A Mathematical Mosaic

255 Pages·1996·4.947 MB·English

by  Ravi Vakil

A MATHEMATICAL MOSAIC
A MATHEMATICAL MOSAIC
//
Patterns & Problem Solving
by
Ravi Vakil
Cover design: Pronk&Associates
Brendan Kelly Publishing Inc. Toronto, Ontario
2122 Highview Drive
Burlington, Ontario L7R 3X4 Illustrations: Taisa Kelly
SEP 81998
Brendan Kelly Publishing Inc.
2122 Highview Drive
Burlington, Ontario L7R 3X4
Copyright © 1996 by Brendan Kelly Publishing Inc.
All rights reserved. No part of this work may be
reproduced or transmitted in any form or by any means,
electronic or mechanical, including photocopying and
recording, or by any information storage or retrieval
system, without permission in writing from the publisher.
ISBN 1-895997-04-6
ATTENTION EDUCATIONAL ORGANIZATIONS
Quantity discounts are available on bulk purchases of this book for educational
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L7R 3X4
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t o ttz z m z m o z y o j m y ja tliE .z
5
M athematics is an intensely social
discipline. My first debt is to
those who have shared their love
of mathematics with me: my teachers,
professors, colleagues, students, and friends. I am especially grateful to the young
men and women who agreed to share their unique perspectives on the joys of
mathematics for the seven profiles included in this book. Most of the ideas contained
within A Mathematical Mosaic have been passed on to me over the years by word
of mouth. Although they are presented here in new and, I hope, engaging ways,
the origins of many of these gems have been lost in mathematical folklore. I have
cited references where I am aware of them, but there are many other sources that I
must gratefully, if silently, acknowledge here.
Michael Roth, a graduate student in mathematics at Harvard University, and Naoki
Sato, a talented undergraduate at the University of Toronto, both gave essential
advice on the final draft. Mark Wunderlich, formerly a student of mine at Harvard
College and now a graduate student in philosophy at the University of Arizona,
provided feedback during the development of much of this material. The book is
far more readable thanks to their efforts; any flaws that remain are solely the
responsibility of the author.
From helping to develop the original concept to suggesting final revisions, Brendan
Kelly transcended his role as publisher. This book has evolved during long
conversations with him, and his influence is visible on every page. I greatly
appreciate his creative vision and commitment to this rather unorthodox project.
Finally, I must thank Alice Staveley of Oxford University for her meticulous editorial
advice and endless patience.
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Publisher’s Introduction
A Celebration of the Human Intellect
A well-known story from the folklore of mathematical history tells how
Archimedes bounded from his bathtub and ran naked through the streets
of ancient Greece yelling, “Eureka!” [I have found it]. Apparently a period
of intense meditation in the bathtub had provided the inspiration for Archimedes’
discovery of the law of hydrostatics which now bears his name. Whether apocryphal
or authentic, this legend captures the ecstasy which accompanies a flash of insight
into a deep problem. It is to those who relish the joy of mathematical discovery
that this book is dedicated. It is for those who chase relentlessly the thrill of eureka
that this book is written.
Through the past three millennia, the mental giants of each generation have pitted
their problem solving skills against the most intractable problems to challenge the
human mind. When a problem yielded like a dragon to a deft sword, the successful
strategy was refined, polished, and added to the growing body of intellectual
technique known as mathematics. One of the greatest problem solvers of all time,
Isaac Newton, characterized his favorite discoveries as “smoother pebbles or prettier
shells than ordinary”.1 From an assortment of the most beautiful pebbles and the
prettiest shells of mathematical technique, Ravi Vakil, author of this book, has
created an intriguing mosaic. It is a mosaic which links various branches of
mathematics through powerful overarching ideas. Ravi himself is a preeminent
problem solver, having won top honors on virtually every mathematics competition
and olympiad. His remarkable achievements are outlined on page 9 and on the
back cover.
Throughout this book, Ravi Vakil profiles several other recent winners of
mathematical competitions and olympiads: young people who have honed their
intellectual gifts to world class levels. As you read these human interest features,
you will observe that these fine young minds all share a passion for the eureka
sensation. It is something they all understand and something that bonds them
together in friendly collegial competitions.
We are pleased to publish this celebration of the human intellect. It is offered as a
salute to the problem solvers of the present and future and as a tribute to the
intellectual giants of the past. To the reader, we issue a challenge. We dare you to
flip through the pages of this book without finding any problems to pique your
curiosity!
Tor the complete quote, see page 8.
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/ do not know what I may appear to the world;
but to myself I seem to have been only like a
boy playing on the seashore, and diverting
myself in now and then finding a smoother
pebble or a prettier shell than ordinary, whilst
the great ocean of truth lay all undiscovered
before me.
— Isaac Newton
Shout tfje Suttjor
PHOTO BY JEWEL RANDOLPH
Ravi Vakil's resume reads like a dictionary
of superlatives. During his high school
years he won every major mathematics
competition. These include first place standing
in the Canadian Mathematical Olympiad for two
successive years, first place standing in North
America on the 1988 USA Mathematical
Olympiad, and two gold medals and one silver
medal in the International Mathematical
Olympiad. He also won first prize in the
Canadian Association of Physicists Competition
and led his high school computer team to three
provincial championships. Ravi was
valedictorian of his 1988 graduating class at
Martingrove Collegiate Institute in Metropolitan
Toronto and was awarded the Academic Gold
Medal and the Governor General's Award for excellence.
During his undergraduate years at the University of Toronto, Ravi raised his brilliant
achievements to new heights. In all four years he placed among the top five
competitors in the prestigious North American Putnam Mathematical Competition,
qualifying him as Putnam Fellow in each year. He graduated in 1992 with B. Sc.
and M. Sc. degrees; for his B. Sc., he was awarded the Governor General's Medal
for the highest graduating marks at the University of Toronto.
Ravi's extraordinary achievements have not prevented him from pursuing his many
other interests. These include squash, debating, student government, journalism,
and Amnesty International. For this breadth of involvement he has received
numerous scholarships including the John H. Moss Memorial Fund Scholarship
for the best all-round graduating student at the University of Toronto.
In spite of these personal successes, Ravi Vakil is a compassionate person with an
eager willingness to help others. He has worked extensively as a coach of the
Canadian Team to the International Mathematical Olympiad from 1989 through
1995. He is also co-founder of Mathematical Mayhem, a mathematical problem­
solving journal for high school and undergraduate students — the only student-run
journal of its kind in the English-speaking world!
Ravi Vakil is currently a Ph.D. candidate in pure mathematics at Harvard University,
working in Algebraic Geometry under the supervision of Dr. Joseph Harris.
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Jforetoorti
The public face of mathematics is sometimes grey and utilitarian: math is a
useful tool, a haphazard collection of recipes and algorithms, a necessary
prerequisite to understanding science. The private face is much more
beautiful: Mathematics as Queen, not servant, of Science. Math is a uniquely
aesthetic discipline; mathematicians use words like beauty, depth, elegance, and
power to describe excellent ideas. In order to truly enjoy mathematics, one must
learn to appreciate the beauty of elegant arguments, and then learn to construct
them.
In A Mathematical Mosaic, I hope to get across a little of what it is that
mathematicians actually do. Most “big ideas” and recurring themes in mathematics
come up in surprisingly simple problems or puzzles that are accessible with
relatively little background. Many of them have been collected here, along with
an inkling of how they relate to the frontiers of modem research. These themes
will also be traced backwards; often ideas centuries old take on new meaning and
relevance as mathematical understanding advances. And if in the process of looking
at these important ideas we get a little playful and irreverent, well, that is also in
keeping with the nature of mathematics.
This book does not purport to teach problem solving, although it might communicate
what is interesting and exciting about grappling with a problem. Instead, the reader
should be left with a large chunk of mathematics to digest slowly.
This is not a book intended to ever be “finished with” or completed. Instead, it
should be read bit by bit. Like all mathematics, it should be read with pencil in
hand, with more time spent in deep thought or frantic scribbling than in actual
reading. Mathematics is not a spectator sport!
You will also notice that, like any mosaic, this book conies in many small pieces of
different sorts. It is loosely arranged in order of difficulty, and the headers often
indicate the general theme of the article. You might enjoy just starting with one
section which especially interests you; from there, you can skip to other sections
which link with your earlier choices. You will soon discover that ideas from one
field carry over into many others. And if you don’t end up at a section that amazes
and perplexes you, I will be sorely disappointed.
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