Order-5-4 square honeycomb

In the geometry of hyperbolic 3-space, the order-5-4 square honeycomb (or 4,5,4 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,5,4}.

Order-4-5 square honeycomb
TypeRegular honeycomb
Schläfli symbol{4,5,4}
Coxeter diagrams
Cells{4,5}
Faces{4}
Edge figure{4}
Vertex figure{5,4}
Dualself-dual
Coxeter group[4,5,4]
PropertiesRegular

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with four order-5 square tilings existing around each edge and with an order-4 pentagonal tiling vertex figure.


Poincaré disk model
Ideal surface

It a part of a sequence of regular polychora and honeycombs {p,5,p}:

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

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

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


Poincaré disk model
Ideal surface

Order-5-6 hexagonal honeycomb
Order-5-6 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{6,5,6}
{6,(5,3,5)}
Coxeter diagrams
=
Cells{6,5}
Faces{6}
Edge figure{6}
Vertex figure{5,6}
{(5,3,5)}
Dualself-dual
Coxeter group[6,5,6]
[6,((5,3,5))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-6 hexagonal honeycomb (or 6,5,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,5,6}. It has six order-5 hexagonal tilings, {6,5}, 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 pentagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(5,3,5)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,5,6,1+] = [6,((5,3,5))].

Order-5-7 heptagonal honeycomb
Order-5-7 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{7,5,7}
Coxeter diagrams
Cells{7,5}
Faces{6}
Edge figure{6}
Vertex figure{5,7}
Dualself-dual
Coxeter group[7,5,7]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-7 heptagonal honeycomb (or 7,5,7 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {7,5,7}. It has seven order-5 heptagonal tilings, {7,5}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many heptagonal tilings existing around each vertex in an order-7 pentagonal tiling vertex arrangement.


Ideal surface

Order-5-infinite apeirogonal honeycomb
Order-5-infinite apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbols{∞,5,∞}
{∞,(5,∞,5)}
Coxeter diagrams
↔
Cells{∞,5}
Faces{∞}
Edge figure{∞}
Vertex figure {5,∞}
{(5,∞,5)}
Dualself-dual
Coxeter group[∞,5,∞]
[∞,((5,∞,5))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-5-infinite apeirogonal honeycomb (or ∞,5,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,5,∞}. It has infinitely many order-5 apeirogonal tilings {∞,5} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-5 apeirogonal tilings existing around each vertex in an infinite-order pentagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(5,∞,5)}, 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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