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

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

Geometry

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


Poincaré disk model
Ideal surface

It a part of a sequence of regular polychora and honeycombs {p,4,p}:

{p,4,p} regular honeycombs
Space S3 Euclidean E3 H3
Form Finite Paracompact Noncompact
Name {3,4,3} {4,4,4} {5,4,5} {6,4,6} {7,4,7} {8,4,8} ...{∞,4,∞}
Image
Cells
{p,4}
{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
{∞,4}
Vertex
figure
{4,p}
{4,3}
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}
{4,∞}

Order-4-6 hexagonal honeycomb

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

In the geometry of hyperbolic 3-space, the order-4-6 hexagonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,6}. It has six order-4 hexagonal tilings, {6,4}, 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 square tiling vertex arrangement.


Poincaré disk model
Ideal surface

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

Order-4-infinite apeirogonal honeycomb

Order-4-infinite apeirogonal honeycomb
TypeRegular honeycomb
Schläfli symbols{∞,4,∞}
{∞,(4,∞,4)}
Coxeter diagrams
↔
Cells{∞,4}
Faces{∞}
Edge figure{∞}
Vertex figure {4,∞}
{(4,∞,4)}
Dualself-dual
Coxeter group[∞,4,∞]
[∞,((4,∞,4))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-4-infinite apeirogonal honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,4,∞}. It has infinitely many order-4 apeirogonal tiling {∞,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal 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 {∞,(4,∞,4)}, 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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