The Art of Insight in Science and Engineering: Mastering Complexity, Sanjoy Mahajan, MIT Press, 2014. $30.00 paper (389 pp.). ISBN 978-0-262-52654-8 Buy at Amazon
Enrico Fermi famously estimated the energy yield of the first fission bomb at the Trinity test site by dropping bits of paper and observing their deflection as the bomb’s blast wave swept past. That anecdote provides a vivid example of the power of estimation. Although physicists widely recognize the utility of back-of-the-envelope calculations, current physics courses provide surprisingly little instruction and practice in that invaluable art.
Sanjoy Mahajan’s The Art of Insight in Science and Engineering: Mastering Complexity is an antidote to that issue. It is a unique and entertaining tour of methods for estimating the solutions to complex problems. Under Mahajan’s skillful guidance, readers assemble a powerful toolbox of techniques, which are illustrated and developed for application to a diverse range of fascinating topics. After working through the book, readers will feel equipped to come up with rapid-fire, approximate solutions to unfamiliar and complex problems, to pose their own questions, and to explore.
Mahajan draws from his extensive teaching experience, which includes courses at MIT and the Franklin W. Olin College of Engineering in Massachusetts. He also was inspired by his work as a graduate teaching assistant for the Order of Magnitude Physics class at Caltech taught by renowned astrophysicists Peter Goldreich and E. Sterl Phinney. Mahajan is also the author of a related book, Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving (MIT Press, 2010). The new book is published under a Creative Commons license, which means it can be freely redistributed or downloaded. In fact, in my own estimation-physics class, I used an early version that I obtained from the author’s website.
The Art of Insight in Science and Engineering is organized into three broad sections and nine chapters, each illustrating different problem-solving techniques. Throughout, Mahajan poses, and subsequently answers, questions that readers can work through independently to help build their proficiency. In addition, numerous exercises are included for further practice.
In the first section, “Organizing Complexity,” Mahajan introduces the divide-and-conquer strategy for breaking up estimation problems into several tractable subproblems. One illustrative example explores how many barrels of oil are imported into the US each year. Readers can estimate the amount of US oil usage from cars, the population of the US, the typical number of miles each person drives in a year, the gas mileage, and the volume of an oil barrel. Another approach described in that section involves conducting simple experiments around the home: For example, a person can estimate his or her maximum mechanical power output by timing a run up a flight of stairs. Also discussed is reasoning by analogy, exemplified by comparing spring–mass systems with electrical RLC circuits and thermal systems.
The second section, “Discarding Complexity without Losing Information,” covers the use of symmetry principles and conservation laws, scaling relations, and dimensional analysis. It discusses characteristic atomic and molecular binding energies and size scales; it also covers solids, their elastic moduli and sound speeds through them. The author presents an approximate analysis of fluid drag forces, which is a powerful and ubiquitous example. The fluid drag analysis is applied to estimate the maximum speed of a cyclist on flat ground, the gas mileage of an automobile, and the power expended in the flight of birds and airplanes. In one classic application of dimensional analysis, Mahajan estimates the first fission bomb’s energy yield from time-stamped blast photographs of the event and a corresponding scale bar.
The final section, “Discarding Complexity with Loss of Information,” starts with a declaration: “When the going gets tough, the tough lower their standards: Approximate first, and worry later.” The chapters include “Lumping,” in which varying quantities are approximated by their typical or characteristic values, and “Easy Cases,” such as considering a system’s behavior in simple, limiting circumstances. The author discusses a wealth of interesting topics, including piano and xylophone physics, acoustic monopole radiation, electric dipole radiation—with a discussion of blue skies and red sunsets—gravitational radiation, the size of neutron stars, the time required to bake a fish, and the temperature of the Sun’s core.
The Art of Insight in Science and Engineering is a wonderful and fun book that fills a gap in existing science curricula. I would recommend it as a primary text for an estimation class or as a valuable supplement for other science and engineering courses. It provides an enjoyable read for a broad audience and it should be comprehensible to first-year science and engineering students, although certain aspects may be fully appreciated only by more advanced readers.
Adam Lidz is an associate professor at the University of Pennsylvania in Philadelphia. He conducts theoretical research in astrophysics and cosmology.
