Truncated tetraheptagonal tiling

Truncated tetraheptagonal tiling
Truncated tetraheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.8.14
Schläfli symbol tr{7,4} or
Wythoff symbol 2 7 4 |
Coxeter diagram
Symmetry group [7,4], (*742)
Dual Order-4-7 kisrhombille tiling
Properties Vertex-transitive

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.

Images

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Poincaré disk projection, centered on 14-gon:

 

Symmetry

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Truncated tetraheptagonal tiling with mirror lines.      

The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are three small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

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Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
                                                           
                   
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
                                                           
               
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
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]
Omnitruncated
figure
 
4.8.4
 
4.8.6
 
4.8.8
 
4.8.10
 
4.8.12
 
4.8.14
 
4.8.16
 
4.8.∞
Omnitruncated
duals
 
V4.8.4
 
V4.8.6
 
V4.8.8
 
V4.8.10
 
V4.8.12
 
V4.8.14
 
V4.8.16
 
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure                
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual                
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞

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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