Gnomon

By Midhat Gazale

The beaver's tooth and the tiger's claw. Sunflowers and seashells. Fractals, Fibonacci sequences, and logarithmic spirals. These diverse forms of nature and mathematics are united by a common factor: all involve self-repeating shapes, or gnomons. Almost two thousand years ago, Hero of Alexandria defined the gnomon as that form which, when added to some form, results in a new form, similar to the original. In a spiral seashell, for example, we see that each new section of growth (the gnomon) resembles its predecessor and maintains the shell's overall shape. Inspired by Hero, Midhat Gazal --a fellow native of Alexandria--explains the properties of gnomons, traces their long and colorful history in human thought, and explores the mathematical and geometrical marvels they make possible. Gazal is a man of wide-ranging interests and accomplishments. He is a mathematician and engineer who teaches at the University of Paris and whose business career lifted him to the Presidency of AT&T-France. He has a passion for numbers that is clear on every page, as he combines elegant mathematical explanations with compelling anecdotes and a rich variety of illustrations. He begins by explaining the basic properties of gnomons and tracing the term--which originally meant "that which allows one to know"--to ancient Egyptian and Greek timekeeping. Gazal examines figurate numbers, which inspired the Greek notions of gnomon and number similarity. He introduces us to continued fractions and guides us through the intricacies of Fibonacci sequences, ladder networks, whorled figures, the famous "golden number," logarithmic spirals, and fractals. Along the way, he draws our attention to a host of intriguing and eccentric concepts, shapes, and numbers, from a complex geometric game invented by the nineteenth-century mathematician William Hamilton to a peculiar triangular shape that Gazal terms the "winkle." Throughout, the book brims with original observations and research, from the presentation of a cousin of the "golden rectangle" that Gazal calls the "silver pentagon" to the introduction of various new fractal figures and the coining of the term "gnomonicity" for the concept of self-similarity. This is an erudite, engaging, and beautifully produced work that will appeal to anyone interested in the wonders of geometry and mathematics, as well as to enthusiasts of mathematical puzzles and recreations.

• Format: Hardcover
• Language: English
• ISBN: 0691005141
• ISBN13: 9780691005140
• Release Date: April 1999
• Publisher: Princeton University Press
• Length: 280 Pages
• Weight: 1.33 lbs.
• Dimensions: 0.9" x 7.8" x 10.0"

Customer Reviews

Slow going, but worth it

According to Gazale', "Hero of Alexandria defined the gnomon as that figure (a number or a geometric figure) which, when added to another figure, results in a figure similar to the original." Gazale's book is, therefore, about self-similarity in numbers and geometry.The subject sounds simple enough, but I found this to be a pretty tough book. That might be partly due to the fact that I've always had a hard time focusing my attention on number theory. This book has a lot of basic stuff about numbers, and I found much of that subject rather tedious and (dare I say it?) boring. I know that's an ignorant thing to say - after all, mathematics is a beautiful subject in its own right, and there is some really neat stuff in number theory. But it was still a tough book for me to wade through.The introduction is mostly historical background, and a little truncated. It serves primarily to illustrate a few basic concepts in self-similarity. The author continues this theme with a short description of figurate and m-adic numbers. Gazale tends to use more technical language than many casual readers are likely to recognize. Yet this really isn't a book on formal mathematics, either. It's really somewhere in between. Gazale often draws on themes from Martin Gardner's series of articles in Scientific American, and in some ways, his book reflects Gardner's style. And, while much of this book seems focused on abstract details, there are occasional forays that illustrate amazing connections between what looks like pure mathematics and the real world.Chapter 2, titled "Continued Fractions," is foundational. I really enjoyed this section, and think the book is worth having for this chapter alone. Beginning with Euclid's algorithm, Gazale offers a natural introduction to continued fractions. Then, in his characteristic style, he continues to explore every nook and cranny of this fascinating branch of mathematics. Among the most pleasing results of this chapter is his demonstration of the mirrored similarity in the appearance of numbers as they are represented by continued fractions, and as they are represented by our traditional positional number system. For example, he shows that both representations are always convergent and uniquely correspond to a number. However, while infinite periodic representations correspond to rational numbers in the positional system, they correspond to quadratic irrationals in the system of continue fractions. And, while transcendental and irrationals are infinite nonperiodic representations in both systems, there are some beautiful expressions of some transcendental numbers in the system of continued fractions that left me mesmerized.One particularly nice feature is the way the author summarizes the important equations at the back of each chapter. Some of these summaries are several pages long, and they actually do a good job of encapsulating the essential material. In fact, the summaries are so well done that, if you

An extremely original book , full of ideas and discoveries.

A very approachable text that appeals to the academic as well as non academic.The simplicity and power of mathematics is demonstrated by this erudite author who promotes this unique and historical approach of the evolution of math. He successfully descibes the self similar processes in math as well as in life forms. Self similarity is the common thread. Very stimulating.