Truncated order-6 hexagonal tiling

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h₂{4,6}

Truncated order-6 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	6.12.12
Schläfli symbol	t{6,6} or h2{4,6} t(6,6,3)
Wythoff symbol	2 6 | 6 3 6 6 |
Coxeter diagram	= =
Symmetry group	[6,6], (*662) [(6,6,3)], (*663)
Dual	Order-6 hexakis hexagonal tiling
Properties	Vertex-transitive

Uniform colorings

By *663 symmetry, this tiling can be constructed as an omnitruncation, t{(6,6,3)}:

Symmetry

Truncated order-6 hexagonal tiling with *663 mirror lines

The dual to this tiling represent the fundamental domains of [(6,6,3)] (*663) symmetry. There are 3 small index subgroup symmetries constructed from [(6,6,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

The symmetry can be doubled as 662 symmetry by adding a mirror bisecting the fundamental domain.

Small index subgroups of [(6,6,3)] (*663)

Index	1	2		6
Diagram
Coxeter (orbifold)	[(6,6,3)] = (*663)	[(6,1+,6,3)] = = (*3333)	[(6,6,3+)] = (3*33)	[(6,6,3*)] = (*333333)
Direct subgroups
Index	2	4		12
Diagram
Coxeter (orbifold)	[(6,6,3)]+ = (663)	[(6,6,3+)]+ = = (3333)		[(6,6,3*)]+ = (333333)

Related polyhedra and tiling

Uniform hexahexagonal tilings v t e
Symmetry: [6,6], (*662)
= =	= =	= =	= =	= =	= =	= =
{6,6} = h{4,6}	t{6,6} = h2{4,6}	r{6,6} {6,4}	t{6,6} = h2{4,6}	{6,6} = h{4,6}	rr{6,6} r{6,4}	tr{6,6} t{6,4}
Uniform duals
V66	V6.12.12	V6.6.6.6	V6.12.12	V66	V4.6.4.6	V4.12.12
Alternations
[1+,6,6] (*663)	[6+,6] (6*3)	[6,1+,6] (*3232)	[6,6+] (6*3)	[6,6,1+] (*663)	[(6,6,2+)] (2*33)	[6,6]+ (662)
=		=		=
h{6,6}	s{6,6}	hr{6,6}	s{6,6}	h{6,6}	hrr{6,6}	sr{6,6}

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

• 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