Four Colors Suffice

by Wilson, Robin J.

• ISBN13: 9780691120232
• ISBN10: 0691120234
• Format: Paperback
• Copyright: 2004-10-18
• Publisher: Princeton Univ Pr

Summary

On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the "map problem" was solved. The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm. Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.

Author Biography

Robin Wilson is Senior Lecturer of Mathematics at the Open University and Fellow of Keble College, Oxford University. He was formerly Visiting Professor in the History of Mathematics at Gresham College, London, and is a frequent visitor to The Colorado College, Colorado Springs

Table of Contents

Preface	p. xi
The four-colour problem	p. 1
What is the Four-colour Problem?
Why Is It Interesting?
Is It Important?
What Is Meant by 'Solving' It?
Who Posed It, and How Was It Solved?
Painting by Numbers
Two Examples
The problem is posed	p. 16
De Morgan Writes a Letter
Hotspur and the Athenaeum
Mobius and the Five Princes
Confusion Reigns
Euler's famous formula	p. 38
Euler Writes a letter
From Polyhedra to Maps
Only Five Neighbours
A Counting Formula
Cayley revives the problem ...	p. 60
Cayley's Query
Knocking Down Dominoes
Minimal Criminals
The Six-Colour Theorem
... and Kempe solves it	p. 73
Sylvester's New Journal
Kempe's Paper
Kempe Chains
Some Variations
Back to Baltimore
A chapter of accidents	p. 95
A Challenge for the Bishop
A Visit to Scotland
Cycling Around Polyhedra
A Voyage Around the World
Wee Planetoids
A bombshell from Durham	p. 116
Heawood's Map
A Salvage Operation
Colouring Empires
Maps on Doughnuts
Picking Up the Pieces
Crossing the Atlantic	p. 143
Two Fundamental Ideas
Finding Unavoidable Sets
Finding Reducible Configurations
Colouring Diamonds
How Many Ways?
A new dawn breaks	p. 169
Doughnuts and Traffic Cops
Heinrich Heesch
Wolfgang Haken
Enter the Computer
Colouring Horseshoes
Success!...	p. 190
A Heesch-Haken Partnership?
Kenneth Appel
Getting Down to Business
The Final Onslaught
A Race Against Time
Aftermath
... but is it a proof?	p. 214
Cool Reaction
What is a Proof Today?
Meanwhile ...
A New Proof
The Future ...
Notes and references	p. 229
Chronology of events	p. 245
Glossary	p. 249
Picture credits	p. 255
Index	p. 257
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