# Edit Edit an Image Page: Tessellations

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 Image Title*: Upload a Math Image This is a tiling in the Alhambra in Spain, one of the many beautiful designs laid out by the Moors in the 14th Century. 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, semi-regular, 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: {{{!}} border=0 cellpadding=10 cellspacing=10 {{!}} [[Image:triangle2.gif|frame|Tessellation with triangles.]] {{!}}{{!}} '''Equilateral Triangles''' : Equilateral triangles can form a regular tessellation, since they are a regular polygon and they can be arranged with no space in between shapes. Other types of triangles can also be used to make irregular tessellations since they are not regular polygons. {{!}}- {{!}} [[Image:squ1.gif|frame|Tessellation with squares.]] {{!}}{{!}} '''Squares''' :Squares form a very simple tessellation. {{!}}- {{!}} [[Image:hexagon2.gif|frame|Tessellation with rectangles.]] {{!}}{{!}} '''Regular Hexagons''' :A regular hexagon is a six-sided regular polygon, and it also tessellates. {{!}}} ===Semi-regular Tessellations=== '''Semi-regular 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 semi-regular tessellations.
[[Image:tess312.gif]] [[Image:tess63.gif]] [[Image:tess36.gif]] [[Image:tess48.gif]] [[Image:tess461.gif]] ===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. [[Image:tess1.gif|200px]] ==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 jig-saw 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. {{SwitchPreview|ShowMessage=Click to show gallery displaying examples of tessellations in nature. |HideMessage=Click to hide gallery |PreviewText=|FullText= This gallery showcases some examples of tessellations in art and the world at large.
file=Tess_gallery.swf|width=500|height=400
}} ==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. {{{!}} border=0 cellpadding=10 cellspacing=10 align=center {{!}} [[Image:Penrose1.gif|frame|The two quadrilaterals that make up the Penrose tiling. They are only allowed to be placed so that the red and blue lines match up.]] {{!}}{{!}} [[Image:Penrose2.gif|thumb|400px|A larger example of Penrose tiling, which can repeat infinitely without repetition or symmetry.]] {{!}}} ==Tessellations in Non-Euclidian 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 semi-regular tessellation on a sphere. In the image below, the hexagons are white and the pentagons are black. [[Image:Soccerball.png|200px]]

Tessellations can also be formed on [[Hyperbolic Geometry | hyperbolic surfaces]]. For more information, check out the separate page on [[Hyperbolic Tilings]] Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other ''For more information, visit:'' *http://mathforum.org/~sanders/geometry/GP07Tessellations.html *http://mathworld.wolfram.com/Tessellation.html Yes, it is.