Measurement Paul lockhart M E A S U R E M E N T M E A S U R E M E N T Paul Lockhart The Belknap Press of Harvard University Press Cambridge, Massachusetts • London, England 2012 Copyright © 2012 by the President and Fellows of Harvard College All rights reserved Printed in the United States of America Library of Congress Cataloging-in-Publication Data Lockhart, Paul. Measurement / Paul Lockhart. p. cm. Includes index. ISBN 978-0-674-05755-5 (hardcover : alk. paper) 1. Geometry. I. Title. QA447.L625 2012 516—dc23 2012007726 For Will, Ben, and Yarrow C O N T E N T S Reality and Imagination 1 On Problems 5 Part One: Size and Shape 21 In which we begin our investigation of abstract geometrical figures. Symmetrical tiling and angle measurement. Scaling and proportion. Length, area, and volume. The method of exhaustion and its consequences. Polygons and trigonometry. Conic sections and projective geometry. Mechanical curves. Part Two: Time and Space 199 Containing some thoughts on mathematical motion. Coordinate systems and dimension. Motion as a numerical relationship. Vector representation and mechanical relativity. The measurement of velocity. The differential calculus and its myriad uses. Some final words of encouragement to the reader. Acknowledgments 399 Index 401 R E A L I T Y A N D I M A G I N A T I O N There are many realities out there. There is, of course, the physical reality we find ourselves in. Then there are those imaginary universes that resemble physical reality very closely, such as the one where everything is exactly the same except I didn’t pee in my pants in fifth grade, or the one where that beautiful dark-haired girl on the bus turned to me and we started talking and ended up falling in love. There are plenty of those kinds of imaginary realities, believe me. But that’s neither here nor there. I want to talk about a different sort of place. I’m going to call it “mathematical reality.” In my mind’s eye, there is a universe where beautiful shapes and patterns float by and do curious and surprising things that keep me amused and entertained. It’s an amazing place, and I really love it. The thing is, physical reality is a disaster. It’s way too compli- cated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in this universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line. Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I will never hold a circle in my hand, but I can hold one in my mind. And I can measure it. Mathematical reality is a beautiful wonderland of my own 2 M EASU R EM ENT creation, and I can explore it and think about it and talk about it with my friends. Now, there are lots of reasons people get interested in physi- cal reality. Astronomers, biologists, chemists, and all the rest are trying to figure out how it works, to describe it. I want to describe mathematical reality. To make patterns. To figure out how they work. That’s what mathematicians like me try to do. The point is I get to have them both—physical reality and mathematical reality. Both are beautiful and interesting (and somewhat frightening). The former is important to me because I am in it, the latter because it is in me. I get to have both these wonderful things in my life and so do you. My idea with this book is that we will design patterns. We’ll make patterns of shape and motion, and then we will try to understand our patterns and measure them. And we will see beautiful things! But I won’t lie to you: this is going to be very hard work. Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively. The truth is, I don’t know of any human activity as demanding of one’s imagination, intuition, and ingenuity. But I do it anyway. I do it because I love it and because I can’t help it. Once you’ve been to the jungle, you can never really leave. It haunts your waking dreams. So I invite you to go on an amazing adventure! And of course, I want you to love the jungle and to fall under its spell. What I’ve tried to do in this book is to express how math feels to me and to show you a few of our most beautiful and excit- R EA LITY AN D IM AGINATION 3 ing discoveries. Don’t expect any footnotes or references or anything scholarly like that. This is personal. I just hope I can manage to convey these deep and fascinating ideas in a way that is comprehensible and fun. Still, expect it to be slow going. I have no desire to baby you or to protect you from the truth, and I’m not going to apolo- gize for how hard it is. Let it take hours or even days for a new idea to sink in—it may have originally taken centuries! I’m going to assume that you love beautiful things and are curious to learn about them. The only things you will need on this journey are common sense and simple human curiosity. So relax. Art is to be enjoyed, and this is an art book. Math is not a race or a contest; it’s just you playing with your own imagina- tion. Have a wonderful time! O N P RO B L E M S What is a math problem? To a mathematician, a problem is a probe—a test of mathematical reality to see how it behaves. It is our way of “poking it with a stick” and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: What does it do and why does it do it? So we poke it—only not with our hands and not with a stick. We have to poke it with our minds. As an example, let’s say you’ve been playing around with triangles, chopping them up into other triangles and so forth, and you happen to make a discovery: When you connect each corner of a triangle to the middle of the opposite side, the three lines seem to all meet at a point. You try this for a wide variety of triangles, and it always seems to happen. Now you have a mystery! But let’s be very clear about exactly what the mystery is. It’s not about your draw- ings or what looks like is happening on paper. The question of what pencil-and-paper triangles may or may not do is a scien- tific one about physical reality. If your drawing is sloppy, for example, then the lines won’t meet. I suppose you could make an extremely careful drawing and put it under a microscope, 6 M EASU R EM ENT but you would learn a lot more about graphite and paper fibers than you would about triangles. The real mystery is about imaginary, too-perfect-to-exist triangles, and the question is whether these three perfect lines meet in a perfect point in mathematical reality. No pencils or microscopes will help you now. (This is a distinction I will be stressing throughout the book, probably to the point of annoy- ance.) So how are we to address such a question? Can anything ever really be known about such imaginary objects? What form could such knowledge take? Before examining these issues, let’s take a moment to simply delight in the question itself and to appreciate what is being said here about the nature of mathematical reality. What we’ve stumbled onto is a conspiracy. Apparently, there is some underlying (and as yet unknown) structural interplay going on that is making this happen. I think that is marvelous and also a little scary. What do triangles know that we don’t? Sometimes it makes me a little queasy to think about all the beautiful and profound truths out there waiting to be discov- ered and connected together. So what exactly is the mystery here? The mystery is why. Why would a triangle want to do such a thing? After all, if you drop three sticks at random they usually don’t meet at a point; they cross each other in three different places to form a