BackTiling the plane
In July this year, a team of mathematicians at the University of Washington filled the world's tiling enthusiasts with delight
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An important road near where I live was closed for repairs for over six months earlier this year. It went like this: labourers excavate a huge hole, lay pipes and such like, fill in the huge hole, move down the road and repeat. As you can imagine, this was a huge disruption, not least because the only hospital in the area is on that road.
A couple of months ago, all the holes had been filled in, and the road paved over for regular use again—but with one major difference. Before the dig, the road surface was the usual tar, with the usual potholes and untidy edges. Still, it was relatively serviceable. Now, the surface is made up of the ubiquitous “paver blocks"—those infernal interlocking things. And it is a job done shoddily beyond belief. There’s one section where the surface is so warped that cars bounce along like flimsy toys.
How a road is out of commission for half a year, then returns far worse than it used to be, could be a tale out of the old Soviet Union. But it’s about today’s India. Yet this article is not a lament; instead, it is about those infernal paver blocks, which actually aren’t all that infernal.
They come in several different shapes. They are tough, versatile, easy to mass-produce and extremely useful. They fit into each other as you lay them, leaving no gaps. Apart from interruptions like trees or manholes, and even if shoddily done, you can use them to cover a flat surface of essentially any size or shape.
And get this: behind those shapes is some serious mathematics. The idea of “tiling the plane" is one that mathematicians have grappled with forever. That is, what simple shape can you give a tile so that identical copies will fit together to cover a plane surface, leaving no holes?
I’m betting you grapple with this problem daily. Putting stuff into your fridge, no doubt you’ve noticed how the usual circular pots and pans leave plenty of space that can’t be used. Put three together so that they touch, and there’s that odd empty shape in the middle. You’d be better off transferring the food to those translucent plastic boxes; they use shelf space far more efficiently.
Indeed, circles fail spectacularly at tiling the plane. In contrast, any triangle will do the job. An equilateral triangle—meaning all three sides are equal—produces a particularly orderly pattern. Its angles are also equal, 60 degrees. This means you can arrange six equilateral triangles to meet at a point, resulting in a “regular" hexagon—all of its sides are equal too, and the tiling happens by repeating this operation.
Therefore, we know that the regular hexagon also tiles the plane. So does one more regular shape: the square, of course. And you can easily visualize a rectangle of whatever size doing the job: you simply lay down a row of them, then another row, and so on.
This is why those plastic boxes are usually square or rectangular, though I’m not sure if you get hexagonal ones.
But other shapes make for some tiling intrigue. Surprisingly, perhaps, you can’t use a regular pentagon for tiling. Nor an octagon (8 sides), an enneagon (9) or a heptadecagon (17). In fact, the only three regular polygons that will tile a plane are the three I mentioned: triangle, square and hexagon.
But here’s something of a kicker: there are several other (irregular) pentagons that can tile the plane. In fact, searching for them has been something of a mathematical cottage industry since a German mathematician, Karl Reinhardt, found five of them in 1918. (In one of those cases, three of the pentagons combine to form, again, a regular hexagon.)
Along the way, Martin Gardner wrote his July 1975 Scientific American “Mathematical Games" column about tiling. A housewife in San Diego with an interest in art, Marjorie Rice, read that column and started searching on her own for tiling pentagons and actually found four. Makes a nice testament to the appeal and rewards of an interest in mathematics.
By 1985, there were 14 different pentagons known that could tile the plane. No more over the next 30 years, and mathematicians—as mathematicians do—began wondering if that was the definitive list. If it was, of course, they wanted to prove it. But they found no such proof either.
Then in July this year, a team of mathematicians at the University of Washington in Bothell filled the world’s tiling enthusiasts with delight. They had found a 15th tiling pentagon. “The problem of classifying convex pentagons that tile the plane is a beautiful mathematical problem that is simple enough to state so that children can understand it," Casey Mann, one of the mathematicians, told The Guardian.
Take a look at a picture of Mann’s team’s work on tiling a surface: it’s like a series of sleek paper planes flying across the frame. Beautiful indeed. Turn it into a paver block and it might make even a shoddily resurfaced road look good.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers explores the joy of mathematics, with occasional forays into other sciences.
Comments are welcome at dilip@livemint.com To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza
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