Penrose Tilings look somewhat like a mosaic, but are mathematically so much cooler!
A Penrose Tiling is a way of tiling a surface which is named after British mathematician and physicist Roger Penrose. The tiling has some special properties, which include:
- It is non-periodic: You can think of a periodic tiling as one that repeats itself in a certain way. A penrole tiling is non-periodic. This means that you can’t slide one section over to another to get it to match. In the tiling below, it is clear that it is a periodic tiling. You can slide any 2×2 set of squares to the left, right, top or bottom and match the pattern:
- It is self-similar: This is somewhat difficult to explain if you don’t know what a fractal is. Basically, it means that patterns start reappearing on grander scales as you keep tiling. To see this, have a look at the picture below. From (1) to (2) to (3) you can see the simple rule to generate the pattern. That is, split all the white squares into four equal squares, then shade the top left and bottom right of those squares purple. The pattern is called a fractal and fractals are self similar. If you look at pattern (1), you can find that pattern all over pattern (3), only smaller:
- It is a quasi-crystal: The Penrose tiling is very ordered. If a crystal had this pattern, wavelengths of light that pass through this crystal would diffract. Diffraction is the process of spreading apart of light as it passes through a small gap, like atoms in a crystal. Additionally, this diffracted light would create a pattern with 5-fold symmetry.
There are actually three types of Penrose tilings possible. We call these P1, P2 and P3. Before I talk about the different tilings, I need to explain one other aspect of Penrose tilings: the concept of matching rules. If you were to take a tile set and just tile in any way you like, you may end up with a pattern that is periodic. In order to avoid this, so-called matching rules exist that tell you how to tile correctly, so as to avoid ending up with a periodic pattern. Usually, we show these tiling rules as numbers on vertices or edges, or patterns on the face.
So, let’s start with the P1 tiling (also known as the Pentagonal tiling). The below is an example of a P1 pattern:
It uses a total of four different tiles: pentagons (red, grey, navy blue), one star (green), one parallelogram (yellow), and one “boat” (light blue). We give the pentagons different colors because they have three different matching rules associated with them. The matching rules are below (numbers with a bar above them must match to numbers without a bar):
The second type of tiling is the P2 tiling. We sometimes call this the Kite and Dart Tiling. This is the tile set that I have in my collection. An example of a pattern we can make using this tile set is below:
It only uses two different tiles: A kite (green) and a dart (yellow).
The matching rule is below:
Finally, we have the P3 tiling (the rhumbus tiling). An example of that tiling is below:
This tiling uses two different types of rhombuses: one with angles 36°, 144°, 36°, and 144° and the other with angles 72°, 108°, 72°, and 108°. The matching rule for this tiling is as follows:
You will notice that the edges have notches to show how to arrange the pieces. This was initially an idea by Penrose himself, who notched the edges to force the matching rules. You can see the notched versions of the tiles in the image below:
The one last thing I want to talk about is the concept of “Inflation” and “Deflation”.We sometimes call it “Composition” and “Decomposition”. This relates to the concept of self-similarity that I was talking about earlier and might help you see it better. You can also use this technique to help you with tiling if you forget the matching rules. Consider the pentagon below:
The large pentagon consists of six smaller pentagons and a few gaps. Now, look what happens if we tile this large pentagon in the same way as the pentagons inside it:
Do you notice anything special about it? Particularly, the gaps the shape has? The gaps actually look like the diamonds, stars and boats that we need in the original tiling. In fact this was the way that Penrose was able to initially find the P1 tiling. He was able to apply this principle of decomposition to a tiling and thus figure out what other smaller shapes he needed to finish the tiling.
Penrose tilings are really interesting, with a lot of other links in other fields of mathematics such as group theory. You can create tile sets at home fairly easily. Just download the shapes from an online source, print them and cut them out. Have fun!