Order-6 Pentagonal Tiling on:
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Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Isohedral tilings Order-6 tilings Pentagonal tilings Regular tilings
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-6 pentagonal tiling is a regular tiling of the hyperbolic plane
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
. 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 .
Uniform coloring
This regular tiling can also be constructed from 5,5,3)symmetry alternating two colors of pentagons, represented by t1(5,5,3). :
Symmetry
This tiling represents a hyperbolickaleidoscope
A kaleidoscope () is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a regular symmetrical pattern when v ...
of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
is called *33333 with 5 order-3 mirror intersections.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices withSchläfli 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 ...
, and Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...
, progressing to infinity.
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)
*
See also
*Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of the s ...
*Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive ( transitive on its v ...
*List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
External links
* *Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch {{Tessellation Hyperbolic tilings Isogonal tilings Isohedral tilings Order-6 tilings Pentagonal tilings Regular tilings