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Order-6-4 triangular honeycomb

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Order-6-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,6,4}
Coxeter diagrams
=
Cells {3,6}
Faces {3}
Edge figure {4}
Vertex figure {6,4}
r{6,6}
Dual {4,6,3}
Coxeter group [3,6,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,4}.

Geometry

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It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1}, Coxeter diagram, , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,4,1+] = [3,61,1].

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It a part of a sequence of regular polychora and honeycombs with triangular tiling cells: {3,6,p}

{3,6,p} polytopes
Space H3
Form Paracompact Noncompact
Name {3,6,3}

 
 {3,6,4}

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

 ... {3,6,∞}

Image
Vertex
figure
{6,3}

 
{6,4}
{6,5}
{6,6}
{6,∞}

Order-6-5 triangular honeycomb

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Order-6-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbol {3,6,5}
Coxeter diagram
Cells {3,6}
Faces {3}
Edge figure {5}
Vertex figure {6,5}
Dual {5,6,3}
Coxeter group [3,6,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,5}. It has five triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

Order-6-6 triangular honeycomb

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Order-6-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,6,6}
{3,(6,3,6)}
Coxeter diagrams
=
Cells {3,6}
Faces {3}
Edge figure {6}
Vertex figure {6,6}
{(6,3,6)}
Dual {6,6,3}
Coxeter group [3,6,6]
[3,((6,3,6))]
Properties Regular

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


Poincaré disk model
Ideal surface

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

Order-6-infinite triangular honeycomb

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Order-6-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,6,∞}
{3,(6,∞,6)}
Coxeter diagrams
=
Cells {3,6}
Faces {3}
Edge figure {∞}
Vertex figure {6,∞}
{(6,∞,6)}
Dual {∞,6,3}
Coxeter group [∞,6,3]
[3,((6,∞,6))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,6,∞}. It has infinitely many triangular tiling, {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular 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 {3,(6,∞,6)}, Coxeter diagram, = , with alternating types or colors of triangular tiling cells. In Coxeter notation the half symmetry is [3,6,∞,1+] = [3,((6,∞,6))].

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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