Who doesn't like spotting weird shapes in your bathroom's tiling when you are bored? Or you must have tried arranging random scraps of paper in a gapless sheet of continued white, right? Tessellations are all around us. One of the most visually stimulating topic in the mathematical universe and you are amidst them everyday: from the everyday square tilings on the floor to the special designs on a turtle's shell. It is considered the backbone for abstract artwork while being the most handy tool for any kind of architect.
Look above. A simple tessellation can become such an intricate and eye-pleasing pattern. Tessellations are the most crowded bridge of art and mathematics. However, we all know that these kinds of "tilings" have repeating patterns always. Give it a thought, have you seen any kind of tiling that does not have a SINGLE that repeats all over? Even for the above photo. Look how we can shift or translate just one part of this tiling and form the entire tiling.
This seems kind of obvious right? There is obviously a repeating shape, else how could it go on forever. That is what I thought and what most of the mathematical community did too.
That is... until David Smith, a mathematics hobbyist, who had not worked on formal mathematics before, booted up the humble PolyForm Puzzle Solver on his computer and messed around with some shapes in his free time. He managed to create a unique 13-sided tile, which he named the "Hat", that amused him.
This fedora-shaped tile was an interesting spectacle because it was managing to tile infinitely without gaps, yet it was not the usual "normal-looking" polygon. Having messaged his friend who was a professional mathematician, David realised that he had found an elusive aperiodic monotile.
Aperiodic Monotiles
In the world of mathematics, one of the most fascinating and long-standing problems has been the quest for an aperiodic monotile. An aperiodic monotile is a shape that can tile a plane, but only in a non-repeating pattern. In simpler terms, a tiling is considered periodic if it can be shifted or translated and still look the same. A typical wallpaper or tiled floor is part of an infinite pattern that repeats periodically too. Recall the lizards at the beginning, they were an example of periodicity too.
Aperiodic tiles, on the other hand, do not display such a pattern. Mathematicians have long sought a single shape that could tile the plane in an aperiodic fashion, and this is jokingly known as the Einstein problem. An einstein is simply another name for an aperiodic monotile. The term "einstein" comes from the German "ein stein," meaning "one stone" or "one tile."
For many years, the problem was dwelled over by professional mathematicians from across the world. Initially, mathematical tiling pursuits were motivated by a broader question: was there any set of tile that could tile the plane non-periodically? The journey of finding such aperiodic tiles began with Robert Berger. He discovered an aperiodic set of 20,426 tiles. Then the chase changed to: how few tiles would do the trick? In the 1970s, Sir Roger Penrose, a mathematical physicist at the University of Oxford, got the number down to two. However, in November 2022, Smith's discovery became an important step forward in the field of mathematics. It is the latest in a long line of efforts to find a single shape that could tile the plane in an aperiodic fashion.
David Smith's discovery has been hailed as remarkable and groundbreaking, and a new paper authored by Smith and three co-authors with mathematical and computational expertise proves the discovery to be true.
Going Further With and From the "hat"
However, after teaming with professional mathematicians, David showed them another tile he had come across called the "turtle". Unbelievably, David had just found two einstein tiles back-to-back, a feat thought to be impossible for such a hobbyist.
Nonetheless, the researchers focused on the "hat" first. Proving the "hat" tile's aperiodicity was a uniquely geometric approach in and of itself. They managed to assemble four structures, labeled H, T, P, and F. These were called the "metatiles" because they were made from the smaller "hat". Similarly, these metatiles were further connected to form the supertiles. The infographic below should be self-explanatory.
Play around with the "hat" yourself on this Waterloo-powered App.
Having dissected the "hat" and its aperiodicity, they moved on to the "turtle" tile. Notably, this tile was of a similar structure to the initial "hat". This inspired the researchers to find an infinite family of aperiodic monotiles that were each 13-sided with small and large sides. Other than the two "extremes" of this resizing in the GIF, and the absolute midpoint where the long and short sides are equal, every shape is an aperiodic monotile.
Significance of Discoveries
So why is the discovery of an aperiodic monotile so important? For one, it helps us understand the fundamental nature of mathematics and the patterns that underline the world we live in. Tiling is a fundamental concept in geometry, and its study has applications in many areas of science and engineering. The discovery of an aperiodic monotile can help us understand the limitations of tiling and the nature of space and geometry. Finally, such aperiodic monotiles can hold significance in insulation materials and interior design.
Nonetheless, as I mentioned, tiling can be visually pleasing as well. We can use this aperidoic monotile to create an infinite artwork that never repeats. Imagine having an infinite floor pattern but someone in the other corner of the room sees a shape completely different from you. See below, attempts at transforming the tile to art by a YouTuber . Do check his video and download his resources to mess around with the tile yourself.
Additionally, the discovery of the "hat" tile is significant because it was made by a hobbyist rather than a professional mathematician. It is a testament to the fact that anyone can make an impact in the field of mathematics, regardless of their background or education. The hat tile is just one example of the contributions that hobbyists have made to mathematics. Many people assume that mathematics is a subject only for professionals with advanced degrees, but this is simply not true. Anyone with an interest in math can explore the subject, ask questions, and make discoveries that push the boundaries of what we know.
In fact, hobbyists and non-professionals have been making contributions to this field for a long time. Below is the efforts of another hobbyist, Marjorie Rice, who managed to find four new convex pentagons that tile on a plane. Below are two of these four alongside the beautiful artwork they can produce.
Implications for Mathematics Today
More recently, the internet has made it easier than ever for people to share their mathematical discoveries and collaborate with others. There are countless forums, blogs, and social media groups where people can ask questions, share ideas, and get feedback on their work. These online communities are open to everyone, regardless of their level of formal education or professional status.
One of the most exciting things about the discovery of the hat tile is that it shows that there are still many open questions in mathematics waiting to be explored. It's easy to assume that all the big questions in math have already been answered, but this is not true. There are still countless unsolved problems and unexplored areas waiting for someone to come along and make a breakthrough, one must just have the patience and interest to look for them.
And while it's true that many of these problems require a high level of mathematical expertise to solve, there are also many that can be tackled by anyone with an interest in math and a willingness to learn. In fact, some of the most interesting questions in math are those that can be approached from a variety of perspectives, including those of non-experts. Consider taking a simpler problem and working on it alongside a professional if it seems too daunting. They could simplify the formal proof side and you could focus on the intuition side. (It is much easier to prove something you know is true!)
Conclusion
In conclusion, the discovery of the hat tile is an instance of mathematical curiosity and the importance of pursuing knowledge for its own sake. It shows that there are still many open questions in mathematics waiting to be explored, and that anyone with an interest in math can make a contribution to the field. So if you're a hobbyist or student with an interest in math, don't be afraid to ask questions, explore new ideas, and push the boundaries of what we know. Who knows? You might just make the next great mathematical discovery.
Bibliography and Suggested Further Reading
Klarreich, Erica. “Hobbyist Finds Math’s Elusive ‘Einstein’ Tile.” Quanta Magazine, 4 Apr. 2023, www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/.
Smith, David, et al. An Aperiodic Monotile. 20 Mar. 2023, arxiv.org/pdf/2303.10798.pdf.
Waterloo. “An Aperiodic Monotile.” Cs.uwaterloo.ca, 2023, cs.uwaterloo.ca/~csk/hat/. Accessed 9 Apr. 2023.
---. Uwaterloo.ca, 2023, cs.uwaterloo.ca/~csk/hat/app.html. Accessed 9 Apr. 2023.
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