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In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball,HOW TO BUILD YOUR OWN HYPERBOLIC SOCCER BALL MODEL
/ref> is a semiregular tiling of the hyperbolic plane. There are two
Hyperbolic and Spherical Tiling Gallery
* ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch
Geometric explorations on the hyperbolic football by Frank Sottile
{{Tessellation Hyperbolic tilings Isogonal tilings Order-7 tilings Semiregular tilings Triangular tilings Truncated tilings
/ref> is a semiregular tiling of the hyperbolic plane. There are two
hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
s and one heptagon on each vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
, forming a pattern similar to a conventional soccer ball
A football (also known as football ball, soccer ball, or association football ball specifically in the United Kingdom) is the ball used in the sport of association football. The name of the ball varies according to whether the sport is called " ...
(truncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...
) with heptagons in place of pentagons. It has Schläfli symbol of t.
Hyperbolic soccerball (football)
This tiling is called a hyperbolic soccerball (football) for its similarity to thetruncated icosahedron
In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares ...
pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space.
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. :
Related tilings
This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and ,3 Coxeter group symmetry. From a Wythoff construction there are eight hyperbolicuniform tilings
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, s ...
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
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...
* Order-3 heptagonal tiling
* Order-7 triangular tiling
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of .
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is ...
* Tilings of regular polygons
Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...
* List of uniform tilings
References
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
*
External links
* *Hyperbolic and Spherical Tiling Gallery
* ttp://www.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch
Geometric explorations on the hyperbolic football by Frank Sottile
{{Tessellation Hyperbolic tilings Isogonal tilings Order-7 tilings Semiregular tilings Triangular tilings Truncated tilings