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

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

Images


Poincaré disk model
Ideal surface

It a part of a sequence of regular polychora and honeycombs with square tiling cells: {4,4,p}

{4,4,p} honeycombs
Space E3 H3
Form Affine Paracompact Noncompact
Name {4,4,2} {4,4,3} {4,4,4} {4,4,5} {4,4,6} ...{4,4,∞}
Coxeter












Image
Vertex
figure
{4,2}
{4,3}
{4,4}
{4,5}
{4,6}
{4,∞}

Order-4-6 square honeycomb

Order-4-6 square honeycomb
TypeRegular honeycomb
Schläfli symbols{4,4,6}
{4,(4,3,4)}
Coxeter diagrams
=
Cells{4,4}
Faces{4}
Edge figure{6}
Vertex figure{4,6}
{(4,3,4)}
Dual{6,4,4}
Coxeter group[4,4,6]
[4,((4,3,4))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-4-6 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,6}. It has six square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-6 square tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {4,(4,3,4)}, Coxeter diagram, , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,6,1+] = [4,((4,3,4))].

Order-4-infinite square honeycomb

Order-4-infinite square honeycomb
TypeRegular honeycomb
Schläfli symbols{4,4,∞}
{4,(4,∞,4)}
Coxeter diagrams
=
Cells{4,4}
Faces{4}
Edge figure{∞}
Vertex figure{4,∞}
{(4,∞,4)}
Dual{∞,4,4}
Coxeter group[∞,4,3]
[4,((4,∞,4))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-4-infinite square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,∞}. It has infinitely many square tiling, {4,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling 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 {4,(4,∞,4)}, Coxeter diagram, = , with alternating types or colors of square tiling cells. In Coxeter notation the half symmetry is [4,4,∞,1+] = [4,((4,∞,4))].

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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