“A physicist should be able to estimate anything to within an order of magnitude; an engineer should be able to take that and calculate it as precisely as needed.” At least the physicist part of this comparison is true.
Typically our initial contact with estimation comes in the first week of introductory physics when we estimate, a la Fermi, the number of piano tuners in Chicago. Unfortunately, after that promising beginning, we switch permanently from broad overviews and sweeping estimates to detailed studies of bricks, blocks, and inclined planes.
Estimation is important. While we're still taking physics courses, estimating answers can help keep our calculators from leading us astray. If you estimated that it takes a pumpkin a few seconds to fall from the top of the building but you calculated that it is s, something is wrong. In real life, we estimate to answer a question and perform an action. Is my new apparatus sensitive enough to do this measurement? Is it worth getting a solar panel on my new Prius (Guesstimation 2.0)? Should we use paper towels or hand driers in bathrooms (Fermi Column, The Physics Teacher, 12/2011)?
Estimation is the apogee of the physicists' dictum (attributed to Einstein) to make everything as simple as possible but no simpler. Building appropriately simplified models requires significant physical insight and there are often multiple possibilities. For example, rather than calculating the fuel consumed by a 747 flying from NY to LA by performing a finite element analysis calculation of the air flow over a detailed numerical model of the airplane, we could estimate the drag force for a car and then scale to a 747 (Art of Insight, p. 119) or we could estimate the glide ratio r so that the energy required is mg(distance/r) (Guesstimation 2.0).
Unfortunately, other than that first chapter of the introductory physics text, there are few resources for learning or teaching estimation. Consider a Spherical Cow and its sequel, Consider a Cylindrical Cow (John Harte, University Science Books, 1988) is excellent, but focuses on environmental problems. Back-of-the-Envelope Physics (Clifford Swartz, Johns Hopkins, 2003) walks the reader through 100 estimation problems, but without any discussion of estimation techniques. How Many Licks (Aaron Santos, Running Press, 2009) also shows how to solve 70 estimation questions. My own books, Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin (Weinstein and Adam, Princeton, 2008) and Guesstimation 2.0 (Weinstein, Princeton, 2012) focus on two techniques for estimation, breaking the problem down into smaller pieces and estimating each piece by taking the geometric mean of upper and lower bounds (e.g., How many clowns can fit in a clown car? More than 1 and less than 100, so we estimate 10.) They then apply those two techniques, together with a liberal (or should that be a conservative?) dose of physics insight, to a variety of problems, from the number of people picking their nose right now to the size of the landfill needed for a century's trash to the kinetic energy of the dinosaur-killing meteor.
However, there are many more techniques that we physicists use to make our models, including linear models, easiest cases, and statistical reasoning. The Art of Insight in Science and Engineering provides these and more in a delightful writing style with a glorious range of examples and a wide variety of homework problems.
The book is divided into three parts. The first part introduces estimation, with the techniques of breaking it down (systematized using nifty tree diagrams), bounding, and geometric means. It also emphasizes one of the key insights of physics, that radically different problems (e.g., simple pendula and RC circuits, electric and gravitational fields) can have the same mathematical structure and can therefore be solved the same way.
The second part introduces using symmetries and conservation laws, proportional reasoning, and dimensional analysis. Mahajan uses symmetry to analyze the classic problem of how fast to walk home in the rain. Because the volume swept out by the front of your body as you walk home is constant, the total water striking the front of your body will not change; it is only the water hitting the top of your body that varies with speed. He uses conservation of energy to make a simple model for drag, tests it by dropping paper cones, and then applies it to a host of other situations including maximum bicycling speed and automobile fuel efficiency. Proportional reasoning is familiar to most physicists and difficult for most students. After applying proportional reasoning to standard problems, he applies it to the terminal speed of falling paper cones and the fuel consumption of a 747 compared to an automobile.
The most important chapter in this part is “Dimensions.” If you're reading this review, you have almost certainly checked dimensions to avoid gross errors. However, Mahajan shows how to solve problems using dimensionless groups formed from the available variables. For example, the available variables for a pendulum are the period, gravity, length, and mass (T [T], g [LT−2], l [L], and m [M], where [L,T,M] refer to the dimensions length, time, and mass). The only available dimensionless group is T2g/l so we get the familiar result . Less familiarly, gravitational bending of starlight by the Sun will depend on angle, gravity, distance of closest approach, and the speed of light ([1], Gm [L3T−2], r [L], and c [LT−1]). The dimensionless group is Gm/rc2 so we expect that light will be bent by an angle . This is the correct result, with a proportionality constant of 2 for Newtonian gravity and 4 for Einsteinian gravity.1
The final part of the book introduces lumping, probabilistic reasoning, easy cases, and spring models. While these are all part of a physicist's tool box, they are rarely explored explicitly. Lumping consists of breaking a continuous problem into discrete pieces, such as treating the motion of a falling object as free fall followed by terminal velocity and ignoring the transition. Easy cases entails making sure that your solution works for the extremes and interpolates or extrapolates nicely in between (e.g., the range of a projectile must be zero for and 90°). Probabilistic reasoning is a bit unlike the other topics, since it is primarily applied to the specific problem of transport by diffusion.
The book has some minor problems. The copy-editing missed a lot of square root signs and other mathematical symbols (although you can consider that as “a problem left for the reader”). Some of the chapters are less coherently organized than others, which is probably unavoidable in a book of this scope. I'm also not convinced that Mahajan's division of techniques into those that discard information along with complexity and those that discard complexity without discarding information is quite as neat as he claims. However, these are minor quibbles concerning a very ambitious and very impressive book.
The Art of Insight in Science and Engineering definitely lives up to its subtitle Mastering Complexity. It is a thorough exploration of the manifold techniques physicists use to “make everything as simple as possible but no simpler.” While both students and teachers will enjoy reading it, this book could and should serve as the textbook for a capstone course providing the tools to “estimate anything to within an order of magnitude.”
References
Lawrence Weinstein is a University Professor of Physics at Old Dominion University. He is the author of Guesstimation and Guesstimation 2.0 and also edits the monthly Fermi Questions column for The Physics Teacher. In his spare time, he smashes atoms at the Thomas Jefferson National Accelerator Facility for which he was named a Fellow of the American Physical Society.
