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.<\/p>\n

A<\/p>\n

The rules dictate that on either side we can only put B-tiles of length 1.<\/p>\n

BAB<\/p>\n

Now, next to each B we must put A-tiles of length 2.<\/p>\n

AABABAA<\/p>\n

Then we add B-tiles of length 2.<\/p>\n

BBAABABAABB<\/p>\n

And so on. It\u2019s 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\u2019s on either side, and the resulting pattern \u2014 BAB \u2014 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\u2019s nowhere for the BAB string to go, so there\u2019s no way to translate this tiling onto itself.<\/p>\n

This will be true regardless of what tile we start with. If it\u2019s B, the rules immediately lead to the string<\/p>\n

\u2026BBAABAABB\u2026<\/p>\n

And, as before, the pattern ABA will never repeat. Even if you start with something like AAA, the same thing will happen.<\/p>\n

\u2026AAAABBBAAABBBAAAA\u2026<\/p>\n

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.<\/p>\n

You might be unsatisfied with an aperiodic tiling that requires infinitely many tiles, and you wouldn\u2019t 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\u2019t have translational symmetry. An early solution used 20,426 tiles, but within a few years mathematicians had brought that number down to six.<\/p>\n

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.<\/p>\n

There\u2019s only one way to improve upon a two-tile aperiodic tiling, so mathematicians, hobbyists and artists began searching for an aperiodic \u201cmonotile\u201d that would do the job all by itself.<\/p>\n

Last November, David Smith found it. This is the \u201chat,\u201d the first known aperiodic monotile.<\/p>\n

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.<\/p>\n

Proving something can tile the plane but can\u2019t 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.<\/p>\n<\/div>\n

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