Truncated trioctagonal tiling

Summary

Truncated trioctagonal tiling
Truncated trioctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.6.16
Schläfli symbol tr{8,3} or
Wythoff symbol 2 8 3 |
Coxeter diagram or
Symmetry group [8,3], (*832)
Dual Order 3-8 kisrhombille
Properties Vertex-transitive

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

Symmetry

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Truncated trioctagonal tiling with mirror lines

The dual of this tiling, the order 3-8 kisrhombille, represents the fundamental domains of [8,3] (*832) symmetry. There are 3 small index subgroups constructed from [8,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A larger index 6 subgroup constructed as [8,3*], becomes [(4,4,4)], (*444). An intermediate index 3 subgroup is constructed as [8,3], with 2/3 of blue mirrors removed.

Small index subgroups of [8,3], (*832)
Index 1 2 3 6
Diagrams          
Coxeter
(orbifold)
[8,3] =      
(*832)
[1+,8,3] =       =     
(*433)
[8,3+] =      
(3*4)
[8,3] =       =      
(*842)
[8,3*] =       =     
(*444)
Direct subgroups
Index 2 4 6 12
Diagrams        
Coxeter
(orbifold)
[8,3]+ =      
(832)
[8,3+]+ =       =     
(433)
[8,3]+ =       =      
(842)
[8,3*]+ =       =     
(444)

Order 3-8 kisrhombille

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Truncated trioctagonal tiling
 
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagram     
Symmetry group[8,3], (*832)
Rotation group[8,3]+, (832)
Dual polyhedronTruncated trioctagonal tiling
Face configurationV4.6.16
Propertiesface-transitive

The order 3-8 kisrhombille is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 16 triangles meeting at each vertex.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.6.16 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 16 triangles. It is the dual tessellation of the truncated trioctagonal tiling which has one square and one octagon and one hexakaidecagon at each vertex.

Naming

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An alternative name is 3-8 kisrhombille by Conway, seeing it as a 3-8 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

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This tiling is one of 10 uniform tilings constructed from [8,3] hyperbolic symmetry and three subsymmetries [1+,8,3], [8,3+] and [8,3]+.

Uniform octagonal/triangular tilings
Symmetry: [8,3], (*832) [8,3]+
(832)
[1+,8,3]
(*443)
[8,3+]
(3*4)
{8,3} t{8,3} r{8,3} t{3,8} {3,8} rr{8,3}
s2{3,8}
tr{8,3} sr{8,3} h{8,3} h2{8,3} s{3,8}
                                                     
     
    
     
    
     
    
           
     or     
     
     or     
     
    
     
 
 
 
 
 
             
 
Uniform duals
V83 V3.16.16 V3.8.3.8 V6.6.8 V38 V3.4.8.4 V4.6.16 V34.8 V(3.4)3 V8.6.6 V35.4
                                                                 
                     

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram      . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
 
[12i,3]
 
[9i,3]
 
[6i,3]
 
[3i,3]
Figures                        
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals                        
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

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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  • Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
  • Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
  • Hyperbolic and Spherical Tiling Gallery
  • KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
  • Hyperbolic Planar Tessellations, Don Hatch