Truncated order-6 pentagonal tiling

Truncated order-6 pentagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration6.10.10
Schläfli symbolt{5,6}
t(5,5,3)
Wythoff symbol2 6 | 5
3 5 5 |
Coxeter diagram
Symmetry group[6,5], (*652)
[(5,5,3)], (*553)
DualOrder-5 hexakis hexagonal tiling
PropertiesVertex-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


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

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)
[(5,5,3)] reflective symmetry uniform tilings

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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

  • Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
  • Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
  • Hyperbolic and Spherical Tiling Gallery
  • KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
  • Hyperbolic Planar Tessellations, Don Hatch
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