Tetrapentagonal tiling
Tetrapentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.5)2
Schläfli symbol r{5,4} or
rr{5,5} or
Wythoff symbol 2 | 5 4
5 5 | 2
Coxeter diagram or
or
Symmetry group [5,4], (*542)
[5,5], (*552)
Dual Order-5-4 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1{4,5} or r{4,5}.

Symmetry

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A half symmetry [1+,4,5] = [5,5] construction exists, which can be seen as two colors of pentagons. This coloring can be called a rhombipentapentagonal tiling.

 

Dual tiling

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The dual tiling is made of rhombic faces and has a face configuration V4.5.4.5:

 
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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
*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
 
[ni,4]
Figures              
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
*5n2 symmetry mutations of quasiregular tilings: (5.n)2
Symmetry
*5n2
[n,5]
Spherical Hyperbolic Paracompact Noncompact
*352
[3,5]
*452
[4,5]
*552
[5,5]
*652
[6,5]
*752
[7,5]
*852
[8,5]...
*∞52
[∞,5]
 
[ni,5]
Figures              
Config. (5.3)2 (5.4)2 (5.5)2 (5.6)2 (5.7)2 (5.8)2 (5.∞)2 (5.ni)2
Rhombic
figures
       
Config. V(5.3)2 V(5.4)2 V(5.5)2 V(5.6)2 V(5.7)2 V(5.8)2 V(5.∞)2 V(5.∞)2

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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