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Mathematical Geology, Vol. 31, No. 8, 1999

Book Review

The Jungles of Randomness: A Mathematical SafariEdited by Ivars PetersonJohn Wiley & Sons: New York, 1998, 239 p., US$14.95 (paperbound),

ISBN 0-471-29587-6

The concept of randomness is familiar to most mathematical geologists. We use itin many situations, ranging from statistical testing and application of probabilityfunctions, to realizations from geostatistical or other simulations, to uncertaintyanalysis, among many others. I read this book in expectation of broadening my viewof randomness, but it was not quite what I expected. The book’s main approachis to show that random processes can be described with mathematical methods,that deterministic situations can lead to the false appearance of randomness (e.g.,chaos), and that some events with apparently low chance of occurrence are notunlikely or even random at all. Indeed, substantial parts of the book are reallyabout nonrandom subjects.

Chapter 1 introduces concepts of randomness with flipping coins and rollingdice; most demonstrations are based on enumeration of all possible outcomes.From there, the discussion moves to Markov chains, illustrated with examplesof how chains may be used to design the game Chutes and Ladders or to ana-lyze Monopoly. Interestingly, Peterson never defines randomness or probabilityformally, but teaches the concepts by example.

Chapter 2 looks at patterns, and then moves into elements of geometry andgraph theory. Chapter 3 brings up the idea that atoms and molecules combine toform biological cells, but that the likelihood of such complex structures occur-ring by chance is miniscule. Chemical and physical properties, however, controlsuch combinations, with chance having a small part. Many viruses consist oficosahedral shells that surround core DNA or RNA, and these are used as exam-ples. A sizable discussion describes geometric properties of icosahedra and theirfacets.

Chapter 4 introduces oscillators, beginning with groups of fireflies that syn-chronize their flashes, and then moving to other systems. Chapter 5 extends os-cillators to acoustic waves. It investigates drums, particularly the effects on theiracoustic properties for membranes with various shapes, sizes, and even fractal

1007

0882-8121/99/1100-1007$16.00/1C© 1999 International Association for Mathematical Geology

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1008 Book Review

boundaries. It also considers bells with different, distorted shapes that yet havesimilar acoustic properties.

Chapter 6 looks at the detection and prevention of random errors or noise.This includes methods used to correct missing or damaged tracks on music CDs;it is surprising how much missing data can be restored. The chapter also presentsmethods used for validating, for example, airline ticket numbers and UniversalProduct Codes, and it shows how redundancy can be used to prevent errors. Anintroduction to cryptographic methods concludes the chapter.

Chapter 7 introduces chaos theory, showing situations that are completelydeterministic, but yet are so unpredictable and complex as to be assumed ran-dom. The presentation discusses the Tilt-A-Whirl amusement-park ride, dynam-ics of double pendulums, and movement of strips of a special alloy that respondto magnetic fields. Riddled systems, in which both the choice of attractor andthe location on the attractor are uncertain, are also discussed. An interesting re-sult is the finding that small perturbations can be used to control chaotic sys-tems. Peterson discusses chaos theory more extensively in his previous bookNewton’s Clock: Chaos in the Solar System.

Chapter 8 presents observed Brownian motion, its mathematical descriptions,and random walks in one or more dimensions. The Levy random walk sets thelength taken by each step by randomly choosing from a probability distribution.Observed clusters of stars and galaxies, and hunting patterns of the wanderingalbatross, are examples.

Chapter 9 turns to gambling, primarily related to slot machines. Further dis-cussion shows that random methods can be used to solve otherwise intractabledeterministic and statistical problems, with Monte Carlo simulation being a well-known example. A sizable part of the chapter is on generating and testing pseudo-random numbers. The chapter concludes with the possibility of using radioactivedecay and quantum principles to generate random numbers.

Chapter 10 looks at the tendency of people to attempt to impose patternson everyday events. It begins by showing that what are thought to be improbablecoincidences perhaps are not so unlikely after all. It then shows examples of howpatterns may be tested for randomness. Peterson concludes with the statement:“There are fundamental limits to what we can know, what we can foretell, andhow much we can understand of both the physical universe and the universe ofmathematical experience. What we cannot attain we label as random, or unknow-able. But perhaps randomness itself is a crucial unifying principle [, as seen innumerous examples in the book]” (p. 202).

The author is mathematics and science editor ofScience News. This somewhatepisodic book gives the feeling that he collected a number of his articles and wrotetransitions to tie them onto the somewhat tenuous thread of randomness. Overall,the book is well written and clear. A characteristic of this author is that manyeveryday examples are used to introduce and illustrate concepts, and this book

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continues the pattern for many random and nonrandom subjects. I recommendthis book for the reader wishing to get a light overview of various subjects withconnections to randomness. However, a disappointment is that Peterson does notdelve deeply enough; his previous books provided more details and insight intothe mathematical concepts than this one does. It was not written for the readerwanting to learn about applying probability.

Thomas A. JonesExxon Production Research CompanyP. O. Box 2189Houston, Texas 77252-2189