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Maybe you hate the backsplash tiles behind your kitchen sink. I know I do. Not your tiles, I mean – mine. Their pattern is a boring grid; their color, a grubby off-white (if you’ve ever dug up grubs in your garden, you know what color I mean). A few of the tiles are stamped with chintzy brown sunflowers in a listless attempt at cheer. It probably doesn’t help that I hate our kitchen sink as well, a sink wedged in a countertop corner for added inconvenience.
But maybe you’re lucky. Maybe you have a convenient kitchen sink and exciting backsplash tile. If your backsplash tile is laid in a hopscotch pattern, you even have the Pythagorean theorem lurking right behind your sink. To the left is a backsplash much nicer than mine, tastefully tiled in a hopscotch pattern (well, tasteful aside from the peculiar choice of blue grout). To mathematicians, the hopscotch pattern is known as “Pythagorean tiling”, because of how beautifully it illustrates the Pythagorean theorem. Several proofs of the Pythagorean theorem exist, some of which are more intuitive than others. To me, the diagram superimposed on the hopscotch tiling produces one of the most intuitive proofs, especially for children. If you like, you can knock off reading right now and do the proof yourself: the diagram has all the labeling you need.
A right triangle is one containing a right angle (an angle measuring 90˚, or π/2 in radians). Its hypotenuse is the side of the triangle opposite the right angle. For the theorem, the length of the hypotenuse is called c, and the length of the triangle’s other two sides are called a and b.
The area of the square outlined in white is c^2. The area of the dark brown square inside it is (a – b)^2. The four right triangles surrounding the dark brown square each have area 1/2 ab. Because the dark brown square and the four triangles surrounding it add up to make the square outlined in white,
(a – b)^2 + 4 * 1/2 ab = c^2. Now expand (a – b)^2 and rewrite 4 * 1/2 as 2:
a^2 – 2ab + b^2 + 2ab = c^2. Notice 2ab and -2ab cancel:
a^2 + b^2 = c^2. Ta da!
That’s the Pythagorean theorem.
Simple. Kids only need to know how to expand a binomial in order to do this computation, something we learned at age 12, but which we could have been taught sooner. So why isn’t proving the Pythagorean theorem more widely taught?
I’m not sure. The same teacher who taught us binomials taught us the Pythagorean theorem and showed us a proof of it – the proof she happened to remember. Her proof was an asymmetric dissection – I don’t remember which one: the asymmetry made it harder for a kid to believe it would generalize to any right triangle, and very hard to remember the initial setup. A symmetric setup for proving the Pythagorean theorem, on the other hand, makes it easier to visualize how changing the lengths of a and b nonetheless leaves the formula a^2 + b^2 = c^2 (where c is the resulting hypotenuse) unchanged, and the symmetry also makes the setup memorable. I believe I was lucky, though, to have a teacher who bothered to show us any proof at all.
If your kitchen-sink backsplash is like mine, square tiles all of the same size in the typical boring grid, you can still use them to diagram a proof of the Pythagorean theorem, by nesting a square askew inside another square. That’s how Pythagoras did it. Follow the link and you’ll see a diagram different from the one above, but with the same symmetry. I happen to like the proof-by-hopscotch-tile, though, if only because I’m annoyed at my kitchen sink.
This post is the second in a series on kitchen math. The first one in the series is here.