Pentahexagonal tiling

Pentahexagonal tiling
Pentahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (5.62
Schläfli symbol r{6,5} or
Wythoff symbol 2 | 6 5
Coxeter diagram
Symmetry group [6,5], (*652)
Dual Order-6-5 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the pentahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t1{6,5}.

Uniform colorings

Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5
*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
Symmetry mutation of quasiregular tilings: (6.n)2
Symmetry
*6n2
[n,6] 		Euclidean Compact hyperbolic Paracompact Noncompact
*632
[3,6] 		*642
[4,6] 		*652
[5,6] 		*662
[6,6] 		*762
[7,6] 		*862
[8,6]... 		*∞62
[∞,6] 		 
[iπ/λ,6]
Quasiregular
figures
configuration
6.3.6.3
6.4.6.4
6.5.6.5
6.6.6.6
6.7.6.7
6.8.6.8
6.∞.6.∞
6.∞.6.∞
Dual figures
Rhombic
figures
configuration
V6.3.6.3
V6.4.6.4
V6.5.6.5
V6.6.6.6
V6.7.6.7
V6.8.6.8
V6.∞.6.∞

References

  • 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

Wikimedia Commons has media related to Uniform tiling 5-6-5-6.