# Aesthetic Explorations: Tessellations

A tessellation refers to the repeated use of geometric tiles to fill a larger space without any gaps. In this way, tessellations combine both artistic ideals and mathematical problem solving. Initially though, tessellations served a more decorative purpose, dating back all the way to 4000 BCE where Sumerians utilized them in patterned clay walls [1].

A Sumerian temple featuring tessellations [1].

Tessellations were also common in Ancient Rome and Islamic art, but it wasn’t until 1691 that the previously decorative trend found its way into traditional mathematics. In Johannes Kepler’s Harmonices Mundi (Harmonics of the World), polyhedrons are explored in the context of polygons creating more complex three-dimensional shapes about a fixed, central point. This book provided the platform for mathematicians to more deeply explore the previously artistic combinations of polygons known as tessellations [2].

In 1879, Evgraf Fedorov published Basics of Polytopes, in which he classifies 17 wallpaper groups in which tiles can fill a planar surface. Wallpaper groups are a mathematical classification for patterns frequently used in textiles and architecture. As an example, the wallpaper group pm refers to patters with horizontal and/or vertical repetition of tiles [3].

An example of wallpaper group pm [3]. Credit: Owen Jones – The Grammar of Ornament (1856), by Owen Jones, plate 11: “Egyptian Nº 8”, image #20.

The work of Fedorov opened the door for further mathematical analysis of tessellations but wasn’t pursued further artistically until M.C. Escher. Escher was born in 1898 in the Netherlands. While he faced little recognition during his lifetime, he has since become famous for his works combining art, mathematics, and illusions [4].

Based on early sketchers from Escher, it is theorized that his work in tessellations were inspired by textiles seen in a Spanish palace, known as Alhambra [4].

Alhambra tessellations [4].

M.C. Escher referred to tessellations as regular divisions of the plane, which was also the title of a book he published in 1958. In Regular Division of the Plane, Escher included 137 drawings illustrating tessellations [5].

Reptile tessellation [4]. Credit: Pencil, India ink and watercolour drawing Study of Regular Division of the Plane with Reptiles.jpg by M. C. Escher

As can be seen above, Escher was largely inspired by nature. While the geometry behind determining this pattern is impressive, Escher didn’t stop there. He furthered his work by bringing life to mathematical concepts. This idea was furthered in many of his works in Regular Division of the Plane.

Horse and rider tessellation [5]. Credit: Regular Division of the Plane III by M. C. Escher. Woodcut in black on wove paper, 1957–1958.

In summation, tessellations provide in an aesthetic that is pervasive through many aspects of our lives. It combines mathematical problem solving with decorative patterns. The phenomenon can even be found in nature.

Honeycomb is another example of tessellations [1].

Therefore, it requires a balanced and well-versed individual to pursue tessellations in a design context. They must understand patterns, mathematics, and the natural world.

[1] https://en.wikipedia.org/wiki/Tessellation

[2] https://en.wikipedia.org/wiki/Harmonices_Mundi

[3] https://en.wikipedia.org/wiki/Wallpaper_group

[4] https://en.wikipedia.org/wiki/M._C._Escher

[5] https://en.wikipedia.org/wiki/Regular_Division_of_the_Plane

• Well done Jackson,
I saw the tessellations in la Alhambra in person. They are phenomenal. I have never seen such intricate tessellations of creatures like the lizards before. I am pretty impressed. I appreciate that tessellations are often still utilized in tile floors, I have seen many beautiful ones in Mexico in particular.

• Jackson Hootman
January 27, 2020 9:49 am

Thanks Jamie! I’d love to see some pictures from the tile floors you saw in Mexico if you have any. As you have a first hand account of, tessellations are frequent in so many different cultures all over the world. Yet another reason why they are so interesting to me.

• Hi Jackson,

I think it’s super cool how humanity has been fascinated with mathematics and integrated the patterns we see to our buildings and art. I really like how thorough you were with your examples and examples. I also enjoyed how your blog post started with where it’s originated to more modern examples. It also showcases how it goes from basic geometric shapes into more complex designs that can be repeated. Your aesthetic immediately reminded me of the Dome on the Rock in Jerusalem.

I think it would have been nice to talk a little more in-depth or just generally more about how Tessellations affect the modern world, whether it’d be in art or math.

I very much enjoyed your aesthetic and the information you provided.