I just received my copy of Tessalation!, a great new book written by Emily Grosvenor and beautifully illustrated by Maima Widya Adiputri, which I helped fund on Kickstarter. It’s about a girl named Tessa who goes exploring in her backyard and finds all sorts of patterns, represented as fun tessellations. I’ve already had a lot of fun reading it with my four-year-old.

Most of the other math blogs in the blog tour for the book release are about early childhood math education, so I thought I’d write something in a slightly more advanced vein, exploring a bit of the underlying mathematics of making tessellations. My hope is that you’ll learn some things and also come away with ideas of new kinds of tessellations to explore. There is way more than I could ever fit in a single blog post (if you want to explore more, John Golden has a great list of resources on Math Hombre), but let’s see how far we get!

## Regular polygons

Let’s start with using *regular* polygons (that is, polygons whose sides and angles are all equal) to tile the plane. Most everyone is familiar with the idea that we can do this with regular (equilateral) triangles, regular quadrilaterals (*i.e.* squares), and regular hexagons:

- Every vertex of an equilateral triangle has an angle of , so six triangles can meet around every vertex to make a total of .
- Four squares meet around a vertex to make a total of .
- Three hexagons meet to make a total of .

In addition, the triangle and hexagon tilings are closely related, since we can get one from the other by subdividing the hexagons:

It’s easy to see that these are the only regular polygons that will work: regular pentagons have angles of , which does not evenly divide . And anything with more than six sides will have angles bigger than , so more than two of them will not be able to fit around a vertex.

## Modifying a square tessellation to make your own

Now, as explained in the back of Tessalation!, and as reproduced in this blog post on Kids Math Teacher, we can take a square tessellation and modify the squares to produce more intricate tessellations which still follow the same underlying pattern. In particular, if you add some shape to one side, you have to remove it from the opposite side, and vice versa. For example, beginning with a square, we might change the right side like this:

But if we do that we need to change the left side in a symmetric way:

Now the altered squares will still line up in a row:

Likewise, we can make symmetric modifications to the top and bottom, like so:

The resulting thingy can still tile the plane:

## Ants on donuts

So far so good. But if we take a step back to think about what’s really going on here, a whole world of possibilities opens up.

What we’ve really done with the square is match up certain edges, so that matching edges always meet in the tessellation.

Here I’ve marked the top and bottom edge both with a single arrow, and the left and right edges with a double arrow. (I’ve also put a letter “P” in the middle; I’ll explain why later.) In the tessellation, corresponding markings always have to match up. Like this:

Now, instead of matching up the edges of a bunch of copies of the same square, we can think about taking *one* square and gluing matching edges together. First, we glue the top and bottom edges together, resulting in a cylinder; then bend the ends of the cylinder around to match up the left and right edges, resulting in a torus (a donut shape).

Now imagine a very tiny ant who lives by itself on the surface of the torus. The ant is so small that it can’t tell that the surface it lives on is curved. To the ant, it just looks flat. (You may know some tiny creatures in a similar situation who live on a sphere.) Unlike those tiny creatures on the sphere, however, the ant has nothing it can use to draw with, no objects to leave behind, *etc.*, so it has no way to tell whether it has ever been to a particular location before. The ant starts walking around, exploring its world. Occasionally there is a straight line drawn on the ground, extending off into the distance. Sometimes it finds places where two lines cross at right angles. Sometimes it finds places where the ground is black, and after making some maps the ant realizes that these places are shaped like a giant letter “P”. After exploring for quite a while, the ant thinks its world looks something like this:

Or perhaps it lives on a torus? (Or an infinitely long cylinder?) The point is that there is *no way for the ant to tell the difference*. The ant cannot tell whether there are infinitely many copies of the letter “P”, or if there is only one letter “P” that it keeps coming back around to. So **a square tessellation is “what a torus looks like to an ant”**, that is, what we get if we cut open a torus and glue infinitely many copies together so that each copy picks up exactly where the previous copy left off.

But there are lots of ways to cut a torus open so it lays flat! And *all* of them will produce some shape which tiles the plane just like a square. This is another way to think about what we are doing when we modify matching edges of a square—we are really just cutting the torus along different lines.

## Onward

This blog post has gotten long enough so I think I will stop there! But I plan to write another followup post or three, because we have only just scratched the surface. In the meantime, I will leave you with some things to think about. First, what if we match up the edges of a square in a different way?

This is almost like the square from before, but notice that the arrow on the top edge is flipped. This means that we can’t just stack two copies of this square on top of each other, because the edges wouldn’t match:

But we can stack them if we flip one of the squares over, like this:

Finally you can see why I included the letter “P”—it lets us keep track of how the square has been flipped and/or rotated.

Can you complete the above to a tiling of the whole plane? What do such tessellations look like? Is it still possible to modify the edges to make other shapes that tile the plane in the same pattern?

How about this square?

Or this one?

Or this one?

And what about triangles and hexagons? What are different ways you can match up their edges to make tessellations? (Related challenge question: when we glue opposite pairs of sides on a square, we got a torus. If you glue opposite pairs of sides on a hexagon, what shape do you get?)

Happy tessellating!

Are you going to talk about wallpaper groups? I really hope you talk about wallpaper groups.

Well, I wasn’t planning to, since I don’t know a whole lot about them. Do you? Want to write a guest post? I’m happy to help with making pictures. =)

I have always wanted to learn more about wallpaper groups, but I’ve never gotten around to it. You consistently explain complex, high-level math in a beautiful and accessible way.

In December, I worked trough your series on The Lucas-Lehmer test with one of my high school students. He and I then started studying Abstract Algebra, and I’m certain he would find wallpaper groups as interesting as I will.

Isn’t the last one a special case of the one before it?

No, they are different — look closely!

I insist: rotate once, then replace “>” (notice I flipped it), and you get the exact same tile.

I separately verified the same simple solution applies to both. On the other hand, it also applies to a square with four sides labeled ‘>’, ‘>>’, ‘>>>’, ‘>>>>’, so it doesn’t mean that much. (That’s an interesting conclusion, so I think you should include this general case as the last problem)

Formatting swallowed my sentence.

Lets call “” A’, then the first is A A’ B’ B and the second is A’ A A’ A. If you take the first and choose B=A’, you get A A’ A A’, which is a rotated version of the second.

Lets call < A and > A’, then…

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