Order-5 octahedral honeycomb

Order-5 octahedral honeycomb
Type Regular 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]
Properties Regular

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.

Images

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Poincaré disk model
(cell centered)
Ideal surface
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It a part of a sequence of regular polychora and honeycombs with octahedral cells: {3,4,p}

{3,4,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,4,3}

 
 {3,4,4}


 {3,4,5}
 {3,4,6}

 {3,4,7}
 {3,4,8}

 ... {3,4,∞}

Image
Vertex
figure
{4,3}

 
{4,4}
{4,5}
{4,6}
{4,7}
{4,8}
{4,∞}

Order-6 octahedral honeycomb

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Order-6 octahedral honeycomb
Type Regular 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))]
Properties Regular

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

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Order-7 octahedral honeycomb
Type Regular 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]
Properties Regular

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

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Order-8 octahedral honeycomb
Type Regular 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]
Properties Regular

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

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Infinite-order octahedral honeycomb
Type Regular 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))]
Properties Regular

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))].

See also

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References

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  • 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) [1]
  • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
  • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
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