geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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

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 is de ...

s and one

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

on each vertex, forming a pattern similar to a conventional soccer ball (

truncated icosahedron In geometry, the truncated icosahedron is a polyhedron that can be constructed by Truncation (geometry), truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as Ball (association football), footballs (or soccer ...

) with heptagons in place of

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

s. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

of t.

Hyperbolic soccerball (football)

This tiling is called a hyperbolic soccerball (football) for its similarity to the

pattern used on

soccer Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 Football player, players who almost exclusively use their feet to propel a Ball (association football), ball around a rectangular f ...

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 configuration In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...

s (n.6.6), and ,3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

symmetry. From a

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

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.

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

* 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