Tessellations...Are They Math?

The more math I learn the less it fits into the conventional definition I once had of mathematics. This semester I am taking my capstone class, The Nature of Modern Mathematics, and this class has been pushing the boundaries of what I consider mathematics. Our discussions have ranged from talking about mathematicians such as Archimedes, Brahmagupta, and Leonardo of Pisa (also known as Fibonacci) to talking about the Pythagorean Theorem and arithmetic in Roman numerals. I have been learning such a rich history of mathematics but there is one topic in particular that has stood out to me, Islamic tessellations.

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)

There is no denying that these tessellations are masterpieces, but are they mathematical? After some research on tessellations (check out this awesome Wikipedia page!) I have discovered that there are three types of tessellations: regular, semi-regular, and irregular.

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

Johannes Kepler was one of the first people to study regular and semi-regular tessellations in 1619. In 1819 Evgraf Stepanovich Fedorov marked the unofficial beginning of the mathematical study of tessellations when he discovered that every periodic tiling contains one of seventeen different groups of isometrics, now known as wallpaper groups. Some people claim that all seventeen of these wallpaper groups are represented in the Alhambra Palace in Granada, Spain but it is still being debated.

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!