Infinite-order triangular tiling

Infinite-order triangular tiling
Infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 3
Schläfli symbol {3,∞}
Wythoff symbol ∞ | 3 2
Coxeter diagram
Symmetry group [∞,3], (*∞32)
Dual Order-3 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.

The {3,3,∞} honeycomb has {3,∞} vertex figures.

Symmetry

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A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)},     . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

 
Alternated colored tiling
 
*∞∞∞ symmetry
 
Apollonian gasket with *∞∞∞ symmetry
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This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

*n32 symmetry mutation of regular tilings: {3,n}
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
                       
3.3 33 34 35 36 37 38 3 312i 39i 36i 33i
Paracompact uniform tilings in [∞,3] family
Symmetry: [∞,3], (*∞32) [∞,3]+
(∞32)
[1+,∞,3]
(*∞33)
[∞,3+]
(3*∞)
                                                                 
     
=     
     
=     
     
=     
            =
     or     
      =
     or     
     
=     
                   
{∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
Uniform duals
                                                           
                 
V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞
Paracompact hyperbolic uniform tilings in [(∞,3,3)] family
Symmetry: [(∞,3,3)], (*∞33) [(∞,3,3)]+, (∞33)
                                       
                                               
               
(∞,∞,3) t0,1(∞,3,3) t1(∞,3,3) t1,2(∞,3,3) t2(∞,3,3) t0,2(∞,3,3) t0,1,2(∞,3,3) s(∞,3,3)
Dual tilings
                                                               
                                               
   
V(3.∞)3 V3.∞.3.∞ V(3.∞)3 V3.6.∞.6 V(3.3) V3.6.∞.6 V6.6.∞ V3.3.3.3.3.∞

Other infinite-order triangular tilings

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A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

 

See also

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References

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  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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