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Truncated order-6 pentagonal tiling

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Truncated order-6 pentagonal tiling
Truncated order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 6.10.10
Schläfli symbol t{5,6}
t(5,5,3)
Wythoff symbol 2 6 | 5
3 5 5 |
Coxeter diagram
Symmetry group [6,5], (*652)
[(5,5,3)], (*553)
Dual Order-5 hexakis hexagonal tiling
Properties Vertex-transitive

In geometry, the truncated order-6 pentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2{6,5}.

Uniform colorings

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t012(5,5,3)

With mirrors
An alternate construction exists from the [(5,5,3)] family, as the omnitruncation t012(5,5,3). It is shown with two (colors) of decagons.

Symmetry

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The dual of this tiling represents the fundamental domains of the *553 symmetry. There are no mirror removal subgroups of [(5,5,3)], but this symmetry group can be doubled to 652 symmetry by adding a bisecting mirror to the fundamental domains.

Small index subgroups of [(5,5,3)]
Type Reflective domains Rotational symmetry
Index 1 2
Diagram
Coxeter
(orbifold)
[(5,5,3)] =
(*553)
[(5,5,3)]+ =
(553)
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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
[(5,5,3)] reflective symmetry uniform tilings

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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