Snub tetraapeirogonal tiling

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Snub tetraapeirogonal tiling
Snub tetraapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.3.4.3.∞
Schläfli symbol sr{∞,4} or
Wythoff symbol | ∞ 4 2
Coxeter diagram or
Symmetry group [∞,4]+, (∞42)
Dual Order-4-infinite floret pentagonal tiling
Properties Vertex-transitive Chiral

In geometry, the snub tetraapeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{∞,4}.

Images

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Drawn in chiral pairs, with edges missing between black triangles:

  
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The snub tetrapeirogonal tiling is last in an infinite series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
               
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
       
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
Paracompact uniform tilings in [∞,4] family
                                         
             
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
                                         
             
V∞4 V4.∞.∞ V(4.∞)2 V8.8.∞ V4 V43.∞ V4.8.∞
Alternations
[1+,∞,4]
(*44∞)
[∞+,4]
(∞*2)
[∞,1+,4]
(*2∞2∞)
[∞,4+]
(4*∞)
[∞,4,1+]
(*∞∞2)
[(∞,4,2+)]
(2*2∞)
[∞,4]+
(∞42)
     
=    
                       
=    
           
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
       
Alternation duals
                                         
   
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞ V∞.44 V3.3.4.3.∞

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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