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Hyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch Hyperbolic tilings Isogonal tilings Uniform tilings
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 rhombitetrahexagonal tiling is a uniform 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 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 rr. It can be seen as constructed as a rectified tetrahexagonal tiling
In geometry, the tetrahexagonal tiling is a uniform tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol r.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal ...
, r, as well as an expanded order-4 hexagonal tiling
In geometry, the order-4 hexagonal tiling is a List of regular polytopes#Hyperbolic tilings, regular tiling of the Hyperbolic geometry, hyperbolic plane. It has Schläfli symbol of .
Symmetry
This tiling represents a hyperbolic kaleidoscope of ...
or expanded order-6 square tiling.
Constructions
There are two uniform constructions of this tiling, one from ,4or (*642) symmetry, and secondly removing the mirror middle, ,1+,4 gives a rectangular fundamental domain ˆž,3,∞ (*3222). There are 3 lower symmetry forms seen by including edge-colorings: sees the hexagons as truncated triangles, with two color edges, with ,4+(4*3) symmetry. sees the yellow squares as rectangles, with two color edges, with +,4(6*2) symmetry. A final quarter symmetry combines these colorings, with +,4+(32×) symmetry, with 2 and 3 fold gyration points and glide reflections. This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of . :
Symmetry
The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is aLambert quadrilateral
In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest s ...
, with 3 right angles. This symmetry can be seen from a ,4 (*642) triangular symmetry with one mirror removed, constructed as ,1+,4 (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.
:
Related polyhedra and tiling
See also
*Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.
Structure and properties
The square tili ...
*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 (Latin: ''The Harmony of the World'', 1619).
Notation of Euclidean tilings
Eucl ...
*List of uniform planar 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 du ...
*List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
References
* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations) *External links
* *{{MathWorld , urlname=PoincareHyperbolicDisk , title = Poincaré hyperbolic diskHyperbolic and Spherical Tiling Gallery
* ttp://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch Hyperbolic tilings Isogonal tilings Uniform tilings