Order-6-4 triangular honeycomb

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}.

Order-6-4 triangular honeycomb
TypeRegular 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]
PropertiesRegular

Geometry

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].

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

Order-6-5 triangular honeycomb
TypeRegular 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]
PropertiesRegular

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

Order-6-6 triangular honeycomb
TypeRegular 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))]
PropertiesRegular

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

Order-6-infinite triangular honeycomb
TypeRegular 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))]
PropertiesRegular

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

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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