The Jungles of Randomness: A Mathematical Safari

When I was first asked to review Ivars Peterson's The Jungles of Randomness: A Mathematical Safari, I was a bit apprehensive. In spite of what our esteemed editor had said ("it's a 'general reader' type book... largely about probability and statistics"), I was expecting a difficult task. It turned that the only difficult task was finding time to write my review! Reading The Jungles was easy, very informative, and... well... fun! What should you expect from a book with chapters having names like "Call of the Firefly," "Noise Police" and "Complete Chaos?"

The Jungles is about the various manifestations of randomness, from those we have all experienced in everyday life, to those of which we were likely not aware. For example, part way through the book I began to think of randomness as coming in two "flavors." First, there is the randomness that we think of when playing games with dice. The outcome of a certain physical action or process cannot be predicted with certainty, though you can analyze its behavior in the long run. In Chapter 1 Peterson begins his "tour" of the jungle by discussing dice tossing and coin flipping. His dice discussion ranges from the children's game of Chutes and Ladders to one of the classical problems probability, that of rolling a double-six in 24 rolls of two dice. With the coin he introduces an important theme of the book: finding patterns in the occurrences of a random process, and that complete disorder is NOT a characteristic of randomness!

The second flavor of "randomness" works like this. We are confronted with one realization of a large possible set of scenarios, and although it was "presented" to us in a deterministic (non-random) way, we study "our" scenario as if it were randomly chosen from that large set. As an example, suppose you are sitting at a table of six people and you find out that there are three people who are mutual acquaintances. Thinking that it is "quite a coincidence," you ask how likely is it for that to have occurred by chance? In Chapter 2 he shows that you are guaranteed to have at least three people who are mutual acquaintances, or at least three people where no one knows anyone else.

In the last chapter Peterson describes three kinds of randomness. I think I got two of them correctly; but you can read the book and "grade" me on that later. The third flavor of "randomness" results from processes that are really deterministic, but are so complicated that they look random to us. In Chapter 7, "Complete Chaos," the carnival ride Tilt-a-Whirl exhibits deterministic behavior at low and high speeds, but its behavior looks random to us at medium speeds.

The most interesting "real-world" examples found in The Jungles are far from trivial, yet they are generally extensions of the very basic. One of the wonderful qualities of this book is that Peterson makes transitions so smoothly and naturally. In the early chapters he moves from platonic solids to the natural construction of the outer shell of a virus; from the synchronized firefly flashes to coupled oscillators and animal locomotion; and from the harmonics of a violin string to whether two differently shaped drums in three-or-more dimensions can have the same sound. In the later chapters he finishes with error correcting codes after starting with word games; with turbulence and the distribution of matter in the universe after beginning with the one-dimensional random walk; and with our ability to judge (or not!) randomness after starting with random digits. Then finally in Chapter 10, he completes his book-long transition from tossing a die and flipping a coin to coincidences, what makes something random, and a short history of philosophical discussions of "randomness" and structure in mathematics!

Lest you feel that the book is too deep, rest assured that most of the time it is not that way. Although I do not think that The Jungles is a "general reader" book for the "general public," for anyone with some background in mathematics the label applies. I found it great reading, even in bed just before I turned out the light! There are some good reasons for this. First, Peterson writes with an easy, non-technical style. Although in a few places I wanted to see a formula or two, he stuck to prose, pictures and some arithmetic to present the ideas. Second, he begins each chapter with a paragraph or two that literally reads like a novel instead of a textbook. Third, he prepares you for the topics to come (here is that "transition" stuff again). Fourth, throughout each chapter, and especially at the end, he uses quotes from people doing the actual research in the topic to anchor the mathematics to reality, and emphasize that important work is going on as you read! Finally, at the end of each chapter he summarizes what you have seen, and leads you into the next chapter (transitions again!).

There were only a few places in The Jungles where I got bogged down for one reason or another. In the section "Party Puzzles" he referred to colored lines when the diagram was in black-and-white. This was frustrating because he employs some wonderful colored plates in other places. In Chapter 7 when he discussed the behavior of the Tilt-a-Whirl, some plots of the behavior or other illustrations would have been helpful. Then in Chapter 8, one part of the diagram comparing Brownian motion and Levy flights is missing (from my book), and I just don't get the idea of stock options (I was never good at financial stuff, but if this is for the general reader...). However, don't let these isolated problems keep you from reading the book!

Before I wind up this review, let me suggest one thing. Several of my colleagues here at Rowan believe in assigning a mathematical "general reader" book to go along with the hard-core textbook. The students read a chapter a week or so. Examples of this genre include books like Journey Through Genius by William Dunham, Mathematics and the New Golden Age by Keith Devlin, or The Mathematical Experience by Davis and Hersh. This book would be a good candidate for the "parallel" text in a probability course or even an upper level discrete math course. The chapters can easily be read one-at-a-time (or even singly).

In closing, if you like to read lighter books about mathematics, or assign a more general book in your class or seminar, I heartily recommend The Jungles of Randomness. Ivars Peterson is an excellent guide, and you won't need to bring your machete.

Dex Whittinghill (whittinghill@rowan.edu) is an assistant professor of statistics, and in fact an "isolated statistician" in the congenial Department of Mathematics at Rowan University. He is becoming increasingly involved in the statistics education "movement," and is a member of the MAA and the ASA (American Statistical Association).