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,/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) 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 truncated icosahedron pattern used onsoccer
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 withvertex 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 ,3Coxeter 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 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.See also
* Triangular tiling * Order-3 heptagonal tiling * Order-7 triangular tiling * Tilings of regular polygons * List of uniform tilingsReferences
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