Book Reviews - November 2007 - Volume 107 (7)

The Mathematics of Oz: Mental Gymnastics from Beyond the Edge

Author: Clifford A. Pickover
p. 262
Cambridge University Press, 40 West 20th Street, New York, NY 10011-4211, USA
2002, 351 pages, Hardback $29.00

Reviewer: Medhat H. Rahim
Lakehead University, Thunder Bay, Ontario, Canada, P7B 5E1

The book, The Mathematics of Oz (Oz is a metaphor for mystery), is not intended for mathematicians looking for formal mathematical explanations, rather, the author did not want the readers to wade through pages of background before getting to the essential ingredients. Thus, each chapter in this book was designed to be just a few pages in length, and as such, the reader can jump right in to experiment and have fun.

It is a call this book imposes on the reader: prepare yourself for a strange journey as The Mathematics of Oz opens the gates for your imagination where the mysteries, puzzles, and problems range from building a yellow-brick road that crosses America, to zebra numbers and circular primes, to Legion's number - a numbers so big that it makes a trillion pale in comparison. Thus, as the author suggests, just grab a pencil, relax, and then take off on a mind-boggling journey to the ultimate frontier of mathematics, mind, and meaning exploring some of the oddest and quirkiest highways and byways of the numerical world.

This book is for anyone who wishes to deal with new mental worlds, whether she/he is a teacher or a computer programmer. You may wish to use the mathematical brainteasers to stimulate your students and have them design their own puzzles similar to the ones in this text. To retain the sense of adventure and the playful spirit, puzzles with different difficulty levels are randomly tossed through the book. This text with its strange mazes, bizarre consequences, and dizzying arrays of logic problems would entertain readers at all levels of mathematical sophistication.

There are 108 puzzles each is designed as a chapter of about two or three pages. They were devised to assess human intelligence, and to tease the brain of all, even the most passionate puzzle fan. The puzzles feature a host of mathematical topics, such as, geometry and mazes, sequences, series, sets, arrangements, probability and misdirection, number theory, arithmetic, together with several real world problems. The text includes numerous illustrations. In sum, the text is an original, fun-filled, and unusual introduction to numbers and their rules in creativity, computers, games, practical research, and adventures that teeter on the edge of logic and amusement. Certainly, fans of recreational mathematics will enjoy this text that offers a range of challenging puzzles with a detailed and plain language answer keys for all puzzles.

Cogwheels of the Mind: The Story of Venn Diagrams

Author: A.W.F. Edwards
The John Hopkins University Press, 2715 North Charles Street, Baltimore, MD 21218
2004; 128 pages, Hardback $25.00, Paperback $19.00

Reviewer: Darlinda Cassel
The University of Central Oklahoma, Edmond, OK 73034

This short 128-page book is an easy, friendly, and enlightening book to read. The author includes several pictorial representations throughout the book, helping the reader to “see” and understand the concepts. The first few chapters trace the history of the Venn Diagrams. Venn found Euler's diagrams too restrictive so he created a new way to graphically represent propositions and not merely sets. Venn devised diagrams in the late 1880s in an effort to create symbolic logic. In July of that same year, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, introduced the world to the three-rings that illustrate Venn's diagram. Since then Venn diagrams have permeated art, economics, etc. The author does a good job of helping the reader to “see” the everyday usefulness of Venn diagrams. The tennis ball illustration helps the reader to conceptually make sense of Venn's diagrams. The author also mentions the use of Venn diagrams around us. We can see them in religious icons, flags, and stained glass windows. Winston Churchill used the three rings to depict Europe, the British Empire, and the English speaking World within the United Kingdom.

Most people can easily draw the two and three set Venn Diagrams, but Edwards' quest was to solve the quandary of creating Venn diagrams using an arbitrary number of sets. The subsequent chapters, written from the first person's perspective, deal with and explain the different properties of Venn Diagrams. Edwards provides an opportunity for readers to experience his thought processes as he seeks a solution to his quandary. Along the way, the author points out the connections between Venn diagrams and Boolean algebra as well as connections and applications to other areas such as, Gray's Codes, binomial coefficients, hypercubes, and other topics.

In the second appendix, Edwards gives simple instructions, with illustrations, explaining how to create a circular form of Venn diagram. Following his instructions you can rotate the sets to see the “Gray code” form of the diagram. Further rotation exhibits the “binary number” version with five sets. A wide range of readers from junior high to adults would be able to follow the mathematics. The book would be of particular interest to college professors, especially those involved in teaching a History of Mathematics class and/or a Graph theory class.