Tessellations

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Tiling of the Alhambra
Field: Geometry
Image Created By: Tessellations.org

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, 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:

Tessellation with triangles.
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.
Tessellation with squares.
Tessellation with squares.

Squares

Squares form a very simple tessellation.
Tessellation with rectangles.
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.

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.


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.

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.
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.
A larger example of Penrose tiling, which can repeat infinitely without repetition or symmetry.
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.



Tessellations can also be formed on hyperbolic surfaces. For more information, check out the separate page on Hyperbolic Tilings




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