geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°. Regular hexa ...

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 "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...

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

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

s. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

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 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...

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 In geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of , having three regular heptagons around each vertex. Images Related polyhedra and tilings This tiling is topologically r ...

with every heptagon divided into seven triangles by the center point. : Heptakis heptagonal tiling

Related tilings

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with

vertex configuration In geometry, a vertex configurationCrystallography ...

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 refl ...

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 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, se ...

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 ''HyperRogue'' is an independent video game developed by Zeno Rogue. It is a roguelike inspired by the puzzle game ''Deadly Rooms of Death'' and the art of M. C. Escher, taking place in the hyperbolic plane. Gameplay ''HyperRogue'' is a t ...

References

John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...

, 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