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 t_1,2{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)

Related polyhedra and tiling

Uniform hexagonal/pentagonal tilings v t e
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

• 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 6-10-10.

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

External links

• 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

This page was last edited on 12 December 2023, at 20:12