Tiling, sometimes also referred to as tessellation, simply refers to the covering of a plane with simple geometric shapes. Examples of this include checkerboard or honeycomb patterns, which alone are not terribly interesting. What I do find interesting, though, are aperiodic tilings, patterns that do not repeat (the before mentioned patterns would be considered periodic).

The first example of aperiodic tilings comes from Hao Wang, who had conjectured that if a group of tiles can tile the plane, the only way to do it was periodical. However, one of his students, Rober Berger, later discovered a set of tiles that disproved Wang’s theorem, and in doing so discovered the first set of aperiodic tiles. His tiles were a set of 20,426 “Wang” dominoes, each a square made of four colored triangles. In order to tile the plane aperiodically, neighboring triangles had to share the same color.

Berger was able to reduce this number down to 104, and Hans Läuchli was able to get it to just 40. Eventually, other form factors were used, and this number dropped to just two.

The most famous example of aperiodic tilings are Penrose tiles, first discovered by legendary physicist and mathematician Roger Penrose in 1974. There are two main sets of Penrose tiles: kites & darts, and thick & thin rhombuses. When placing these tiles, there are some rules that ensure aperiodicity. This can include colored edges that must be paired with like-colored edges, or notches and cutouts (imagine a jigsaw puzzle).

An interesting aspect of Penrose tiles are the numerous ways to generate them. The most obvious of these is putting them together like puzzle pieces, but this is neither very clever nor interesting. Here are two cooler ways to generate the pattern.

The second way to create Penrose tiles is using what is called a pentagrid. A pentagrid is similar to a normal grid, except that instead of two sets of parallel lines, it has five. To draw a pentagrid, rotate these sets of lines by a multiple of 2π/5. At the intersection of two lines, we can make a diamond with faces normal to these lines.

Now, if we increase the size of the diamonds and slightly move them such that they perfectly tile the plane, we see we have created a Penrose tiling.

It might be difficult to believe that such simple rules can arrange such simple shapes in a way that they will never repeat. Unfortunately, there is no single proof that applies to all aperiodic tilings that shows they never repeat, instead each one is highly individualized. That being said, proving that Penrose tiles are aperiodic is not very hard. Let’s go back to inflation. First, scroll up to the sketch I made, and, beginning with zero, slowly increase the number of iterations, counting the ratio of thin to thick rhombuses (or kites to darts) each time. What you’ll find is that this ratio approaches the golden ratio, approximately 1.618. We can easily prove this by noticing that the number of thin and thick rhombuses follows the fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). We can assign variables to the number of the types of rhombuses, and write a formula for the ratio between them. Due to the fact that they follow the fibonacci sequence, we can also write a formula for the ratio for the next iteration of inflation. Finally, because our tiling is in infinite space, we can set these two formulas equal to each other.

Now, we can easily solve this equation by first dividing the right side by B. Now, we can assign X to A/B, and solve for X.

What we find is that X (or the ratio of thin to thick rhombuses), is equal to the golden ratio, which we know to be irrational. If this ratio was rational, that would mean that we could divide our tiling into smaller subtilings, one that has, for example, four thin rhombuses and five thick ones. After these four thin and five thick rhombuses, our theoretically rational tiling would repeat, and we would find another subtitling with these parameters.