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)[edit]

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
Hyperbolicsoccerball.jpg
A paper construction
of a hyperbolic soccerball

Dual tiling[edit]

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[edit]

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 hyperb. 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
Hexagonal dihedron.png Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.png H2 tiling 237-6.png H2 tiling 238-6.png H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png 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
Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.png Order3 heptakis heptagonal til.png H2checkers 334.png 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.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 73-t0.png Uniform tiling 73-t01.png Uniform tiling 73-t1.png Uniform tiling 73-t12.png Uniform tiling 73-t2.png Uniform tiling 73-t02.png Uniform tiling 73-t012.png Uniform tiling 73-snub.png
{7,3} t{7,3} r{7,3} 2t{7,3}=t{3,7} 2r{7,3}={3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 73-t2.png Ord7 triakis triang til.png Order73 qreg rhombic til.png Order3 heptakis heptagonal til.png Uniform tiling 73-t0.png Deltoidal triheptagonal til.png Order-3 heptakis heptagonal tiling.png Ord7 3 floret penta til.png
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[edit]

References[edit]

  1. ^ HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL

External links[edit]