Truncated order-4 pentagonal tiling

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Truncated pentagonal tiling
Truncated order-4 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.10.10
Schläfli symbol t{5,4}
Wythoff symbol 2 4 | 5
2 5 5 |
Coxeter diagram
or
Symmetry group [5,4], (*542)
[5,5], (*552)
Dual Order-5 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{5,4}.

Uniform colorings

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A half symmetry [1+,4,5] = [5,5] coloring can be constructed with two colors of decagons. This coloring is called a truncated pentapentagonal tiling.

 

Symmetry

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There is only one subgroup of [5,5], [5,5]+, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.

Small index subgroups of [5,5]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram    
Coxeter
(orbifold)
[5,5] =       =     
(*552)
[5,5]+ =       =     
(552)
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*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Truncated
figures
               
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
n-kis
figures
               
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
                                                           
                   
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
                                                           
                 
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55
Uniform pentapentagonal tilings
Symmetry: [5,5], (*552) [5,5]+, (552)
     
=      
     
=      
     
=      
     
=      
     
=      
     
=      
     
=      
     
=      
               
Order-5 pentagonal tiling
{5,5}
Truncated order-5 pentagonal tiling
t{5,5}
Order-4 pentagonal tiling
r{5,5}
Truncated order-5 pentagonal tiling
2t{5,5} = t{5,5}
Order-5 pentagonal tiling
2r{5,5} = {5,5}
Tetrapentagonal tiling
rr{5,5}
Truncated order-4 pentagonal tiling
tr{5,5}
Snub pentapentagonal tiling
sr{5,5}
Uniform duals
                                               
             
Order-5 pentagonal tiling
V5.5.5.5.5
V5.10.10 Order-5 square tiling
V5.5.5.5
V5.10.10 Order-5 pentagonal tiling
V5.5.5.5.5
V4.5.4.5 V4.10.10 V3.3.5.3.5

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.

See also

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