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Saturday, December 9, 2023

Jimmy Kimmel and a thinking hat

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For a very long time, mathematicians have thought about tiling. That is, they search for geometrical shapes that can cover a given area with no gaps, no overlap. In fact, mathematicians are not just concerned with a “given area”—they search for shapes that can “tile the plane”, meaning a flat surface of any shape or size. (See my column “Tiling the plane”, https://www.livemint.com/Opinion/1TPDmsXpheJ8FJCgP70vjK/Tiling-the-plane.html). Maybe your floor has tiles? Square or rectangular ones? Well, so we know that squares and rectangles qualify. So do all triangles and the regular hexagon—one with all its sides equal.

While no other regular polygons will tile the plane, we do know of 15 irregular pentagons that can do the job, the last of those discovered just a few years ago. What’s characteristic about all these mentioned shapes is that when they are used to tile the plane, they form patterns—meaning a block of tiles that repeats across the plane. If that block was turned into a tile, you could use it, in turn, to tile the plane. You’ve undoubtedly noticed this in tiled floors or walls. Mathematicians call such tiling shapes “periodic”.

Which of course means that mathematicians want to know: is there an “aperiodic” shape that will tile the plane? Meaning, one that won’t form repeating patterns? Tiling theorists have investigated this for years, without an answer.

Well, they have come close. Robert Berger wrote a PhD thesis at Harvard University in 1964, in which he described a set of shapes that, when combined in different ways, tile the plane aperiodically. Only, his set had 20,426 shapes—perhaps something of a deterrent to widespread popular acceptance. But a decade later, Roger Penrose announced a set that had just two shapes.

If it took about 10 years to bring 20,000-plus down to two, it took nearly another half-a-century to bring two down to one. And it came from what might seem like a thoroughly unlikely direction. Last year, a retired printer and amateur mathematician in England, Dave Smith, found exactly such an “aperiodic monotile”, to use its mathematical name. It’s also known as an “einstein”—not named for the great man of relativity, but because “ein stein” means “one stone” in German.

Whatever you call it, this shape tiles the plane, and does so with no repeating patterns.

If you look at a picture of the tiling it produces—for example, see https://cs.uwaterloo.ca/~csk/hat/ —it is pretty and orderly, and you might think you detect some patterns that repeat. Until you look a little closer and realize that they are not quite repetitive. Smith’s monotiles fit together in different ways and it’s those differences that ensure nothing repeats. Look even more closely, and you’ll see that some of the monotiles are mirror images of the others.

In fact, Smith reports that when he happened on the shape, he made several cut-outs of it and played the tiling game with them until he was convinced he was onto something. Some of the cut-outs, he flipped over so that he was using the reflection.

Smith’s 13-sided shape has come to be known as the “hat”— because, well, it looks like one. He got in touch with Craig Kaplan, a mathematician he knew at the University of Waterloo, Canada. With some other colleagues, they published a paper about this new einstein (“An Aperiodic Monotile”, https://arxiv.org/abs/2303.10798, 20 March 2023).

They comment that the hat “can form clusters called ‘metatiles’ [and] because the metatiles admit tilings of the plane, so too does the hat.” In fact, they show how these metatiles can be combined into even larger versions of themselves, and that’s how they prove that the hat is indeed an aperiodic tiler.

So, on his show, Jimmy Kimmel “wore” one of these hats—electronically—though he remarked that “it looks more like an upside-down t-shirt.” Be that as it may, there are two things to say about the shape.

One, on still closer examination, the hat is actually made up of four pentagons. That should remind you of my mention above of 15 irregular pentagons that are known to tile the plane. Indeed, this particular pentagon is one of those 15. So you might wonder —if we have known for years that this pentagon tiles the plane, why the fuss now over a hat shape formed by four of the same pentagons?

That’s because the tiling we know, that uses that pentagon, is decidedly periodic. That is, a pattern is immediately apparent. (See the red tiling here: https://upload.wikimedia.org/wikipedia/commons/4/44/PentagonTilings15.svg).

Two, the “immediately apparent” pattern I mentioned just above is just a regular hexagon. That is, three of these pentagons combine to form a hexagon, and the hexagon tiles the plane easily and prettily (think of a beehive). Nobody thought to combine them differently, or, if anyone did, they didn’t come up with anything very interesting, tiling-wise. Smith’s insight was to put four of these pentagons together to form this particular hat shape. Once Smith did that, he had found an aperiodic monotile.

Which gives us some food for thought. Consider that this hat is just a relatively simple shape made up of familiar and even simpler shapes. Dave Smith arranged those in a certain way, and voilà! The hat emerged. As another mathematician remarked, “It’s hiding in plain sight!”

Thus the question: was the hat always there—hiding in plain sight, waiting to be discovered? Or did Smith invent it?

That’s actually a deep, fundamental question to ask about mathematics (see my column “Are we inventors or discoverers?”, https://www.livemint.com/opinion/columns/are-we-inventors-or-discoverers-11643313196116.html).

You see, many of us might wonder why mathematicians are excited about this einstein—indeed, about aperiodic tiling in general. Well, it does make beautiful wallpaper.

But for me, its real beauty is that it stimulates that deep question. You could say, it makes me put on my thinking hat.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.

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