Some delightful news from the world of pure mathematics yesterday: a team of four mathematician (Smith, Myers, Kaplan, Goodman-Strauss) released a preprint on the Arxiv with a proof that aperiodic monotiles, or “einsteins,” exist. Their representative example is the “hat polykite,” which can be built from eight kite shapes. In fact, I was able to build one this morning using my 21st Century Pattern Blocks!
To be clear, the shape in question is the entirely of what’s above, not the individual kites. And what makes it special isn’t that it tiles the plane by itself—lots of shapes do that, including a single kite (or any triangle). What’s special about it is that none of these tilings of the plane are periodic, meaning that none of them repeat in any fundamental way.
If you’d like to play around with this tiling now, try out the Polypad plugin below, or click here to explore in its own tab.
If you’ve noticed some of us math educator/communicator types swooning over this result on social media, there’s a reason. First off, the statement of the problem—do these tiles that, by themselves, tile the plane without admitting repetitions—is relatively easy to understand. And the tile itself is a pretty simple polygon! When you consider that the earliest example of an aperiodic tiling required 20426 different tiles, this is a triumph of simplicity over complexity! There’s something fundamentally beautiful about that.
But understanding why the result is true is another matter. This was an open question for decades, and it was unsolved because it was hard! Folks have probably constructed or sketched these einsteins many times. They’re hiding in plain sight! But how do you know you have one when you have one? That is the heart of the difficulty.
You can play around with these monotiles on polypad,
This whole thing reminds me of a more accessible mathematical tiling demonstration I witnessed on twitter last week, when Libo Valencia tweeted out a problem he and his daughter were exploring: how can one build a regular hexagon using the same number of each of a collection of different shapes?
Morning #MathPlay with my daughter 👨👧 – We built hexagons using the same number of different shapes. I feel @fawnpnguyen & @Trianglemancsd may like this #mathgames 🙂#visualpatterns #tmwyk #ITeachMath #MTBoS #MathIsFun #STEM #ElemMathChat #CodeBreaker pic.twitter.com/NehXIRSrI8
— Libo Valencia 🧮 MathPlay (@MrValencia24) January 16, 2023
The question inspired Hana Murray to explore the question with 21st Century Pattern Blocks. She posted solutions for 3, 4, 6, and 8.
More solutions for building regular hexagons using the same number of different shapes with #UpscalePB #21CenturyPB. pic.twitter.com/XG3acZIG91
— Hana Murray (@MurrayH83) March 18, 2023
At this point, I jumped in with a question: what numbers can this problem be solved for? After 3, will odd numbers ever be possible again? Shyam Drury (@MathsXfer) suggested that since all the examples so far had been multiples of 2 or 3, maybe 9 would prove possible.
This is how new problems emerge in mathematics. We play around, and suddenly we have a question. And if we have a hunch about that question, maybe we have a conjecture. Now our play becomes more focused.
Sure enough, Hana came back with extensive new progress, building hexagons with 5, 9, 15, 36, and 81 of each type of shape.
Once I saw that 5 hexagons, 5 trapezoids, and 5 triangles fit the rhombus, I put the three rhombuses together to create the regular hexagon. After that, I just experimented arranging the blocks into various designs and creating larger hexagons. pic.twitter.com/1NvDGOynms
— Hana Murray (@MurrayH83) March 18, 2023
Another counterexample. The regular hexagons pictured below have 9 blocks of each shape. The first example has 9 triangles, 9 rhombuses and 9 trapezoids. The second example has 9 darts, 9 kites and 9 rhombuses. pic.twitter.com/AuIlesXGuN
— Hana Murray (@MurrayH83) March 19, 2023
This regular hexagon contains 81 blocks of each type. pic.twitter.com/E8LDbO9bfB
— Hana Murray (@MurrayH83) March 19, 2023
To be clear, this question is nowhere near the importance or difficulty of the discovery of the einstein that has math twitter celebrating today. However, it’s worth noting that even in our fun little accessible tiling question, some of the same techniques show up. Hana’s key insight is to build intermediate shapes. Building medium sized rhombuses that use 3 of each shape type give her a single building block (the rhombus) that can construct the larger regular hexagons. If she can use an odd number of the rhombuses, she uses an odd number of each of the component blocks too!
The authors of the einstein paper pull the similar trick, though in much more complex ways (and often with the assistance of a computer). They call their intermediate shapes metatiles.
Interestingly, the substitution rules do not apply to the hats directly. Instead, we derive new metatiles from the clusters, and build a substitution system based on the metatiles. The underlying hats are simply brought along for the
ride.
Here is the deep pleasure of mathematics: a child can pick up blocks and build shapes from them, and experts can tackle problems including the same shapes for decades or centuries, until their resolution is cause for celebration!
And to those of us in between… we can explore our own problems, inspired by the playful genius of the professional mathematicians, and animated by the same instinct for playful exploration as the child. Mathematics is an infinite territory, and it has space for us all.
