May 15 2023 How often do mathematical findings find their way to late-night talk shows? I'm still a little puzzled by why this particular one did. It made a splash in mathematical circles, certainly, but also the regular news in various places - and yes, even Jimmy Kimmel mentioned it (at the start of this March 30 show, https://www.youtube.com/watch?v=h-YkSTHz_PM).
I've written about tiling before. What caught my attention about this particular tiling discovery - I use that word deliberately - is that it is essentially built on tilings we know about. So how was this one just "hiding in plain sight", and what questions might that put in our minds? Let me know what you think! Jimmy Kimmel and a thinking hat, https://www.livemint.com/opinion/columns/the-discovery-of-an-aperiodic-monotile-the-mathematics-of-tiling-and-the-fascinating-hat-shaped-shape-hiding-in-plain-sight-11683221525265.html cheers, dilip --- Jimmy Kimmel and a thinking hat When the shape made an appearance on Jimmy Kimmel's late night talk show last month, I knew this was big news. He didn't seem to fully grasp what the fuss was about, but he did "wear" the shape, in a manner of speaking. A mathematical discovery, making the leap into popular culture - how often does that happen? 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, meaning 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 about a decade later, Roger Penrose announced a set that had only two shapes. If it took about ten years to bring 20,000+ 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 that 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 cutouts of it and played the tiling game with them until he was convinced he was onto something. Some of the cutouts, 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 in 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. -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. 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