Jump to content

Order-6-4 triangular honeycomb

Add links
From Wikipedia, the free encyclopedia
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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

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
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

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

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

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) [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)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Order-6-4_triangular_honeycomb&oldid=1269701728"