The Beauty of Everyday Mathematics is a collection of thirteen problems of the type one may encounter in everyday life, hence the title. The mathematics is generally at an elementary level, and the problems are easy to understand and easy to solve. The goal of the book is to show you how and where mathematics fit into your life. Let's take a look at some of the problems to see this more clearly.
Problem 1: The Soda Can Problem. What is the level of liquid in a soda can so that the can is stable and will not easily tip over? Herrmann begins with a lesson on the center of gravity, and he shows you how that applies to a (partially) filled soda can. He does an admirable job of thinking through the problem before applying the math. He tells you to imagine the can as a cylinder filled with liquid. Thus the center of gravity is at the center of the can. Then, if the can is empty, the center of gravity is back at the center of the can. But, if there's liquid in the can, the center of gravity can be below the center of the can and hence more stable than with the center of gravity at the center of the can. What level of liquid puts the center of gravity at its lowest point? Herrmann shows you how to find it with two different methods: via a thought experiment and analytically. It's a great problem and the solution requires a bit of calculus with simple integrals.
Problem 4: The Sketch Problem. This is an old problem, but it's a good one for budding mathematicians to see the pitfalls of a sketch. Herrmann asserts that 90o = 100o and he proves it with the help of a sketch. The logic seems right at first glance, and that's his point. The logic is fine but the reader is duped from the start because the sketch is all wrong. The mathematics requires some geometry but is otherwise easy to follow. Herrmann makes his point well about how far to trust a sketch and how easily one can be deceived with a poor diagram.
Problems 5 and 6: The Parallel Parking Problem and The Parking Garage Problem. These are just what they say they are: What are the mathematics required to parallel park a car and what are the mathematics needed to park a car in a straight-in-space either front first or rear first? Herrmann walks the reader through a good model of the car, its turning radius, the geometry of the spaces, and how the car would have to maneuver to fit into the spaces. He even gives a list of various cars, and angles and minimum space needed to park them. I thought these two problems were quite interesting because of the current commercials for cars that park themselves. I can't say if those cars use mathematics, but to see the problem so clearly solved was quite interesting.
Problem 8: The Slippery-Ice or Bread-Slicing Problem. The author asks and then answers the question: Why do locked or spinning wheels make a car slide easily? And, why does a person move a knife across bread to cut down the loaf? The book starts with Newton's law (F = ma) for a car of a given mass with lateral forces and friction. There is a second order differential equation to solve, which Herrmann does with Laplace Transforms and then he derives a simplified equation describing the motion. It's in this final equation where the force of friction is absent (surprise!) that we see the key. At immediate start of a skid, there is no lateral force of friction. Hence, any slight lateral force will push car to the side thus making the car slide.
Problem 11: The Beer Coaster Problem. Suppose there are two beer coasters, one on top of the other. At what point, in moving one across the other is the area of overlap exactly half the area of a single coaster? It's a straightforward problem, but the solution requires solving a transcendental equation. Herrmann gives you a fixed point procedure for the solution as well as Newton's procedure. It's a wonderful problem because it is so simple to state yet requires sophisticated tools to solve.
The problems are varied and their solutions require a variety of mathematics. Some problems and solutions require differential equations, as above, some use integral calculus, geometry, matrices, and other techniques. The book covers a wide range of mathematics, albeit at a shallow depth. It's quite a short book, only 134 pages with lots of diagrams, equations, and explanatory text.
There are some errors, such as equation (7.1): “0+a = 0 for all numbers a.” This particular mistake is repeated in a subsequent equation.
This book offers interesting problems and thoughtful solutions. It's worth your time to read and then, if you like it, pass along for others to enjoy.
David S. Mazel received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.