Fixed point theoremsvital tools for mathematicianscan seem esoteric, but fixed points actually crop up every now and then in unexpected places which you might not readily associate with mathematics
Years ago, I ran across the Time-Life book Mathematics on my parents’ bookshelf. Plenty in it thrilled me and left me thirsting for more. It features explorations and explanations of chance and approximation, topology and the golden ratio and plenty more. But towards the end of the book is a pair of images that I always stopped at in wonder, because I always found them oddly mysterious. They may constitute the reason I quietly abducted that selfsame book—now considerably worse for wear—and today it resides proudly on my shelf.
The first image has two sheets of paper, identical except one is yellow and the other blue. They are both printed with a 12 x 10 grid, filled with the numbers from 1 to 120. The yellow sheet floats an inch or two above the blue one, so you can see the numbers on both.
In the second image, the yellow sheet is crumpled and lies forlornly on the blue one. Here’s the text that accompanies the pair:
“First, a numbered paper sheet is placed over an exact duplicate so that all points on both the sheets are aligned. Then the top sheet is crumpled above the bottom sheet. One point on the crumpled sheet must still be over its starting point. Here it is a point in the region of the number 78."
Sure enough, the yellow square numbered “78" rests almost directly on its blue counterpart. I’d look at this and marvel. How is it possible to crumple one sheet — that is, to totally destroy its neatly gridded order — and still have one point in that sheet remain in the same place? Such an apparently counter-intuitive vision, but this is just what happens. It’s just one example of what a “fixed-point theorem"—of which there are several in mathematics—suggests. This one, Brouwer’s Theorem, makes a simple point and I quote from the Time-Life book again: “When a surface is ‘transformed into itself’ [in this case, by crumpling the yellow sheet and dropping it on the blue sheet] in this way, one point on the surface will remain where it was."
A little underwhelmed, are you? Well, may be this will help get your juices flowing a little faster. Take a map of Mumbai, or Banaskantha, or wherever you are as you read this. Drop it on the floor, crumpled or otherwise. Brouwer’s Fixed Point Theorem tells us that there is some point on the map that is positioned directly over the real point on the ground that it represents.
This is not so hard to imagine, actually. If you’re standing in Bhuleshwar to drop your map, say, you can probably pick out with reasonable accuracy the spot on the map—somewhere near where it says ‘Bhuleshwar’ in large letters— that corresponds to the spot on the ground where you’ve dropped it. That’s what Brouwer’s Theorem means.
For mathematicians, fixed point theorems are vital tools. Using them, they can show that differential equations have solutions, or that there are equilibria in game theory problems. But if that’s too esoteric, fixed points actually crop up every now and then elsewhere too—in unexpected places you might not readily associate with mathematics.
An implication of the fixed point concept is that you can’t cover a sphere with hair without there being at least one fixed point from which the hair radiates outward.
Like, for example, a little puzzle my father first sprang on me. It always foxed me—until I learned to look at it a in certain way. Imagine a priest who lives at the bottom of a hill. At the top is a temple to the god he venerates. Once a year, he makes a pilgrimage to the temple: sets out at daybreak, stops for lunch and tea, reaches the temple just as the sun goes down. He stays a few days to do his rituals, then returns: sets out again at daybreak, stops for lunch and tea, reaches home just before sundown because descending is a little quicker than ascending.
Question: is there a point on the hill at which he is at the same time on both days? (Note: if there is one, we’re not interested in what that time is, but whether there is such a point at all, such a moment at all). That is, is there a fixed point for the two journeys? Think it over for a bit, I’ll give you the answer at the end of this column.
For another example off the top of my head, take the top of your head. Or better yet, the nearest head you can find and examine. One implication of the concept of a fixed point is that you cannot cover a sphere—or someone’s head—with hair, without there being at least one fixed point from which the hair radiates outwards. You know that point: it’s the whorl—sometimes two—that’s readily apparent on any head, especially a male’s. You’ll also find it on female heads, except that female heads tend to have longer hair than male ones, and long hair makes such a whorl less obvious. A fixed point, mathematically speaking, on your head! Who would have imagined that? Yet we see it every day without even thinking about it.
The spirit of this radiating hair is captured in the so-called “hairy ball theorem" (yes, that is in fact its name—mathematicians do have a sense of humour). If you have a ball covered with hair—a tennis ball, for example, or that head you’re examining once again—and you try to comb all the hair flat, you’ll necessarily leave one or more tufts of hair standing, somewhere on the ball. This will be familiar to anyone who has cut a kid’s hair short. Where the whorl is, you’ll usually find a small clump of hair standing straight, defying all attempts to smoothen it with a comb.
This is the basis for an intriguing thought experiment. Think of the winds on our planet as strands of hair, in the sense that they go from one point to another—like a hair does. According to the hairy ball theorem, as long as there are winds that blow, there has to be at least one point on our Earth where there’s no wind at all. A “tuft", you might say. You’ve heard of the eye of a storm—well, this no-wind spot is one such. Thus the rather startling realization that at any given time, there’s a storm raging somewhere on earth.
In truth, and for reasons I won’t get into here, zero winds somewhere on the planet do not imply that there’s a cyclone happening right there. Still, this thought experiment remains a nice demonstration of what fixed points mean.
And if you broaden the idea of a fixed point to mean simply keeping some things invariant through some manipulations, you’re on the threshold of plenty more mathematical exploration. A mathematics professor I know told me about the magic he does in his classes to introduce fairly deep concepts. Some of it involves cards, and some of those use a little manoeuvre called the Hummer shuffle. Named for an eccentric genius called Bob Hummer, it’s simple indeed: just turn over the top two cards of your deck so that they switch from face-down to face-up (or vice-versa). Now cut the deck however you choose, do the Hummer shuffle again—in fact, do both moves as many times as you like in any order. When you stop, you will have a deck that has some unpredictable number of face-up cards. But even if that’s unpredictable, there is one feature of your card deck that has stayed invariant through all your shuffling and cutting: the number of face-up cards is even. Give a mathematically-minded magician that little fact, and she’ll use it to invent any number of awe-inspiring card tricks. And if you do those tricks for a class of wide-eyed college students, you open the door to more profound mathematics that follows.
And what about the priest and his pilgrimage? Is there a spot on the hill that he reaches at the same time on both days?
The best way to answer that question is to think of two priests on the same day, one walking up the hill and the other walking down. At some time during the day, they will bump into each other. There you are: that’s the spot we’re searching for. They’re both right there right then.
Perhaps they will sit down and have lunch together. I’m told there’s an excellent restaurant there: good meatball dishes, though watch for hairs. It’s called The Fixed Point.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His latest book is Jukebox Mathemagic: Always One More Dance. His Twitter handle is @DeathEndsFun
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