Truncated order-7 triangular tiling

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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 CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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,[1] 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.

Comparison of truncated icosahedron and soccer ball.png
A truncated icosahedron
as a polyhedron and a ball		 Uniform tiling 63-t12.png
The Euclidean hexagonal tiling
colored as truncated
triangular tiling		 160px
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.

Order3 heptakis heptagonal til.png

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.

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.

See also

References

  1. ↑ HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL
  • 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)
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External links

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