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

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 modelIdeal 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].

Related polytopes and honeycombs

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

{3,6,p} polytopes
SpaceH3
FormParacompactNoncompact
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 modelIdeal 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 modelIdeal 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 modelIdeal 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

  • 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 , ISBN 0-486-40919-8 (Chapter 10, 2016-06-10 at the Wayback Machine) 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)
  • Roice Nelson, Henry Segerman (2015)

External links