Truncated order-7 triangular tiling

Truncated order-7 triangular tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	7.6.6
Schläfli symbol	t{3,7}
Wythoff symbol	2 7 | 3
Coxeter diagram
Symmetry group	[7,3], (*732)
Dual	Heptakis heptagonal tiling
Properties	Vertex-transitive

In geometry, the Order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t_1,2{7,3}.

Construction

This tiling is called a 'hyperbolic soccerball for its similarity to the truncated icosahedron pattern used on soccer balls. Portions of it can be constructed in 3-space as a nonplanar surface.

Truncated icosahedron as a polyhedron and a ball

a paper construction

Dual tiling

The dual tiling is called an order-3 heptakis heptagonal tiling, named for being constructible as an order-3 heptagonal tiling with every heptagon divided into seven triangles by the center point.

Related tilings

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732)							[7,3]+, (732)
{7,3}	t{7,3}	r{7,3}	t{3,7}	{3,7}	rr{7,3}	tr{7,3}	sr{7,3}
Uniform duals
V73	V3.14.14	V3.7.3.7	V6.6.7	V37	V3.4.7.4	V4.6.14	V3.3.3.3.7

See also

• Triangular tiling
• Order-3 heptagonal tiling
• Order-7 triangular tiling
• Tilings of regular polygons
• List of 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.

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
• PDF with instructions
• The first paper hyperbolic soccerball
• A rather large hyperbolic soccerball