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{3,7}.

Hyperbolic soccerball (football)

This tiling is called a hyperbolic soccerball (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space.

File:Comparison of truncated icosahedron and soccer ball.png A truncated icosahedron as a polyhedron and a ball

File:Uniform tiling 63-t12.svg The Euclidean hexagonal tiling colored as truncated triangular tiling

File:Hyperbolicsoccerball.jpg A paper construction of a hyperbolic soccerball

Dual tiling

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

File:Heptakis heptagonal tiling.svg

Related tilings

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6
Sym. *n42 [n,3]	Spherical				Euclid.	Compact		Parac.	Noncompact hyperbolic
	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]
Truncated figures	File:Hexagonal dihedron.svg	File:Uniform tiling 332-t12.svg	File:Uniform tiling 432-t12.png	File:Uniform tiling 532-t12.png	File:Uniform tiling 63-t12.svg	Error creating thumbnail:	File:H2-8-3-trunc-primal.svg	File:H2 tiling 23i-6.png	File:H2 tiling 23j12-6.png	File:H2 tiling 23j9-6.png	File:H2 tiling 23j-6.png
Config.	2.6.6	3.6.6	4.6.6	5.6.6	6.6.6	7.6.6	8.6.6	∞.6.6	12i.6.6	9i.6.6	6i.6.6
n-kis figures	File:Hexagonal Hosohedron.svg	File:Spherical triakis tetrahedron.svg	File:Spherical tetrakis hexahedron.svg	File:Spherical pentakis dodecahedron.png	File:Uniform tiling 63-t2.svg	File:Heptakis heptagonal tiling.svg	File:H2-8-3-kis-dual.svg	File:H2checkers 33i.png
Config.	V2.6.6	V3.6.6	V4.6.6	V5.6.6	V6.6.6	V7.6.6	V8.6.6	V∞.6.6	V12i.6.6	V9i.6.6	V6i.6.6

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.

In popular culture

This tiling features prominently in HyperRogue.

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-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)

External links

• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
• Geometric explorations on the hyperbolic football by Frank Sottile