter diagrams
↔
Cells {∞,6}
Faces {∞}
Edge figure {∞}
Vertex figure {6,∞}
{(6,∞,6)}
Dual self-dual
Coxeter group [∞,6,∞]
[∞,((6,∞,6))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-infinite apeirogonal honeycomb (or ∞,6,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,6,∞}. It has infinitely many order-6 apeirogonal tiling {∞,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-6 apeirogonal tilings existing around each vertex in an infinite-order square tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(6,∞,6)}, Coxeter diagram, , with alternating types or colors of cells.

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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