A never-ending story

Keys To Infinity
March 14, 1997

This book is a collection of puzzles, quotes, facts about numbers and mathematical and scientific analysis, all intended to convey a sense of the infinite. What we see between the covers is the human mind at work - many minds in fact, since Clifford Pickover includes responses to his puzzles attributed to colleagues around the world - giving us access to the infinite through language, computational process and thought experiment.

The book conveys a sense of a group of people enjoying light-hearted and challenging intellectual activity that touches on many of the popular mathematical ideas of our age. Readers are encouraged to join in by trying to solve numerical puzzles or by speculating on life in various outlandish yet systematic transformations of our everyday experience, such as the consequences of constructing a ladder to the moon, or of living on a world made up of an infinite repetition of our world, each one joined to the next at the poles.

Fractals get a look in, of course, supported by a number of beautiful colour plates.

The key to this book, and I would say to infinity itself, lies in the use of language to define uniformities and processes. Rather than discussing these issues explicitly and in the process losing most of the potential readership, the book exhibits a myriad ways of defining patterns, and of concrete processes based on these patterns showing how we get to grips with infinity in practice. For example, the problem of which powers of ten are the product of two numbers each of whose decimal form contains no zeros, is not solved, but addressed by giving a brief reason why such numbers must be the product of 2 and 5, each raised to a given power, for example, 1033= 233x 533. A computer program is then presented that searches for powers of five containing no zeros. Another example invites us to consider the physical process of drawing a pentagon around a first so that the first is inscribed in the second. This operation is easily achievable at home if the first pentagon has one-inch sides. A simple mathematical analysis of the length of line required to draw such pentagons leads to a table showing that repetition of this operation only 30 times would require drawing lines equal to the distance from the earth to the sun. Repetition of the operation only 65 times yields a currently accepted value for the radius of the universe. Such examples convey a wonderful sense of the ability of repetition of pattern to exceed our expectations in space, time and number, and to confound our imagination.

It is the concrete nature of the patterns, related to everyday objects, numbers and distances by example, quotes and comparisons, that gives this book its delightful and instructive character. The literary device of including solutions and speculations from colleagues, works to relieve the author of an authoritative voice and to ensure that the reader gets involved in the evaluation of each suggestion. The encouragement to apply one's mind is almost irresistible, a factor that, together with the organisation of the book into short chapters, makes it good to use when trying to stimulate young and inquisitive minds. There are many surprises here also for the computer hobbyist with a Pentium in the house and little more than unimaginative games to exercise it on; this book will show that repetition does not have to be boring.

Lincoln Wallen is fellow incomputation, St Catherine's College, Oxford.

Keys To Infinity

Author - Clifford A. Pickover
ISBN - 0 471 11857 5
Publisher - John Wiley
Price - £17.99
Pages - 332

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