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,/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'' h ...
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 n ...
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 squa ...
) with heptagons in place of pentagon
In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is ...
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 mor ...
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 squa ...
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 aheptagonal 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 topologicall ...
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 uniformtruncated
Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones.
Truncation may also refer to:
Mathematics
* Truncation (statistics) refers to measurements which have been cut of ...
polyhedra with vertex configuration
In geometry, a vertex configurationCrystallography ...
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
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, 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 inHyperRogue
''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 tur ...
.
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 t ...
* Tilings of regular polygons
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his '' Harmonices Mundi'' ( Latin: ''The Harmony of the World'', 1619).
Notation of ...
* List of uniform tilings
This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.
There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their dual ...
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 branch ...
, 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