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When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible Paperback – 22 Jul 2007


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Amazon.com: 8 reviews
125 of 128 people found the following review helpful
Off the Charts 10 Mar 2004
By Dennis S. Bernstein - Published on Amazon.com
Format: Hardcover
Nahin's book is a tour de force about the deep intellectual threads that surround the notion of optimality. In physics, engineering, and mathematics, while touching on a wide range of applications, he asks over and over again: What is the optimal solution and why does it matter? Since I've spent most of my professional career thinking about optimality in one form or another, I was skeptical about how much new I would find in this book. But I was astounded to find something new and interesting on virtually every page. Some examples:
--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.
22 of 22 people found the following review helpful
More would be better ! 2 Aug 2006
By Michael LaDeau - Published on Amazon.com
Format: Hardcover
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.
11 of 14 people found the following review helpful
Delightful read, just avoid the the diagrams 14 Nov 2006
By Kaushik Basu - Published on Amazon.com
Format: Hardcover
Science writing needs to avoid two obvious traps: the pedantic discourse for the supposed layman, or the oversimplified analogy bordering on the erroneous. Paul Nahin deserves full credit for circumventing these effectively in his book about optimization. He is clear about the background he assumes on his readers part, though he doesn't always provide adequate references for those who don't. He does, however, offer pertinent citations for those readers who wish to wade deeper. I only wish he was more careful with his equations, and even more so with his diagrams, which often confuse rather than clarify... but the good things first.

The choice of topics, their sequence and the examples signify not only their historical importance, but places several in a modern context, with an emphasis on numerical solutions. I especially liked the approach he takes in section 1.7 with the numerical-graphical technique. The muddy wheel in section 3.6 demonstrates how an interesting (and real!) problem can yield to analytical techniques. However, numerical methods can be stretched at times, which is evident in the justification of eliminating one of the two values for a minimal surface on page 268. One would have appreciated a physical explanation based on analytical techniques. From a historical perspective, the use, discovery, exploration, development and the final definition of the derivative, (in that sequence!) and how Fermat played a seminal role in it clears several misconceptions even before the Newton-Leibniz imbroglio.

There are two particular examples that I would like to underline, both for their simplicity and their beauty of exposition. The first is the projectile problems in sections 5.4 and 5.5. They demonstrate a wonderful but simple extension to a topic typically explored in the classroom or in textbooks, but which has its roots in a very real problem- the strategy of the discus throw or basketball shot. The analysis of rainbows in section 5.8 is astounding. It is detailed to the extent necessary, and answers some very basic questions about a beautiful phenomenon. [...]

Where Nahin stumbles at times is maintaining that delicate balance between challenging his readers and fleshing out every mathematical step in its symbolic detail. While I am sure it has helped increase his readership from Hawking's half-lost-for-every-equation conjecture, it is often a waste of space, or worse still, displays a lack of economy and elegance. See, for example, the algebra on page 249. There is also a failure to mention the physical interpretation of beta as the angle for the parametric equation for the cycloid (page 217) which would make things simpler while proving the tautochronous property (page 222-227).

While I have commended the excellent treatment of rainbows in the context of derivatives, I must add that I find it appalling how inaccurate some diagrams in the book are, and in one case, even incorrect. The case in point being the figure 5.10 on page 184, "Illuminating a raindrop." While it is mentioned that it is rotated relative to figure 5.9, the arrows on 5.10 completely misrepresent its purpose. There should be no arrows along the line that makes an angle to the center. This is just a reference normal line. There should be arrows on the horizontal line above the one that passes through the center, and this should be extended to a line with arrows and subtending the correct refracted angle with the normal line.

How important is scale? Is it important to represent geometrical figures to scale when possible? Is it detrimental to the perception or solution of the problem if little attention is given to scale? If you have answered any of the questions in the affirmative, look at figure 3.5 on page 81 and figure 7.2, 7.3 and 7.4 on pages 281-283, but especially figure 7.2. I shall not waste my words here, but it really saddens me to see that geometrical figures that lends itself to a very satisfying visual treatment has so little attention paid to accuracy and scale. With that, I rest my case.
1 of 1 people found the following review helpful
Pretty darn amazing 12 Mar 2008
By Jonathan Fischoff - Published on Amazon.com
Format: Paperback
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.
4 of 6 people found the following review helpful
A fabulous book! 11 April 2007
By Nicholas Warren - Published on Amazon.com
Format: Hardcover
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.
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