Order-5 octahedral honeycomb

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

Order-5 octahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,4,5}
Coxeter diagrams
Cells{3,4}
Faces{3}
Edge figure{5}
Vertex figure{4,5}
Dual{5,4,3}
Coxeter group[3,4,5]
PropertiesRegular

Images

Poincaré disk model
(cell centered)
Ideal surface

It a part of a sequence of regular polychora and honeycombs with octahedral cells: {3,4,p}

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

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


Poincaré disk model
(cell centered)
Ideal surface

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

Order-7 octahedral honeycomb
Order-7 octahedral honeycomb
TypeRegular honeycomb
Schläfli symbols{3,4,7}
Coxeter diagrams
Cells{3,4}
Faces{3}
Edge figure{7}
Vertex figure{4,7}
Dual{7,4,3}
Coxeter group[3,4,7]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,7}. It has seven octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-7 square tiling vertex arrangement.


Poincaré disk model
(cell centered)
Ideal surface

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

In the geometry of hyperbolic 3-space, the order-8 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,8}. It has eight octahedra, {3,4}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-8 square tiling vertex arrangement.


Poincaré disk model
(cell centered)

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

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


Poincaré disk model
(cell centered)
Ideal surface

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

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See also
  • Convex uniform honeycombs in hyperbolic space
  • List of regular polytopes

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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