Penrose and the search for “Einstein”: the story of the aperiodic monotile

In 1961 Hao Wang conjectured that there exists a finite number of tiles of different shapes which can only fill an infinite plane by never repeating. To see what “no repetition” means in this case, imagine you have two copies of the infinite plane, one on top of the other. No repetition means there is no amount by which you can move the upper plane so that it maps directly onto the one below it. This problem is known as aperiodic tiling.

In 1966 one of Wang’s students, Robert Berger, produced the first set of “Wang tiles” – a set which consisted of 20,426 different shapes! Over the next few decades this number was slowly whittled down until finally, in 1997, Oxford’s Roger Penrose showed that there is a single pair of tiles (now known as the kite and dart) which tile the plane only aperiodically. These became known as “Penrose tiles”. These were so well-known that later in 1997 Penrose sued the Kimberly Clark Corporation over the pattern on their quilted toilet paper which closely resembled Penrose tiling. The matter was settled out of court.

Since 1997 there has been a quest for an aperiodic monotile, a single tile which fills an infinite plane without repeating. Several candidates have been proposed but they all challenge in some way the concept of a tile, tiling, or aperiodicity. On 20 March 2023, a paper was published in arXiv claiming to have found the first clear “Einstein” (or aperiodic monotile)[1]. David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss have proposed the “hat polykite” and it is beautiful.

So how do you show that something infinite never repeats? The authors showed that no matter how you arrange the tiles, it is always possible to divide it up so that each tile belongs to one of four possible clusters (shown below) and that the edges on adjacent clusters can only match up in certain ways. This first step in the proof is done on a computer so instead of proving this mathematically, the authors just had to prove their algorithm was correct. The next step is to show that when these clusters are tiled with each other they form groups called metatiles, which have the same shapes as the clusters. We can do this repeatedly to find larger and larger “supertiles”. Thankfully this is where the proof stops because the authors would have run out of words for “bigger tiles”. Since there is no translational symmetry among the original clusters, there is no translational symmetry among the metatiles, or the supertiles.

It is rather surprising how simple the basics of the proof are but what may be even more surprising is that the hat polykite suggested by the authors is just one member of a continuous family of shapes that are all aperiodic and that all tile the plane in the same way. And the hat polykite is not even the simplest! An even simpler member of the family is shown below. The authors have provided a hypnotising animation which transitions smoothly between the members of this family of shapes (available on the website accompanying their paper).

This paper is historic for a number of reasons. Firstly and most obviously, it concludes a search which has been ongoing for decades. Secondly, the conclusion of the search was made possible only by the power of modern computers. While it has been known for a long time that computers would spark many revolutions in every branch of maths and science, this is a clear-cut piece of evidence that this is already the case and will almost certainly continue to be the case as computers become more powerful. Thirdly, the website which accompanies the paper is a wonderful example of do-it-yourself science. The paper is easy to read, the animation is available on Youtube, and the source code used by the authors is free to download. The power of computers lies not just in opening up new branches of maths and science, but also in opening up existing branches to those that couldn’t previously access it. This paper is a ground-breaking achievement. Congratulations to Smith, Myers, Kaplan, and Goodman-Strauss.

All image credits to “An Aperiodic Monotile” by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss)

[1] “Einstein” here is not named after the famous physicist, but is rather a coincidence. It translates directly from German as “one stone”.