Order-6-4 square honeycomb

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}.

Order-4-6 square honeycomb
TypeRegular honeycomb
Schläfli symbol{4,6,4}
Coxeter diagrams
Cells{4,6}
Faces{4}
Edge figure{4}
Vertex figure{6,4}
Dualself-dual
Coxeter group[4,6,4]
PropertiesRegular

Geometry

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

It a part of a sequence of regular polychora and honeycombs {p,6,p}:

Order-6-5 hexagonal honeycomb

Order-6-5 pentagonal honeycomb
TypeRegular honeycomb
Schläfli symbol{5,6,5}
Coxeter diagrams
Cells{5,6}
Faces{5}
Edge figure{5}
Vertex figure{6,5}
Dualself-dual
Coxeter group[5,6,5]
PropertiesRegular

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

Order-5-6 hexagonal honeycomb
TypeRegular 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)}
Dualself-dual
Coxeter group[6,5,6]
[6,((6,3,6))]
PropertiesRegular

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

Order-6-infinite apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbols{∞,6,∞}
{∞,(6,∞,6)}
Coxeter diagrams
Cells{∞,6}
Faces{∞}
Edge figure{∞}
Vertex figure {6,∞}
{(6,∞,6)}
Dualself-dual
Coxeter group[∞,6,∞]
[∞,((6,∞,6))]
PropertiesRegular

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

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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