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

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area?

By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes--with values becoming as small (or as large) as possible--and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot.

Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge, When Least Is Best is full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

--Preface: Torricelli's funnel, which has finite volume and can be filled, but has infinite surface area and cannot be painted; and a slick proof that an irrational number raised to an irrational power can be rational.

--Chapter 1: An optimization problem that is not amenable to calculus, but whose solution can be discerned by some clever insight; an optimization problem that is amenable to calculus, but whose solution can be arrived at by algebra; and the use of the arithmetic mean-geometric mean inequality in optimization.

--Chapter 2: The ancient isoperimetric problem of Dido on maximal area, how it remained unsolved until modern times; the fact that there exists a figure in the plane whose area is equal to the area of the period at the end of this sentence and which contains a line segment one million light years in length that can be rotated 360 degrees within the figure (the shape of the figure is a little hard to picture); and the fact that there are two consecutive prime numbers the gap between which is greater than a googolplex (don't ask what they are).

--Chapter 3: Optimization problems involving the viewing of a painting, the rings of Saturn, folding envelopes, carrying a pipe around a corner in a hallway, the maximum height of mud ejected from a wheel, and other daily concerns.

--Chapter 4: Snell's law, the path of light, and the feud between Descartes and Fermat.

--Chapter 5: The power of the calculus, the aiming of basketballs and cannon, Kepler's wine barrel, United Parcel Service package size constraints, L'Hospital's pulley problem, and the geometry of rainbows.

Chapter 6: Galileo's work on the descent of a particle sliding along the arc of a circle; the discovery of the minimum-time brachistochrone curve by Jacob Bernoulli arrived at by an argument based on the path of light in a variable-density medium, his feud with Newton, and Newton's anonymously published solution to the problem; the isochronous property of both the circle and brachistochrone, which states that the descent time is independent of the starting location along the cure (a point mentioned in chapter 96 of Moby Dick and which left me wondering which paths are isochronous since a straight line is clearly not); the fact that the brachistochrone is about 1.5% faster than the circular arc and that a brachistochrone tunnel dug from New York to Los Angeles would entail a travel time of a mere 28 minutes assuming frictionless sliding and no propulsion; the fact that 45 degrees maximizes range of a golf ball but 56.466 degrees maximizes arc length; the Euler-Lagrange equation of the calculus of variations and its proof formulated by Lagrange at age 19; the hyperbolic cosine shape of the catenary loaded by its own weight as compared to the parabolic shape of a string under uniform loading; the rigorous solution of the isoperimetric problem by Weierstrass; and the theory of soap bubble shapes by Plateau who was blinded by an optics experiments he performed during his Ph.D. research; and a brief illustration of optimal control theory

Chapter 7: Hofmann's solution of Steiner's problem on minimum distance inside a triangle and its use by Delta Airlines to save money on its phone bill; the traveling salesman problem, linear programming, a tutorial on dynamic programming along with a brief bio of IEEE Medal of Honor awardee Richard Bellman with emphasis on the fact the IEEE is an engineering society.

For a control audience, the connections between control and optimization are addressed by the lengthy discussion on the calculus of variations and the tutorial on dynamic programming. My only (minor) disappointment was the lack of more discussion about the nature of optimality in mechanics, that is, the least action principle, the specialization of Hamilton's principle to conservative systems. This underlying principle of mechanics is not, in fact, a statement of optimality but rather one of stationarity.

This book is clearly the result of immense effort. The author's notes suggest that most of the book was written in a single year, which is amazing. Not only are many topics covered, but mathematical details abound. The author, who is known for popular treatments of technical subjects (An Imaginary Tale: The Story of i, Dueling Idiots and Other Probability Puzzlers, The Science of Radio, Oliver Heaviside: Sage in Solitude, Time Travel), just seems to get better and better.

The book was produced with painstaking care. While there are surely errors somewhere, I spotted exactly zero. I would guess that the book has roughly half as many figures as pages, all drawn with great accuracy. To say the price of the book is reasonable would be an understatement.

Who might find this book of interest? The book is really a popular book of mathematics that touches on a broad range of mathematical problems associated with optimization. Some mathematical sophistication, and certainly calculus, is needed to follow the details. But much in this book could be digested by students in high school, even before calculus. The flavor and richness of the subject matter cannot help but whet the curiosity of neophytes. Undergraduate and graduate engineering students of all disciplines will find something that relates to their coursework.


More would be better !   August 2, 2006
Michael LaDeau (Texas)
17 out of 18 found this review helpful

Mr. Nahin states in his preface that 1st year undergraduate math and physics is enough to manage "a lot of mathematics in this book." He is fairly on the mark, discounting my comments about chapter six below. As usual, the reader must keep pencils and scrap paper ready to fully appreciate this book. I hoped to find a book based on applications of math and physics, an engineer's approach. This is one such fascinating book.

I was familiar with the AM-GM inequality technique to find extremas. However, Mr. Nahin dispenses of this method early and shows the reader so much more. And in this book, there is a constant exercise of looking at problems a different way.

If you like geometric solutions along with the typical lines of algebraic manipulations, you'll love this book. The first five chapters are packed with problems and solutions with excellent graphic representations. Integration requirements increase throughout.

In finding extremas in chapter six, the author goes beyond ordinary calculus with the calculus of variations including the Euler-Lagrange differential equation and Beltrami's identity. The focus problem is the minimal decent time curve. It is in section 6.4 that the author truly breaks from his stated reader requirements of "high school algebra, trigonometry, and geometry, as well as the elementary integration techniques." I think most authors of this book's scope typically underestimate reader requirements. As for my part, I did not understand the calculus of variations technique on the first reading. After reading sections 6.4 through 6.8 again, I gained an appreciation of how the method works. After one more reading of these sections, I might know just enough to be dangerous. These challenging sections are well written, but a struggle within the stated reader requirements.

Chapter 7 found me in more comfortable ground where great geometric solutions to problems are shown and there is a keen introduction to linear programming.

In various cases, Mr. Nahin works through problems with results generated by computer programs. These are not my favorite problems because I lack access to the high end (very expensive) programs that he uses.

This book is well written and engaging; and it is easier to manage than An Imaginary Tale. This is my second book by Mr. Nahin, and I view him as a favorite author of technical books. In this review, I intentionally avoided mentioning specific problems covered because I do not want to spoil the surprises. I found them all quite fascinating. The reader will see so many real world physics in a different light. I highly recommend this book.



A fabulous book!   April 11, 2007
Nicholas Warren (New York, NY USA)
3 out of 5 found this review helpful

I've recently finished the above book and can't tell the reader just how much I enjoyed it, particularly the connections made that often get lost in the usual silo approach to math topics. Plus, Professor Nahin explains the math extremely well and makes it fascinating. I wish there were more similar books at this level.

I appreciate the time it must take to put together a book like that but, if Professor Nahin was ever thinking about the next topic(!), how about Linear Algebra and the connections with geometry and calculus -- something along the lines of books by W.W. Sawyer, but more advanced? I know he would do a superb, and valuable, job.


excellent - I want all his books   April 24, 2004
pete
16 out of 26 found this review helpful

Finally, a solid book that challenges the lay reader just like the best math teachers do - by showing the elegance and power of mathematical reasoning.

This is top shelf material. Nahin is one heck of writer and must be one hell of a teacher! Bravo!

Already ordered his book on the history of imaginary numbers.

6 stars: ******


Pretty darn amazing   March 12, 2008
Jonathan Fischoff (Chapel Hill, NC United States)
This book does so many things well, that I would get bored trying to explain them all. What really impressed me was the explanation of the Euler-Lagrange equation. What is incredible about the treatment is that it is so easy to understand but doesn't leave out any of the math. For anyone trying to teach themselves the calculus of variations I recommend this book as an intro before jumping into a textbook.