Tessellations
Tiling of the Alhambra 

Tiling of the Alhambra
 This is a tiling in the Alhambra in Spain, one of the many beautiful designs laid out by the Moors in the 14th Century.
Contents
Basic Description
Tessellations, more commonly referred to as tilings, are patterns which are repeated over and over without overlapping or leaving any gaps. Tessellations are seen throughout art history from ancient architecture to modern art.
Tessellations can be regular, semiregular, or irregular.
Regular Tessellations
Regular tessellations are made up of polygons which are regular and congruent . We say that a shape can tessellate if it can form a regular tessellation, and there are only three regular polygons which can tessellate on the Euclidean plane:
Equilateral Triangles
 
Squares
 
Regular Hexagons

Semiregular Tessellations
Semiregular tessellations, also known as Archimedean tessellations, are formed by two or more regular polygons whose arrangement at every vertex are identical. Below are examples of semiregular tessellations.
Irregular Tessellations
Irregular tessellations encompass all other tessellations, including the tiling in the main image. Many other shapes, including ones made up of complex curves can tessellate. The image below is an example of an irregular tessellation.
Tessellations in Real life
Tessellations are a combination of math, art and fun, in this regard there are numerous applications in real life ranging from the patterns on floors to jigsaw puzzles. Tessellations are observed in some works of great artists like M.C. Escher. Examples of beautiful tessallations in nature are cracking patterns in dried mud or pottery, cellular structures in Biology and and crystals in metallic ingots.
This gallery showcases some examples of tessellations in art and the world at large.
A More Mathematical Explanation
Asymmetric Tessellations
Asymmetric tessellations are ones that have no translational symmetry. [...]Asymmetric Tessellations
Asymmetric tessellations are ones that have no translational symmetry. A well known example was discovered by Roger Penrose and is known as the Penrose Tiles. The Penrose Tiles are a pair of quadrilaterals that can tile a plane infinitely without repeating. This is also called aperiodic tiling.
Tessellations in NonEuclidian Geometry
Shapes can be tessellated on surfaces other than the plane, such a spheres. A soccer ball is covered in hexagons and pentagons, which form a semiregular tessellation on a sphere. In the image below, the hexagons are white and the pentagons are black.
Tessellations can also be formed on hyperbolic surfaces. For more information, check out the separate page on Hyperbolic Tilings
Teaching Materials
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References
For more information, visit:
 http://mathforum.org/~sanders/geometry/GP07Tessellations.html
 http://mathworld.wolfram.com/Tessellation.html
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