When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible, Paul J. Nahin, 2004, xviii + 370 pp , $29.95, ISBN 0-691-07078-4, Princeton University Press.
Have you ever asked how high mud can fly off the edge of a rotating wheel? Well, Paul Nahin has asked and answered this along with many other classical and not so classical optimization questions.
When Least is Best is actually a history book for those who love doing mathematical derivations, or a Nahin puts it, “This book is for readers with calluses on their fingers because they read with pencil and paper in hand!”. Assuming that the reader has a good knowledge of freshman calculus and physics, Nahin explores interesting and practical max/min problems, techniques, and associated personalities from Queen Dido’s isoperimetric problem (c. 800 BC) to George Dantzig’s simplex algorithm and Richard Bellman’s dynamic programming methods (c.1950). The author is an electrical engineer so do not expect many formal proofs, but do expect clear and crisp exposition, a nice selection of literary references (for example, from Søren Kierkegaard: “You can only understand life backwards, but we must live it forwards.”), lots of diagrams, and lots of algebra, trigonometry, and inequalities.
Here is a sample of other topics: finding a can with fixed volume and minimal surface area using the Arithmetic Mean-Geometric Mean Inequality, using Jensen’s Inequality to cut a string so as to minimize the area enclosed by two prescribed shapes to be formed from the pieces of string (regarding minimal surface area, do not overlook the Chardin painting of boys blowing soap bubbles on the dust jacket), using computer analysis to find the minimum passage width needed to get a real-life pipe (or sofa) around a corner, how Fermat used a metaphysical principle to correct Descartes’ misunderstanding of the law of refraction, why cylindrical packages may a good choice when using a parcel service, and a rather nice pulley problem from L’Hospital’s calculus book. Rainbows are thoroughly analyzed, with explanations for the shapes, locations, and color patterns of the first three rainbows (yes, there are at least three, although you will never see the third one).
When Least is Best is a good read (perhaps skimming over some of the derivations on the first time around) and useful reference for any mathematics instructor at the calculus level or higher. If you suggest this book to a bright student, ask them to find and discuss any minor errors; I won’t tell you the few errors I found, but will merely suggest they could be a starting point for a good teacher-student conversation.
Clifford H. Wagner, Associate Professor of Mathematics and Computer Science, The Pennsylvania State University at Harrisburg