This blog is named after one of my favorite areas of Mathematics. An aperiodic tiling is a specific arrangement of a given set of shapes that covers an infinite, two-dimensional plane without translation symmetry. This means that the tiling never repeats itself, while still covering an infinite amount of surface area.
To me, the mere fact that aperiodic tilings exist is pretty cool. Taking only a finite collection of component parts, you can produce something that is infinite, and infinitely variable. However, it turns out that most sets of tiles allow aperiodic tilings. As an example, consider the set of tiles consisting of 2x1 rectangles and 2x2 squares. This set allows an aperiodic tiling, since in the construction of a tiling, every time you need to place a tile, you can place either the square or the rectangle. If, at any point in the construction, the tiling is in danger of becoming periodic, just place the “wrong” tile down.
Interestingly, there are sets of tiles which only allow aperiodic tilings. The first such set of tiles to be discovered arose from the research of the logician Wang Hao. Note that, while Wang had an undergraduate degree in Mathematics, his masters was in Philosophy, and his doctorate in Logic, an area of study which Philosophers and Mathematicians have been fighting over almost since its inception. This is another reason why I find aperiodic tilings so fascinating—they’re a simple geometrical concept that happens to encode a form of the Halting Problem.
Wang Hao was researching the Domino Problem. Succinctly, the Domino Problem is the question of whether a given set of square tiles with colored edges can tile the plane such that only edges with the same color touch. Wang showed that the Problem can be solved by an algorithm (a simple set of instruction) if and only if there did not exist aperiodic tilings of the tiles. Wang thought that there was an algorithm for the problem, and hence, no aperiodic tilings, but his student, Robert Berger, proved that such an algorithm could not exist [1], and provided, somewhat offhandedly, a set of 20,426 tiles which was aperiodic.
Since then, several mathematicians have come up with sets of aperiodic tiles that are a bit simpler than Berger’s original set of 20,426. Perhaps the most widely known sets are the Penrose tilings, named after their discoverer, Roger Penrose. My favorite Penrose tiling is the P3 tiling, also known as the “Rhombus Tiling.” It consists of the following two tiles:
The arc and dots produce matching rules, since tiles must be placed so that adjacent features have the same color. Now, instead of needing 20,426 different kinds of tiles to create an aperiodic set, we need only two. This is very pleasing to the intellect, since two is the lowest possible number of tiles that could make up such a set (it would be difficult to create a tiling of a single shape which never repeated).
All three types of Penrose tilings share a few remarkable properties. Take a look at the following ratios of sides to diagonals in the Penrose rhombi:
Where the phi you see above stands for the golden ratio. The reason the golden ratio shows up in these rhombi is because they are constructed from a pentagon, and the ratio of side-length to chord-length in a pentagon is golden. Similarly, the ratio of thin to thick rhombi in the P3 tiling approaches the golden ratio as you add more tiles.
Neat mathematical properties aside, the P3 tiling is just plain pretty.
Anyways, I hope you’ve enjoyed this brief excursion into mathematics. While most of the content of this blog will be focused on my digital artwork and electronics projects, I’ll probably write a few more posts in this vein, especially once school starts up again.
References
- Berger, Robert (1966), “The Undecidability of the Domino Problem”, Memoirs of the American Mathematical Society 66.