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Game, Set and Math
Enigmas and Conundrums
By Ian Stewart
Dover Publications, Inc.Copyright © 1989 Ian Stewart
All rights reserved.
Mother Worm's Blanket
"Bother!" said Mother Worm.
"Something the matter, dear?"
"It's our sweet little Wermentrude. I know I shouldn't criticize the child, but sometimes — well — her blanket's come off again! She'll be chilled to the bone!"
"Anne-Lida, worms don't have bones."
"Well, chilled to her endodermic lining, then, Henry! The problem is that when she goes to sleep, she wriggles around and curls up into almost any position, and the blanket falls off."
"Does she move once she's asleep?"
"No, Henry, she sleeps like a log."
She even looks like a log, thought Henry Worm, but did not voice the thought. "Then wait until she's asleep before you cover her up, dearest."
"Yes, Henry, I've thought of that. But there is another problem."
"Tell me, my pet."
"What shape should the blanket be?"
It took a while for Henry to sort that one out. It turned out that Mother Worm wanted to make a blanket which would completely cover her worm-child, no matter how she curled up. Just the worm, you understand: not the area she surrounds. The blanket can have holes. But, being thrifty, Mother Worm wished the blanket to have as small an area as possible.
"Ah," said Father Worm, who — as you will have noticed — is something of a pedant. "We may choose units so that the length of the little horr- ... dear little Wermentrude is 1 unit. You're asking what shape is the plane set of minimal area that will cover any plane curve of length 1. And no doubt you also wish to know what this minimal area is."
"Hmmmmmmmm. Tri-cky ..."
When you start thinking about Mother Worm's blanket, the greatest difficulty is to get any kind of grip. The problem tends to squirm away from you. But as Henry explained to his wife — in order to distract her attention from his inability to answer the question — there are some general principles that can form the basis of an attack. Suppose that we know where some points of the worm are: what can we say about the rest? He pointed out two such principles (box 1.1): they depend upon the fact that the shortest distance between two points is the straight line joining them.
"Excellent," said Father Worm. "Now, Anne-Lida my dear, we can make some progress. An application of the Circle Principle shows that a circle of diameter 2 will certainly keep Wermentrude warm. Lay the centre of the blanket over Baby's tail, my dear: the rest of her cannot be more than her total length away! How big is the blanket? Well, a circle of diameter 2 has an area of ?, which you'll recall is approximately 3.14159...."
"That's enough, Henry! I've already thought of something much better. Suppose that you (mentally!) chop Wermentrude into two at her mid-point. Each half lies inside a circle of radius ½ centred on her mid-point. If I place a circular blanket of radius ½ — that is, diameter 1 — so that its centre lies over Baby's mid-point, it will cover the dear little thing."
What's the area now? Remember pi-r-squared?
In fact this is the smallest circle that will always cover Baby, because if she stretches out straight she can poke out of any circle of diameter less than 1. But might a shape different from a circle be more economical? "It had better be," groaned Father Worm, who would have to pay for the blanket, as he retired to his study. Two hours later he emerged with several sheets of paper and announced that Anne-Lida's proposal, a circle of diameter 1, is at least twice as large as is necessary.
"Good news, my dear. A semicircle of diameter 1 is big enough to cover Baby no matter how much the little pest — er, pet — squirms before snoozing."
That cuts the area down even more: to what?
As I said, Henry Worm is a pedant. He won't say anything like that unless he's absolutely certain it's true. So he hasn't just spent his time doing experiments with semicircles: he has a proof that the unit semicircle (a semicircle of diameter 1) always works. It isn't an easy proof, and if you want to skip it I wouldn't blame you. But proof is the essence of mathematics, and you may be interested to see Father Worm's line of reasoning. If so, it's in box 1.2.
"Very clever, Henry," sniffed Anne-Lida. "But I think the same idea shows that you can cut some extra pieces off the semicircle. You see, when P and Q are closer together than b, the distance between X and Y is less than 1. That must leave room for improvement, surely?"
"Hrrrumph. You may well be right, my dear. But it gets very complicated to work out what happens next." And Henry rapidly changed the topic of conversation. My more persistent readers may wish to pursue the matter, because Anne-Lida is right: the unit semicircle is not the best possible shape. Indeed, nobody knows what shape Baby Worm's blanket should be. The problem is wide open. Remember, it must cover her no matter what shape she squirms into; and you must give a proof that this is the case! If you can improve on π/8, let me know.
Later that evening, Henry suddenly threw down his newspaper, knocking over a glass of Pupa-Cola and soaking the full-size picture of Maggot Thatcher on the front page. "Anne-Lida! We've forgotten to ask whether a solution exists at all!" You can't keep a good pedant down. But he has a point. Plane sets can be a lot more complicated than traditional things like circles and polygons. The blanket may not be convex: in fact it might have holes! For that matter, what do we mean by the "area" of a complicated plane set?
"My God," said Henry. "Perhaps the minimal area is zero!"
"Don't be silly, dear. Then there would be no blanket at all!"
Henry poured a replacement and sipped at it with a superior smirk. "Anne-Lida, it is obviously time I told you about the Cantor set."
"What have those horrible horsey snobs got to do with ..."
"Cantor, my dear, not canter. Georg Cantor was a German mathematician who invented a very curious set in about 1883. Actually, it was known to the Englishman Henry Smith in 1875 – but 'Smith set' wouldn't sound very impressive, would it? To get a Cantor set you start with a line segment of length 1, and remove its middle third. Now remove the middle third of each remaining piece. Repeat, forever. What is left is the Cantor set." (Figure 1.6)
"I don't see how there can be anything left, Henry."
"Oh, but there is. All the end-points of all the smaller segments are left, for a start. And many others. But you are right in one way, my dear. What is the length of the Cantor set?"
"Its ends are distance 1 apart, Henry."
"No, I meant the length not counting the gaps."
"I have no idea, Henry. But it looks very small to me. The set is mostly holes."
"Yes, like Wermentrude's sock."
"Are you criticizing me? I'm going to darn her sock tomorrow! Of all the..."
"No, no, my dear; nothing was further from my mind. Hrrumph. The length reduces to 2/3 the size at each stage, so the total length after the nth stage is (2/3)n. As n tends to infinity, this tends to 0. The length of the Cantor set is zero." Anne-Lida worked out the first few powers of 2/3 on her calculator — it wasn't a pocket calculator because worms don't have pockets — and nodded in agreement.
"Now the Cantor set, despite being mostly holes, has a remarkable property," Henry continued relentlessly. "Given any number between 0 and 1, there are two points in the Cantor set whose distance apart is exactly equal to that number. Er — I don't think you'd want to see a proof, my dear, so let us merely assume the result is true, yes? Good. Now, suppose that Baby can only curl up into rectangles ..."
"Henry, you know very well she's as wriggly as a baby worm ..."
"Pretend she's been playing tailball and is very stiff in the joints." Wermentrude, I must add, goes to a non-sexist equal opportunity awormative action school which discourages differences between boy and girl worms — not that you'd notice — and girl worms play tailball just like the boys. Nevertheless, Anne-Lida objected.
"You know very well worms don't have joints, Henry!"
"Oh, for heaven's sake! Pretend that they do, all right? Just to please me!"
"Very well," said Anne-Lida huffily. "Since you insist."
"Thank you. Because Wermentrude's length is 1, the height and the width of the rectangle are between 0 and 1. So I can find two points in the Cantor set whose distance apart is equal to the height, and two more whose distance apart is equal to the width. Now I consider the Cantor tartan."
"Cantor isn't a Scottish name, Henry!"
"Very well, the MacCantor tartan. I take a set of horizontal lines of unit length, spaced vertically according to the Cantor set, together with a set of vertical lines, spaced horizontally the same way (figure 1.7). Now, in the horizontal set I can find two lines whose distance apart is equal to the height of the rectangle, and in the vertical set two lines whose distance apart is equal to its width. So — as J. R. Kinney noticed in 1968 — the MacCantor tartan can be placed so that it covers Baby Worm's rectangle."
"You mean the perimeter, not the inside of the rectangle."
"Naturally. The blanket must cover Wermentrude, not the space she encloses."
"That's not a blanket, Henry: it's a net."
"If you wish, I shall rename this chapter 'Baby Eel's Net'. But then your name won't be mentioned ..."
"No, no, Henry. I now realize it is a cellular blanket."
"Excellent. It also has area zero. The horizontal part has area 0X1=0, and so does the vertical part, because the Cantor set has length 0."
"So for rectangular worms," said Anne-Lida, "there exists a blanket of area zero that will cover them all! What a bizarre result!" She paused. "But of course that's because rectangles are very special."
"Well, yes and no," said Henry Worm. "I've been reading about the problem, and it turns out that in 1970 D. J. Ward constructed a blanket of area zero capable of covering any polygonal worm. A worm made up of finitely many straight line segments, that is. The blanket is an incredibly messy tangle, of course — mostly holes."
"Curiouser and curiouser, my dear. And what of smooth worms, like our lithe and flexible Wermentrude?"
"Well, for a while mathematicians began to wonder whether there might exist a zero-area universal blanket for smooth worms — speaking in the vermicular, of course. But in 1979 J. M. Marstrand proved that no blanket of area zero can cover all smooth worms."
"Remarkable. It must have taken some very difficult geometry to prove that."
"I gather the main idea was to use the concept of the entropy of a totally bounded metric space, my pet."
"Fascinating, Henry! Do tell me more."
"Well — hrrrumph — I don't think you'd really want to know about that, Anne-Lida. Ergodic theory is kind of tricky ..."
"You don't know, do you Henry?"
"Well ... No. But at any rate, we know that the minimal area for Baby's blanket must be greater than zero."
Mother Worm can be a pedant too. "Do we, though, Henry? I mean, might there not be a blanket of area that works, and one of area ¼, and one of area 1/8, and so on — areas greater than zero but becoming as small as we please? Then the minimum area would be zero, but it wouldn't actually correspond to a blanket." Can you think of a simple problem about minimal areas for which this kind of thing happens? Here's a hint: Mother Gnat's tent.
But Father Worm knew when he was beaten, and was already talking about the analogous problem in three dimensions: Baby Worm's sleeping-bag. What is the minimal volume that will hold a worm of length 1 in ordinary three-dimensional space? And that problem is virtually unexplored. Can you make any progress worming your way towards a solution?
The circle of radius r = 1/2 has area πr2 = π(1/2)2 = π/4, is about 0.785. Easy! Yes, but this is just the worm-up problem ...
Halving that to get a semicircle leads to π/8, or about 0.393.
Here's an example of an area-minimizing problem which has solutions of arbitrarily small non-zero area, but does not have a solution with area zero. Mother Gnat is making a tent so that her daughter Gnathalie can go camping. Gnathalie is tiny, no more than a point; she always sleeps hovering a little way off the ground. The tent must cover her head to keep the rain off and reach down to the ground to keep out draughts. What is the smallest area of tent that will do the job?
The answer is that any area greater than zero will work, but zero itself will not.
To see why, imagine a point gnat G, some distance — which we may take to be 1 unit — above a plane. The problem of Mother Gnat's tent boils down to this: what is the smallest area of a surface whose boundary lies in the plane, and which passes through G? Consider a sharp conical surface (figure 1.8) whose base is a circle of radius r units. Then the surface area of the cone is πr, and we can make this as small as we want by choosing r small enough. For instance if r = 0.000000001 then the area is πr = 0.00000000314....
But to get zero area we must let r = 0, and then the cone becomes a line segment joining G to the plane. But a line segment isn't a surface!
This example shows that area-minimizing problems may not have solutions: that is, the "minimal" area may not be attainable.
Baby Worm's sleeping-bag: do you want to minimize the surface area or the volume? Your choice! Similar arguments can get you to a hemisphere of radius r = 1/2, with volume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], about 0.262; and surface area 3πr2 (why?) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], about 2.356. But it must be possible to improve on those figures.CHAPTER 2
The Drunken Tennis-Player
The tennis season has started up again.
A few weeks ago, I spent the afternoon at the local tennis-club, playing an enjoyable match with my friend Dennis Racket. He won in straight sets, 6-3,61,6-2. Afterwards, as we sank a few beers in the bar, a thought struck me.
"Dennis: how come you always beat me?"
"I'm better than you, old son."
"Yes, but you're not that much better. I've been keeping score and I reckon that I win one-third of the points. But I don't win one-third of the matches!"
"Let's face it, you don't win any matches against me." He took a quick swig at his beer. "That's because you don't win the crucial points, the ones that really matter. I mean, remember when you were leading 40-30 with the set at three games to two? You could have levelled the score at three all. Instead, you ..."
"Served a double fault. Yes, Dennis, I know all about that. But I reckon I still win about one in three of the crucial points! No, there must be another explanation."
"I'd like another beer, that's for sure," said Dennis. "My round. I'll be right back." He heaved himself to his feet and began to negotiate his way through the crowd towards the bar. I heard him shouting over the hubbub. "Elsie! Two pints of Samuel Smith's and a packet of peanuts!" With a glass in each hand, he began to make his way back. There were so many people that he went two steps sideways for every step forwards.
Then it hit me.
That's why Dennis always wins!
He sat down, and I decided to share my sudden insight. "Dennis, I've worked it out! Why you always win! I was watching you coming back from the bar, and I suddenly thought: drunkard's walk!"
"Actually, my son, they stagger. Anyway, I've only had two pints!"
I hastened to reassure him that my choice of phrase was nothing personal. The drunkard's walk — less colourfully called the random walk - is a mathematical concept: the motion of a point which moves along a line, going either left or right, at random. Or on a square grid, taking steps randomly north, south, east, or west. In 1960 Frederik Pohl wrote a science fiction story called Drunkard's Walk, and he described it like this:
Cornut remembered the concept with clarity and affection. He had been a second year student, and their house-master was old Wayne; the audio-visual had been a marionette drunkard, lurching away from a doll-sized lamp-post with random drunken steps in random drunken directions.
To simulate the simplest random walk, all you need is a 30 cm ruler and two coins. One coin acts as a marker, the other as a random number generator. Place the marker coin on the ruler at 15 cm. Toss the other one. If it comes down "heads", move the marker coin 1 cm to the right; if "tails", move it left (figure 2.1).
According to probability theory, after n moves you will be on average a distance [square root of n] cm away from the middle. (Try it!) Despite this, your chances of eventually returning to the middle are 1 (certainty). On the other hand, on average it takes infinitely long to get there. Random walks are subtle things. With a random walk on a square grid, you still have probability 1 of returning to the centre; but in three dimensions the probability of getting back to the centre is about 0.35. A drunkard lost in a desert will eventually reach the oasis; but an inebriated astronaut lost in space has roughly a one in three chance of getting home. Maybe they should have told ET that.
Excerpted from Game, Set and Math by Ian Stewart. Copyright © 1989 Ian Stewart. Excerpted by permission of Dover Publications, Inc..
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