# The Many Symmetries of Tessellations ft. Harry’s Tiles

Tessellations

Before we get started, I need some help. I keep hearing about Harry’s Tiles *everywhere I go*, and I can’t for the life of me figure out what people are talking about?! Let me give you some examples:

*“Wow, did you see that Vogue featured Harry’s Tiles?”* I thought Vogue was about fashion, not interior design?

*“I’m such a big fan of Harry’s Tiles music!”* So like, clinks? I don’t get the appeal.

*“I think I’m in love with Harry’s Tiles…”* Wow, they must be really beautiful!

Naturally, I wanted to figure out what all the hype was about. I re-read all seven of the Harry Potter books, and didn’t see anything about tiles. Then, I googled it–as one does when one is at a loss– but all I found was a blatantly fake tiling company. So, I decided to just look into tiles, and hoped I’d find some famous Harry associated with it. And, what do you know, tiles are actually really interesting!

So today we’re going to talk about a sort of generalization of tiling: tessellations! Maybe along the way we’ll solve the puzzle of who exactly Harry is and why his tiles are so famous.

**An Introduction to Tessellations**

So, what are tessellations? Well, you can think of it like perfect patterns!

More specifically, a tessellation (or tiling) of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Chances are you can just walk over to the bathroom and stare at the floor to see an example of a tessellation. But, because I’m nice and don’t want people to look at you weirdly, here are some other examples:

You may have noticed that the third example is a little different from the other two–we’ll talk about that later!

The word “tessellation” comes from the Latin word “tessellātus,” meaning small square stones or cubes, also made from clay or glass. These were used to create tessellata, i.e. beautiful Roman mosaic floors/tilings.

This Latin word comes from the Greek word “tessera,” meaning four, as in how squares have four sides. The word’s etymology suggests that the very first tilings were made from square shapes, which makes sense given that squares are the easiest shape to tile with!

The first examples of such tessellations can be traced back to the Sumerians in around 4000 BCE, when they used clay tiles to adorn their homes and temples. Since then, tessellations have been found in several civilizations, including alongside the Egyptians, Arabs, Romans, Greeks, Byzantines, Japanese, Chinese, and Moors. (No mention of Harry here, but it’s okay, we’ve just gotten started!)

Before we talk about the impact of tessellations on art and architecture, let’s look at the deep mathematics of tessellations!

**Types of Tessellations**

I think we can all agree that tessellations are pretty satisfying to look at–everything fits together so perfectly! You can see that there are repeating motifs/patterns throughout tessellations, and this plays into the symmetry of most tessellations. Hold up, what is symmetry? Well, one example is reflectional symmetry: if you hold an object in the mirror and it looks exactly the same, it has symmetry! Here’s a quick puzzle that may help: What do the letters A, W, T, Y, U, I, O, H, X, V, M have in common with each other that the other English alphabet letters do not have?

Given what we were just talking about, it is indeed symmetry! Each of the given letters is symmetrical about the vertical line that splits it in half, whereas a letter like P is not.

There are three main types of symmetry in a plane: translation, rotation, and glide reflection. If you don’t know what these mean, don’t worry! They aren’t the focus of this article. Instead, let’s talk about the general classifications of tessellations.

The first type is **regular tessellations**. No, they aren’t the boring ones. In math, the word “regular” describes any shape that is **equilateral**, meaning all its sides have equal length, and **equiangular**, meaning all its interior angles are equal. There are exactly three regular shapes that make up regular tessellations (ones that you are likely familiar with): equilateral triangles, squares, and regular hexagons. You can see these three regular tessellations in the following picture!

The next type is **semiregular tessellations**, also known as Archimedean tessellations (after the famous Greek mathematician Archimedes). In regular tessellations, there’s only one type of shape being used. But for these tessellations, we use more than one kind of regular shape! Variety is truly the spice of life. However, there is the rule that the same shapes in the same order must surround each vertex (i.e. corner) of the shapes.

There are exactly 8 semiregular tessellations, shown here!

Another type is **monohedral tessellations**. Here, we can notice that “mono” means one and “hedral” means shape, and we would be correct in thinking that these tessellations are made up of only shape! However, these shapes don’t have to be regular–they can be rotated or flipped to fit into the tessellation, but their size can’t change. In fact, every triangle and quadrilateral can be tessellated in at least one way! (Try drawing a triangle and quadrilateral, and see if you can figure out how to tile it.)

Here are some examples!

So, if we’re no longer talking about regular polygons (meaning shapes), can shapes other than triangles, quadrilaterals, and hexagons tessellate? Wow, what a good question, yes!

In fact, pentagons can tessellate! There are currently 15 known classes of pentagons that can tessellate, and, as expected, these are called **pentagonal tessellations**. If you’re interested, all fifteen of them can be generated here: https://demonstrations.wolfram.com/PentagonTilings/.

They are also shown here:

Okay, cool! So heptagons (7-sided shapes), octagons (8-sided), nonagons (9-sided), decagons (10-sided), etc. can tessellate too, right? Sorry to break the bad news to you bud, but shapes with more than six sides don’t tessellate unless one of their angles is greater than 180 degrees, i.e. if they are a **concave** polygon.

Now, to be completely honest, I’m feeling a bit lazy right now. I’ve drawn one tessellation, but I don’t have the energy to create an entirely new one…Thankfully I don’t have to! There is an underlying connection between some of these geometric tessellations: **duality**! To create a dual from your tessellation, draw a dot to mark the center of each shape and connect them to the neighboring shapes’ centers. Then, erase the original pattern, and voila! There’s your new tessellation. Here are some visuals to demonstrate:

(As a fun little puzzle, see if you can figure out what the notation means. Hint: choose a shape and look at its neighbors!)

Finally, let’s talk about **aperiodic tessellations**! Most tessellations are grounded in their repetition, but it’s not a requirement. What? Aren’t tessellations made from repeated shapes?! Well yes, but the arrangement of these shapes are unique, so these patterns are called **aperiodic** (…because they aren’t periodic). A famous example is the Penrose Tiling, named after English mathematical physicist Sir Roger Penrose. These tilings display five-fold symmetry, which is something that doesn’t appear in any of the (periodic) tessellations we’ve seen so far!

**Tessellations in Islamic Art and Architecture**

In fact, aperiodic tessellations have rich, vibrant history. They can be found extensively in medieval Islamic architecture, and were used in Islamic art and buildings at least half a millenium before they emerged in the West! For example, you can see this five-fold symmetrical aperiodic tessellation on the exterior of an 1197 tomb tower in Maragha, Iran, the** Gunbad-i Qabud.**

Now, I looked around this building, but unfortunately still couldn’t find a Sign of the (Harry’s) Tiles. Regardless, let’s just stop our crying and keep going!

In fact, this concept of tessellations, i.e. beautiful geometric objects, were perfect for Islamic art as Islam forbids representing living figures in art. Hence, Islamic artists fully explored and developed the abstract beauty of tessellations over centuries of incredible artwork.

Islamic architecture’s embrace of tessellations inspired the works of an artist that this article would be incomplete without: Maurits Cornelis Escher, known as **M.C. Escher**, a 20th century Dutch graphic artist.

**M.C. Escher**

As the story goes, M.C. Escher visited the Alhambra, a fortress-turned-palace constructed by the Muslim Moors in the 14th century–an extraordinary display of tessellations. As an aspiring artist, Escher copied these patterns into his notebooks and later excitedly shared his sketches with his brother. His brother then showed him a 1924 paper by Hungarian Mathematician George Pólya. This paper claimed that there were 17 ways a pattern can be used to tile a flat surface, called** wallpaper groups**. This is a mathematical classification based on those symmetries we talked about earlier!

He studied these and between 1936 and 1942, he created 43 colored drawings by employing these patterns. However, it was not geometric shapes that he played with–he explored the humanity of tessellations. Geometric shapes are really only recognizable by classification, but M.C. Escher’s goal was unique recognizability, so he created tessellations of people, animals, plants, and everyday objects–things people could empathize with. His works are based in the innate geometry of tessellations, but he furthered these shapes into the art he is so well-known for. Here are some examples, taken from https://mcescher.com/gallery/symmetry/ !

He created this unique art by modifying monohedral tessellations, and inspired a whole line of artists that followed him. Though he had no formal training in math, his art is rooted in the concepts of dimension, topology, and space, and some researchers believe that parts of his work may have predicted deep features of modern cosmology! With his art, he tells poignant and moving stories, as seen in “Liberation”, where birds evolve from regular, restrictive triangles to flying free in the sky.

With his work, he adds a human and vibrant dimension to the mathematical concept of tessellations while creating captivating visuals that enamor all audiences.

“We do not know space. We do not see it, we do not hear it, we do not feel it. We are standing in the middle of it, we ourselves are part of it, but we know nothing about it.”— M.C. Escher

# Tessellations in Other Spaces

Speaking of space, so far we’ve only really talked about two-dimensional tessellations in Euclidean geometry (i.e. our typical, everyday geometry–like the one taught in high school). So, what about three-dimensional tessellations? Well, given that we use two-dimensional shapes, aka polygons, to create two-dimensional tessellations, it’s only natural that we use 3-dimensional figures, **polyhedra**, to tile three-dimensional tessellations. A simple example would be the standard cube (whose 2D analog is the square)! We call general n-dimension tessellations **honeycombs**, which are created by tiling **polytopes** (if this means nothing to you, don’t worry, I feel you–I have an existential crisis every time I try to think about the fourth dimension). I wonder which dimension we can find Harry’s Tiles in…

But now, let’s move out of our comfort zone of Euclidean geometry to go into (what is creatively named) **non-Euclidean geometry! **Actually, let’s look at two geometries in it: **hyperbolic** geometry and **spherical** geometry. I’m not going to get too much into the details of them, but the main thing you should know is that these geometries play with the concept of parallel lines, which is what differentiates them from our traditional geometry. In Euclidean geometry, we know that, given a point and a line in a plane, there is exactly one line through the point that is parallel to the given line (i.e. never touches it). However, in hyperbolic geometry, there are *at least two* different lines that pass through this point that are parallel to the given line. On the other hand, in spherical geometry, there are *no* such parallel lines. Crazy, right?! It’s kind of hard to wrap your head around, but them’s the parallels. So, how in the world do tessellations work in these strange worlds? Is it even possible to create one? In hyperbolic geometry, it is indeed! We have seen that there are only three regular tessellations in a Euclidean plane, but in a hyperbolic plane there are infinitely many regular tessellations! (For semiregular tessellations, there are also many more combinations compared to a Euclidean plane.) And, of course, our good friend M.C. Escher has some beautiful art to depict this for us:

(Source: https://mcescher.com/gallery/mathematical/ — I absolutely recommend going through his works–they’re mesmerizing with their stunning detail and composition.)

Now, whenever you spot a tessellation, you can appreciate it in its entirety and for its beauty. I hope you don’t go peeking your head into beehives to do this, but if you do, don’t say I didn’t warn you! As for Harry’s Tiles, after my fantastical expedition in the world of tessellations, I still didn’t find any mention of him. Last I heard Harry’s Tiles left in One Direction, walking on a Fine Line, so we may never find out where Harry and his tiles have gone. But, I think it was worth the journey because now we can appreciate *all *tiles.

By the way, before you go, would you want to form a tessellation with me? Because we fit together perfectly. ;)

Until next time! If you found this interesting, make sure to check out the next column! If you have any questions or comments, please email me at apoorvapwrites@gmail.com.

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