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When Least Is Best

How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible
  • Paul J. Nahin
Language: English
Published/Copyright: 2003
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Volume 114 | Princeton Science Library

About this book

A mathematical journey through the most fascinating problems of extremes and how to solve them

What is the best way to photograph a speeding bullet? How can lost hikers find their way out of a forest? Why does light move through glass in the least amount of time possible? When Least Is Best combines the mathematical history of extrema with contemporary examples to answer these intriguing questions and more. Paul Nahin shows how life often works at the extremes—with values becoming as small (or as large) as possible—and he considers how mathematicians over the centuries, including Descartes, Fermat, and Kepler, have grappled with these problems of minima and maxima. Throughout, Nahin examines entertaining conundrums, such as how to build the shortest bridge possible between two towns, how to vary speed during a race, and how to make the perfect basketball shot. Moving from medieval writings and modern calculus to the field of optimization, the engaging and witty explorations of When Least Is Best will delight math enthusiasts everywhere.

Author / Editor information

Paul J. Nahin is the author of many popular math books, including How to Fall Slower Than Gravity and Hot Molecules, Cold Electrons (both Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.

Reviews

"Nahin has written a beautifully clear, fascinating book on a topic which is truly vital to so many areas of science and I would recommend anyone who enjoys puzzle solving and having new tools to tackle old (or new) problems should read it."---Jonathan Shock, Mathemafrica

"Terrific fun to read!"—Bonnie Shulman, MAA Online

"A tour de force."—Dennis S. Bernstein, IEEE Control Systems Magazine



"Highly recommended."—Clark Kimberling, Mathematical Intelligencer

"A valuable and stimulating introduction to problems that have fascinated mathematicians and physicists for millennia."—D. R. Wilkins, Contemporary Physics



"Anyone with a modest command of calculus, a curiosity about how mathematics developed, and a pad of paper for calculations will enjoy Nahin's lively book."—Donald R. Sherbert, SIAM Review

Topics

  • Publicly Available

    Frontmatter

    i
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    Contents

    ix
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    Preface to the 2021 Edition

    xiii
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  • Publicly Available

    Preface to the 2007 Paperback Edition

    xix
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  • Publicly Available

    Preface

    xxvii
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  • 1. Minimums, Maximums, Derivatives, and Computers
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    1.1 Introduction

    1
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    1.2 When Derivatives Don’t Work

    4
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    1.3 Using Algebra to Find Minimums

    5
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    1.4 A Civil Engineering Problem

    9
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    1.5 The AM-GM Inequality

    13
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    1.6 Derivatives from Physics

    20
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    1.7 Minimizing with a Computer

    24
  • 2. The First Extremal Problems
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    2.1 The Ancient Confusion of Length and Area

    37
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    2.2 Dido’s Problem and the Isoperimetric Quotient

    45
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    2.3 Steiner’s “Solution” to Dido’s Problem

    56
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    2.4 How Steiner Stumbled

    59
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    2.5 A “Hard” Problem with an Easy Solution

    62
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    2.6 Fagnano’s Problem

    65
  • 3. Medieval Maximization and Some Modern Twists
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    3.1 The Regiomontanus Problem

    71
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    3.2 The Saturn Problem

    77
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    3.3 The Envelope-Folding Problem

    79
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    3.4 The Pipe-and-Corner Problem

    85
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    3.5 Regiomontanus Redux

    89
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    3.6 The Muddy Wheel Problem

    94
  • 4. The Forgotten War of Descartes and Fermat
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    4.1 Two Very Different Men

    99
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    4.2 Snell’s Law

    101
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    4.3 Fermat, Tangent Lines, and Extrema

    109
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    4.4 The Birth of the Derivative

    114
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    4.5 Derivatives and Tangents

    120
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    4.6 Snell’s Law and the Principle of Least Time

    127
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    4.7 A Popular Textbook Problem

    134
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    4.8 Snell’s Law and the Rainbow

    137
  • 5. Calculus Steps Forward, Center Stage
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    5.1 The Derivative: Controversy and Triumph

    140
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    5.2 Paintings Again, and Kepler’s Wine Barrel

    147
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    5.3 The Mailable Package Paradox

    149
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    5.4 Projectile Motion in a Gravitational Field

    152
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    5.5 The Perfect Basketball Shot

    158
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    5.6 Halley’s Gunnery Problem

    165
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    5.7 De L’Hospital and His Pulley Problem, and a New Minimum Principle

    171
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    5.8 Derivatives and the Rainbow

    179
  • 6. Beyond Calculus
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    6.1 Galileo’s Problem

    200
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    6.2 The Brachistochrone Problem

    210
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    6.3 Comparing Galileo and Bernoulli

    221
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    6.4 The Euler-Lagrange Equation

    231
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    6.5 The Straight Line and the Brachistochrone

    238
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    6.6 Galileo’s Hanging Chain

    240
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    6.7 The Catenary Again

    247
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    6.8 The Isoperimetric Problem, Solved (at last!)

    251
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    6.9 Minimal Area Surfaces, Plateau’s Problem, and Soap Bubbles

    259
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    6.10 The Human Side of Minimal Area Surfaces

    271
  • 7. The Modern Age Begins
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    7.1 The Fermat/Steiner Problem

    279
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    7.2 Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs

    286
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    7.3 The Traveling Salesman Problem

    293
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    7.4 Minimizing with Inequalities (Linear Programming)

    295
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    7.5 Minimizing by Working Backwards (Dynamic Programming)

    312
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    Appendix A. The AM-GM Inequality

    331
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    Appendix B. The AM-QM Inequality, and Jensen’s Inequality

    334
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    Appendix C. “The Sagacity of the Bees” (the preface to Book 5 of Pappus’ Mathematical Collection)

    342
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    Appendix D. Every Convex Figure Has a Perimeter Bisector

    345
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    Appendix E. The Gravitational Free-Fall Descent Time along a Circle

    347
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    Appendix F. The Area Enclosed by a Closed Curve

    352
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    Appendix G. Beltrami’s Identity

    359
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    Appendix H. The Last Word on the Lost Fisherman Problem

    361
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    Appendix I. Solution to the New Challenge Problem

    364
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    Acknowledgments

    367
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    Index

    369

Publishing information
Pages and Images/Illustrations in book
eBook published on:
May 18, 2021
eBook ISBN:
9780691220383
Pages and Images/Illustrations in book
Main content:
392
Other:
99 b/w illus.
https://doi.org/10.1515/9780691220383
eBook ISBN: 9780691220383
Keywords for this book
Mathematician; Mathematics; Calculation; Geometry; Cartesian coordinate system; Inequality of arithmetic and geometric means; Derivative; Quantity; Surface area; Maxima and minima; Result; Special case; Euler–Lagrange equation; Snell's law; Diameter; Square root; Equation; Isoperimetric problem; Jensen's inequality; Notation; Line segment; Right triangle; Differential calculus; Pythagorean theorem; Illustration; Ray (optics); Trigonometric functions; Tangent; Hypotenuse; Potential energy; Vertex angle; Kinetic energy; Linear programming; Curve; Coordinate system; Change of variables; Variable (mathematics); Dimension; Iteration; Fermat's principle; Science; Integer; Addition; Real number; Circumference; Parametric equation; Parabola; Convex function; Chain rule; Summation; Refraction; Perimeter; Catenary; Pumping station; Quadratic equation; Mathematical problem; Arc (geometry); Pierre de Fermat; Instant; Almost surely; Scientific notation; Logarithm; Cylinder (geometry); Clockwise; Hyperbolic function; Length; Differential equation; Cycloid; Equilateral triangle; Elementary function
Audience(s) for this book
College/higher education;Professional and scholarly;
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