Gnomon [Hardcover]

Midhat Gazale (Author)

Book Description

Publication Date: April 19, 1999

The beaver's tooth and the tiger's claw. Sunflowers and seashells. Fractals, Fibonacci sequences, logarithmic spirals. These diverse forms of nature and mathematics are united by a common factor: all involve self-repeating shapes, or gnomons. Almost 2000 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 Gazale - a fellow native of Alexandria - explains the properties of gnomons, traces their long and colourful history in human thought, and explores the mathematical and geometrical marvels they make possible. The text should appeal to anyone interested in the wonders of geometry and mathematics, as well as to enthusiasts of mathematical puzzles and recreations.

Editorial Reviews

Amazon.com Review

How are the great pyramids like seashells? Ask mathematician Midhat J. Gazale, then brace yourself for a heady ride through the wilds of self-similar geometry in Gnomon: From Pharaohs to Fractals, his paean to the roiling mysteries that lie beneath the tranquil surfaces of such objects. The great mathematician Hero of Alexandria defined a gnomon as an object that, when added to another, creates a new object similar in form to the original. Gazale, also of Alexandria, goes much further and uses 20th-century concepts to fully explore "gnomonicity"--the property of self-similarity.

Be prepared for slow going: Gnomon is densely packed with information and concepts foreign to all but the professional mathematician, but Gazale's enthusiasm and brilliant illustrations win the day. Whether he's moving on from the familiar golden rectangle to his own "silver pentagon" or rooting around in the numbers underlying the groovy fractal images popping up on T-shirts worldwide, he takes care to explain to the reader not just what's going on mathematically but what all this abstraction really means to us. Few science books, and even fewer mathematics books, achieve that kind of depth. --Rob Lightner

Review

"Gnomon entices mathematically inclined readers to embark upon a highly rewarding voyage of discovery. . . . Dr. Gazal explores and explains a myriad of instances in which manifest themselves in nature and human contrivance. Fascinating stuff." -- Arno Penzias, winner of the 1978 Nobel Prize for Physics

"Gnomon offers a stimulating collection of diagrams, photographs, Escher prints, Penrose tiles and more. It also features some interesting quotations by scientists, mathematicians and literary figures about geometrical forms." -- Susan Duhig, Chicago Tribune

"Hero of Alexandria defined the gnomon as that form which, when added to some form, results in a new form, similar to the original." Put more simply, the author defines a gnomon as a self-repeating shape. The author elaborates on and applies the idea of the gnomon throughout ten chapters that deal with a variety of topics, including the Fibonacci sequence, the golden number, whorled figures, spirals, and fractals. -- Michael W. Good, Mathematics Teacher, February 1st, 2000

A book that, even if at times demanding, will enhance our understanding of numbers and make us appreciate their history. -- Review

[W]ill enhance our understanding of numbers and make us appreciate their history. -- Eli Maor, American Mathematical Monthly

See all Editorial Reviews

Product Details

• Hardcover: 280 pages
• Publisher: Princeton University Press (April 19, 1999)
• Language: English
• ISBN-10: 0691005141
• ISBN-13: 978-0691005140
• Product Dimensions: 9.5 x 6.4 x 0.9 inches
• Shipping Weight: 1.3 pounds
• Average Customer Review: 3.2 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #853,066 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

18 of 18 people found the following review helpful:

4.0 out of 5 stars Slow going, but worth it, February 7, 2001

By 

Duwayne Anderson (Saint Helens, Oregon) - See all my reviews
(REAL NAME)   

This review is from: Gnomon (Hardcover)

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 read the book, you probably will be able to go back and use the formulas in the back of the chapters without having to refer back to the text.

If you ever wanted to know about the Fibonacci sequence, I can hardly imagine a book that will satisfy you better than this one. The first thing you will learn is that the Fibonacci sequence you met in grade school is just a small subset of a more general form. Then, in a whirlwind of mathematical activity you will see the general recursive formula (which depends, of course, on the seed and gnomonic numbers). This is followed by explicit formulae for the terms of the sequence and even a demonstration of how some of these equations, in the limit, model the behavior of wave propagation in an electronic transducer ladder, and the movement of a ganged series of pulleys.

A continuing source of amazement is the way in which the mathematical themes in this book are so interconnected. That's fitting, I suppose, for a book called Gnomon. In the chapter on whorled figures we see many of the other subjects in this book reappearing.

The book also has an excellent chapter on the golden number, and another on the silver number. The golden number, as you may know, shows up all over the place, and not just in Gazale's book. Here, he connects it with the Fibonacci sequence, whorled spirals, and golden rectangles. And for every example using the golden number, not surprisingly, there is another using the silver number. It's fascinating to read of all the ways in which these numbers shows up, and try to contemplate the underlying order that makes it happen this way.

Things get a little more abstract toward the end of the book (but no less interesting), and a bit harder to read. There are some very interesting developments with spirals and the rotation matrix, along with some interesting construction techniques for making your own spirals with paper and pen.

The last chapter, on fractals, exercised my little gray cells more than the rest of the book. This is not your typical discussion about fractals, with pretty pictures and non-technical explanations about self-similarity at any level. While Gazale does not dive in with the sort of mathematical rigor to which a pure mathematician would aspire, he claims to have written an unusual chapter on the subject, derived directly from number-theoretic considerations. This chapter will keep you busy following all the ins and outs of some pretty involve matrix mathematics. If you like the fast-Fourier algorithm, I think you'll love it.

This book is definitely not for everyone. But if you really, really like mathematics, and especially number-theoretic mathematics, I think you will like it. It will most definitely exercise your mind, but then again, that's what a good book on mathematics is supposed to do. Isn't it?

15 of 16 people found the following review helpful:

3.0 out of 5 stars It would be better if this guy could write, August 25, 1999

By A Customer

This review is from: Gnomon (Hardcover)

I'm a regular reader of all sorts of books on math, and so Gnomon seemed a natural for me. I have a master's degree in computer science (bachelors, too, but i digress) and this sort of thing is right up my alley. The book doesn't really cover any new ground, but it does gather separate things into one volume, which makes it nice as a reference. The biggest problem with the book is with the actual text. This guy can't write. Yes, the material is technical, yes it's slow going, but that is no excuse for poorly structured arguments and incoherent organization of the material.

It's all here, but you'll need to work through it slowly and try to infer what he means because he leaves out a bunch of foundation work.

6 of 7 people found the following review helpful:

5.0 out of 5 stars An extremely original book , full of ideas and discoveries., July 5, 1999

By A Customer

This review is from: Gnomon (Hardcover)

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.