So what exactly are tessellations? Tessellations are tilings of the plane using one or more geometric shapes, called tiles. The important part though, is that those tiles cannot overlap, nor can there be any gaps between the tiles. In order to understand Islamic tessellations, it is important to understand some beliefs of the Islamic faith. Most interpretations of Islamic law discouraged the portrayal of humans or animals in art for fear that it would cause people to idolize those humans or animals. Therefore Islamic art centers around three main elements: calligraphy in Arabic script, floral and plant-like designs, and geometrical designs.

The Islamic geometric designs often included a lot of repetition and variations. Even though the designs may have only consisted of a couple shapes, the patterns those shapes created couldn't have more variety. The Islamic tessellations also included a lot of symmetry, and many designs included translations and rotations of tiles as well. Here are some famous examples of Islamic tessellations.

The Alhambra in Granada, Spain (Google Images) |

The Alhambra in Granada, Spain (Google Images) |

The Alhambra in Granada, Spain (Google Images) |

Alcazar in Sevilla, Spain (Google Images) |

Alcazar in Sevilla, Spain (Getty Images) |

Alcazar in Sevilla, Spain (Getty Images) |

Alcazar in Sevilla, Spain (Getty Images) |

Regular tessellations have both regular tiles and identical regular corners and vertices. There are only three shapes that can form regular tessellations, equilateral triangles, squares, and regular hexagons.

Semi-regular tessellations are tessellations that use regular tiles of more than one shape with every corner identically arranged. There are eight semi-regular tilings.

Irregular tessellations are tessellations made from shapes such as pentagons that are not regular. M.C. Escher was an artist who was famous for creating tessellations that used tiles shaped like animals, humans, or other objects.

Two Lizards (1942) M.C. Escher |

There is no doubt that there is a lot of sophisticated patterning behind tessellations. Not only are they visually stunning but they are also rich in mathematics. I encourage you to try creating some of your own tessellations, here are some great resources to get you started!

John Golden: Islamic Tessellation Helper

Victoria and Albert Museum: Maths and Islamic Art & Design

The Metropolitan Museum of Art: Islamic Art and Geometric Design

Also here are links to isometric and square paper.

Happy Tessellating!

Of course I like, especially all those great images from the Alcazar and the Alhambra. I think you could benefit here by tying back to your question/point of stretching your idea of mathematics. (How might you describe your old idea? new?) Either in the conclusion arguing why these are math, or in the examples, pointing out the math. Or in the history - why was Kepler interested? (I didn't know that, actually.) [consolidated]

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