Tessellations
From Math Images
| Tiling of the Alhambra |
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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.
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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:
- 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.
- Squares
- Squares form a very simple tessellation.
- 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.
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.
For more information
- Another page on the Math Form site has a basic description of how to figure out which polygons tessellate.
- Mathworld has a more advanced description of tessellations, and has several interesting images.
A More Mathematical Explanation
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.
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