Order-3-7 hexagonal honeycomb

In the geometry of hyperbolic 3-space, the order-3-7 hexagonal honeycomb or (6,3,7 honeycomb) a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,7}.

Order-3-7 hexagonal honeycomb

Poincaré disk model
TypeRegular honeycomb
Schläfli symbol{6,3,7}
Coxeter diagrams
Cells{6,3}
Faces{6}
Edge figure{7}
Vertex figure{3,7}
Dual{7,3,6}
Coxeter group[6,3,7]
PropertiesRegular

Geometry

All vertices are ultra-ideal (existing beyond the ideal boundary) with seven hexagonal tilings existing around each edge and with an order-7 triangular tiling vertex figure.

Ideal surface

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
Closeup

It a part of a sequence of regular polychora and honeycombs with hexagonal tiling cells.

Order-3-8 hexagonal honeycomb
Order-3-8 hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{6,3,8}
{6,(3,4,3)}
Coxeter diagrams
=
Cells{6,3}
Faces{6}
Edge figure{8}
Vertex figure{3,8} {(3,4,3)}
Dual{8,3,6}
Coxeter group[6,3,8]
[6,((3,4,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-8 hexagonal honeycomb or (6,3,8 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,8}. It has eight hexagonal tilings, {6,3}, 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-8 triangular tiling vertex arrangement.


Poincaré disk model

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [6,3,8,1+] = [6,((3,4,3))].

Order-3-infinite hexagonal honeycomb
Order-3-infinite hexagonal honeycomb
TypeRegular honeycomb
Schläfli symbols{6,3,∞}
{6,(3,∞,3)}
Coxeter diagrams
↔
↔
Cells{6,3}
Faces{6}
Edge figure{∞}
Vertex figure{3,∞}, {(3,∞,3)}
Dual{∞,3,6}
Coxeter group[6,3,∞]
[6,((3,∞,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-3-infinite hexagonal honeycomb or (6,3,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,3,∞}. It has infinitely many hexagonal tiling {6,3} 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 triangular tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {6,(3,∞,3)}, Coxeter diagram, , with alternating types or colors of hexagonal tiling 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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