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Thursday, March 26, 2015
Science 27 March 2015: Vol. 347 no. 6229 p. 1426 DOI: 10.1126/science.aaa5153
We now live in a world obsessed with data, in which paper and pencil have been traded for code and algorithms. As a result, we often spend less time getting a feel for problems we are tackling than we would have 35 years ago. It was therefore very refreshing to read a book that encourages the reader to do just that.
The Art of Insight in Science and Engineering acts as a step-by-step guide that enables the reader to tackle fundamental scientific problems through simple back-of-the-envelope calculations. The main objective of the book is not to promote a thorough understanding of an underlying theory or to allow us to come to an exact solution but rather to encourage us to use our instincts and knowledge of the fundamental concepts to come to an approximate and reasonable solution. “Approximate first, and worry later,” says the book's author, Sanjoy Mahajan. “Otherwise you never start, and you can never learn that the approximations would have been accurate enough—if only you had gathered the courage to make them.”
To gain insight into a variety of problems, Mahajan has devised a series of reasoning tools. He uses real-life examples to illustrate each tool's utility, showing the reader how to calculate everything from the energy needed for a plane to take off to the time needed to cook a fish to perfection (which is about 10 minutes, for those interested). Each is also cleverly illustrated by practical exercises that reinforce our understanding of the concepts.
The tools in the first section of the book enable us to make a problem more manageable. The divide-and-conquer strategy, for instance, tells us to divide a problem into subparts that can be solved or approximated easily and then aggregated back to the main answer (e.g., mass = density × volume).
In the second section, the reader is encouraged to simplify a problem by combining some of its elements. One way this can be done is by looking at a problem's symmetry. Here, Mahajan recalls the story of the German mathematician Carl Friedrich Gauss, who managed to calculate the sum 1 + 2 + … + 100 quickly when he was only 3 years old by using symmetry.
The final section has tools to help us discard some elements of complexity and purposefully omit information to reach a conclusion, using techniques including probabilistic analysis and what Mahajan refers to as “lumping.” As an example, instead of integrating complex velocity and acceleration profiles to estimate the airborne time of a falling object, Mahajan suggests lumping the profiles into rectangles and computing their areas to generate rough approximations.
Naturally, the book is not perfect. Mahajan clearly comes from a physics background, and he relies mostly on physics problems. The traditional electrical and spring models are used extensively, and so are the fundamentals of fluid mechanics. Moreover, toward the end of the book, Mahajan tackles much more complex problems, looking at sound, light, and gravitational radiation, for instance, which may be difficult for nonphysicists to appreciate. Nevertheless, these more advanced examples show that the overall approach is also applicable and even pertinent to more complex problems.
Senior undergraduate and graduate students will likely enjoy the book because it encourages them to think beyond the equation, and it may help build mental connections between many concepts learned in classes. Researchers and other professionals stand to benefit from it as well, because it may encourage them to think about their own problems in a slightly different way. Teachers will undoubtedly enjoy the book because it should equip them with a battery of techniques to improve their classes. Whichever applies to you, this book is definitely worth adding to your reading list.