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Order-5 hexagonal tiling

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Order-5 hexagonal tiling
Order-5 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 65
Schläfli symbol {6,5}
Wythoff symbol 5 | 6 2
Coxeter diagram
Symmetry group [6,5], (*652)
Dual Order-6 pentagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-5 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,5}.

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This tiling is topologically related as a part of sequence of regular tilings with order-5 vertices with Schläfli symbol {n,5}, and Coxeter diagram , progressing to infinity.

Spherical Hyperbolic tilings

{2,5}

{3,5}

{4,5}

{5,5}

{6,5}

{7,5}

{8,5}
...
{∞,5}

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings

{6,2}

{6,3}

{6,4}

{6,5}

{6,6}

{6,7}

{6,8}
...
{6,∞}
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

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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