But let’s get back to tiling the line. There’s an important distinction we need to make.
Consider the following new rules for our tiles A and B.
- Next to an A, on either side you can place an A or a B.
- Next to a B, on either side you can only place an A.
Can we still tile the line with these rules? An easy way to see that the answer is yes is to notice that the previous tiling also satisfies the new rule set.
…ABABABABABABA…
But the new rules allow more flexibility, and this leads to more tilings of the line.
For example, these are both valid configurations under the new rules:
AAABABA
ABABAAABAAB
And these can be extended infinitely in either direction in infinitely many ways.
In addition to giving us lots of new tilings of the line, the new rules allow us to generate tilings that, unlike our first example, don’t repeat. For example, consider the following tiling:
…ABAABAAABAAAAB…
What’s the pattern here? Start with an A, then place a B to the right, then two A’s on the right, then a B, then three A’s, then a B, then four A’s, and so on. On the left, just keep adding A’s:
…AAAAAABAABAAABAAAABAAAAAB…
Do this, and you’ll end up with a tiling that can’t be translated onto itself so that everything matches up.
A simple way to see this is to observe that there is a unique leftmost B in this tiling, so where will it go after translation? If you translate to the left, there’s no B for it to match up with. But if you translate to the right, there’s no B coming from the left to match up with it.
The new rules therefore allow both tilings that have translational symmetry and tilings that don’t. There are tilings of the plane that work like this, too.
For example, we’ve already seen a tiling with squares that has translational symmetry, but we can also use the square to construct tilings that don’t have this property.
This is a very different situation from the monohedral tilings that use equilateral triangles or regular hexagons. In those tilings, the repetitive structure is unavoidable. The geometry of the tiles themselves forces the tiling to have translational symmetry. We call such tilings “periodic.”
In contrast, the square allows for patterns that repeat and patterns that don’t. This leads to a natural, irresistible question for mathematicians: If there are tilings of the plane that are forced to have this repetitive structure, are there tilings that are forced to avoid it? With this question, formulated in the 1960s, the hunt for “aperiodic tilings” was on.
For our search, we’ll make one more trip back to the line. Our final tiling of one-dimensional space will use an unusual-looking set of tiles:
A-tiles: A, AA, AAA, AAAA, …
B-tiles: B, BB, BBB, BBBB, …
Notice that this tile set is infinite. If this seems like cheating, you’re thinking like a mathematician. We’ll come back to that later, but for now, here are the two rules for putting our infinitely many tiles together:
- Next to an A-tile of length n, you can only put a B-tile of length n on either side.
- Next to a B-tile of length n, you can only put an A-tile of length n + 1 on either side.
As always, our question is: Can we tile the line with these tiles and rules? Well, suppose we start with an A-tile of length 1.
A
The rules dictate that on either side we can only put B-tiles of length 1.
BAB
Now, next to each B we must put A-tiles of length 2.
AABABAA
Then we add B-tiles of length 2.
BBAABABAABB
And so on. It’s easy to see we can go on forever in either direction, which means we can indeed tile the line with these new tiles and rules. And relevant to our search, this tiling does not have translational symmetry. Notice that the single A we placed at the start gets immediately surrounded by B’s on either side, and the resulting pattern — BAB — will never appear again. In the infinitely long string that represents our tiling, every other A that appears will be next to at least one other A. This means there’s nowhere for the BAB string to go, so there’s no way to translate this tiling onto itself.
This will be true regardless of what tile we start with. If it’s B, the rules immediately lead to the string
…BBAABAABB…
And, as before, the pattern ABA will never repeat. Even if you start with something like AAA, the same thing will happen.
…AAAABBBAAABBBAAAA…
Whatever you start with, the initial tile will always be the only A- or B-tile of that particular length, which will prevent any translational symmetry from emerging. This happens to be exactly what we were looking for: a set of tiles and rules that allows us to tile the line but will never allow translational symmetry.
You might be unsatisfied with an aperiodic tiling that requires infinitely many tiles, and you wouldn’t be alone. When mathematicians started seriously looking for aperiodic tilings of the plane, they wanted to find a finite set of tiles that could tile the plane but couldn’t have translational symmetry. An early solution used 20,426 tiles, but within a few years mathematicians had brought that number down to six.
A breakthrough occurred in the 1970s when Roger Penrose, the British mathematician and physicist, discovered the famous two-tile set that now bears his name. Penrose tiles are a pair of simple quadrilaterals that, with a careful set of rules, tile the plane without allowing translational symmetry.
There’s only one way to improve upon a two-tile aperiodic tiling, so mathematicians, hobbyists and artists began searching for an aperiodic “monotile” that would do the job all by itself.
Last November, David Smith found it. This is the “hat,” the first known aperiodic monotile.
Smith, a recreational mathematician, artist and tiling enthusiast, discovered the hat the way much mathematics is discovered: by playing around and seeing what happened. Smith would later connect with the researchers Craig Kaplan, Chaim Goodman-Strauss and Joseph Samuel Myers, who together verified that this was indeed the long-sought aperiodic monotile.
Proving something can tile the plane but can’t have translational symmetry is no easy task, but some of the techniques they used are hinted at in our simple examples. For instance, one way to show that equilateral triangles can tile the plane is to notice that they come together to form larger structures, in this case hexagons, which are known to tile the plane. The hat tile also comes together to form larger, regular structures, which can be used to understand how it tiles the plane.
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