When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible

by Paul J. Nahin

Overview

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.

Product Details

ISBN-13:	9780691218762
Publisher:	Princeton University Press
Publication date:	05/18/2021
Series:	Princeton Science Library , #114
Pages:	406
Product dimensions:	5.50(w) x 8.50(h) x (d)

About the Author

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.

Table of Contents

	Preface	xiii
1.	Minimums, Maximums, Derivatives, and Computers	1
1.1	Introduction	1
1.2	When Derivatives Don't Work	4
1.3	Using Algebra to Find Minimums	5
1.4	A Civil Engineering Problem	9
1.5	The AM-GM Inequality	13
1.6	Derivatives from Physics	20
1.7	Minimizing with a Computer	24
2.	The First Extremal Problems	37
2.1	The Ancient Confusion of Length and Area	37
2.2	Dido's Problem and the Isoperimetric Quotient	45
2.3	Steiner's "Solution" to Dido's Problem	56
2.4	How Steiner Stumbled	59
2.5	A "Hard" Problem with an Easy Solution	62
2.6	Fagnano's Problem	65
3.	Medieval Maximization and Some Modern Twists	71
3.1	The Regiomontanus Problem	71
3.2	The Saturn Problem	77
3.3	The Envelope-Folding Problem	79
3.4	The Pipe-and-Corner Problem	85
3.5	Regiomontanus Redux	89
3.6	The Muddy Wheel Problem	94
4.	The Forgotten War of Descartes and Fermat	99
4.1	Two Very Different Men	99
4.2	Snell's Law	101
4.3	Fermat, Tangent Lines, and Extrema	109
4.4	The Birth of the Derivative	114
4.5	Derivatives and Tangents	120
4.6	Snell's Law and the Principle of Least Time	127
4.7	A Popular Textbook Problem	134
4.8	Snell's Law and the Rainbow	137
5.	Calculus Steps Forward, Center Stage	140
5.1	The Derivative: Controversy and Triumph	140
5.2	Paintings Again, and Kepler's Wine Barrel	147
5.3	The Mailable Package Paradox	149
5.4	Projectile Motion in a Gravitational Field	152
5.5	The Perfect Basketball Shot	158
5.6	Halley's Gunnery Problem	165
5.7	De L'Hospital and His Pulley Problem, and a New Minimum Principle	171
5.8	Derivatives and the Rainbow	179
6.	Beyond Calculus	200
6.1	Galileo's Problem	200
6.2	The Brachistochrone Problem	210
6.3	Comparing Galileo and Bernoulli	221
6.4	The Euler-Lagrange Equation	231
6.5	The Straight Line and the Brachistochrone	238
6.6	Galileo's Hanging Chain	240
6.7	The Catenary Again	247
6.8	The Isoperimetric Problem, Solved (at last!)	251
6.9	Minimal Area Surfaces, Plateau's Problem, and Soap Bubbles	259
6.10	The Human Side of Minimal Area Surfaces	271
7.	The Modern Age Begins	279
7.1	The Fermat/Steiner Problem	279
7.2	Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs	286
7.3	The Traveling Salesman Problem	293
7.4	Minimizing with Inequalities (Linear Programming)	295
7.5	Minimizing by Working Backwards (Dynamic Programming)	312
Appendix A.	The AM-GM Inequality	331
Appendix B.	The AM-QM Inequality, and Jensen's Inequality	334
Appendix C.	"The Sagacity of the Bees"	342
Appendix D.	Every Convex Figure Has a Perimeter Bisector	345
Appendix E.	The Gravitational Free-Fall Descent Time along a Circle	347
Appendix F.	The Area Enclosed by a Closed Curve	352
Appendix G.	Beltrami's Identity	359
Appendix H.	The Last Word on the Lost Fisherman Problem	361
	Acknowledgments	365
	Index	367

What People are Saying About This

From the Publisher

"This is a delightful account of how the concepts of maxima, minima, and differentiation evolved with time. The level of mathematical sophistication is neither abstract nor superficial and it should appeal to a wide audience."—Ali H. Sayed, University of California, Los Angeles

"When Least Is Best is an illustrative historical walk through optimization problems as solved by mathematicians and scientists. Although many of us associate solving optimization with calculus, Paul J. Nahin shows here that many key problems were posed and solved long before calculus was developed."—Mary Ann B. Freeman, Math Team Development Manager, Mathworks

Sayed

This is a delightful account of how the concepts of maxima, minima, and differentiation evolved with time. The level of mathematical sophistication is neither abstract nor superficial and it should appeal to a wide audience.
— Ali H. Sayed, University of California, Los Angeles

Mary Ann B. Freeman

When Least Is Best is an illustrative historical walk through optimization problems as solved by mathematicians and scientists. Although many of us associate solving optimization with calculus, Paul J. Nahin shows here that many key problems were posed and solved long before calculus was developed.
— Mary Ann B. Freeman, Math Team Development Manager, Mathworks

Recipe

"This is a delightful account of how the concepts of maxima, minima, and differentiation evolved with time. The level of mathematical sophistication is neither abstract nor superficial and it should appeal to a wide audience."—Ali H. Sayed, University of California, Los Angeles

"When Least Is Best is an illustrative historical walk through optimization problems as solved by mathematicians and scientists. Although many of us associate solving optimization with calculus, Paul J. Nahin shows here that many key problems were posed and solved long before calculus was developed."—Mary Ann B. Freeman, Math Team Development Manager, Mathworks

Editorial Reviews

"This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting. . . . [Nahin shows] obvious delight and enjoyment—he is having fun and it is contagious."—-Bonnie Shulman, MAA Online

"When Least is Best is clearly the result of immense effort. . . . [Nahin] just seems to get better and better. . . . The book is really a popular book of mathematics that touches on a broad range of problems associated with optimization."—-Dennis S. Bernstein, IEEE Control Systems Magazine

From the Publisher

"Nahin delivers maximal mathematical enjoyment with minimal perplexity and boredom. . . . [He lets] general readers in on the thrill of riding high-school geometry and algebra to breakthrough insights. . . . A refreshingly lucid and humanizing approach to mathematics."

Booklist

"[When Least is Best is] a wonderful sourcebook from projects and is just plain fun to read."

Choice

"Nahin delivers maximal mathematical enjoyment with minimal perplexity and boredom. . . . [He lets] general readers in on the thrill of riding high-school geometry and algebra to breakthrough insights. . . . A refreshingly lucid and humanizing approach to mathematics."

Booklist

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. His enthusiasm is infectious, his writing style is active and fluid, and his examples always have a point. . . . [H]e loves to tell stories, so even the familiar is enjoyably refreshed.

SIAM Review - Donald R. Sherbert

A valuable and stimulating introduction to problems that have fascinated mathematicians and physicists for millennia.

Contemporary Physics - D.R. Wilkins

This book is highly recommended.

Mathematical Intelligencer - Clark Kimberling

When Least is Best is clearly the result of immense effort. . . . [Nahin] just seems to get better and better. . . . The book is really a popular book of mathematics that touches on a broad range of problems associated with optimization.

IEEE Control Systems Magazine - Dennis S. Bernstein

This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting. . . . [Nahin shows] obvious delight and enjoyment—he is having fun and it is contagious.

MAA Online - Bonnie Shulman

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. His enthusiasm is infectious, his writing style is active and fluid, and his examples always have a point. . . . [H]e loves to tell stories, so even the familiar is enjoyably refreshed.
— Donald R. Sherbert

SIAM Review

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

Contemporary Physics

This book is highly recommended.
— Clark Kimberling

Mathematical Intelligencer

When Least is Best is clearly the result of immense effort. . . . [Nahin] just seems to get better and better. . . . The book is really a popular book of mathematics that touches on a broad range of problems associated with optimization.
— Dennis S. Bernstein

IEEE Control Systems Magazine

This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting. . . . [Nahin shows] obvious delight and enjoyment—he is having fun and it is contagious.
— Bonnie Shulman

MAA Online

This book is highly recommended.
— Clark Kimberling

Mathematical Intelligener