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From Pharaohs to Fractals
Midhat J. Gazalé

Honorable Mention for the 1999 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers

Hardcover | 1999 | This edition is out of print | ISBN: 9780691005140


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 Gazalé--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.

Gazalé 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. Gazalé 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 Gazalé terms the "winkle." Throughout, the book brims with original observations and research, from the presentation of a cousin of the "golden rectangle" that Gazalé 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.


"[A] crisp introduction ... the general reader can read it like a coffee-table book, enjoying the pictures. 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

"Midhat Gazalé describes clearly the concepts underlying gnomic patterns such as contained fractions, Fibonacci sequences, whorls, spirals, and fractals. Gazalé provides many interesting illustrations of symmetry in plants, animals, tiling patterns, and electrical circuits."--American Scientist

"A book that, even if at times demanding, will enhance our understanding of numbers and make us appreciate their history."--Eli Maor, American Mathematical Monthly


"Midhat Gazalé's Gnomon, handsomely illustrated by the author, is a splendid introduction to the surprising properties of gnomons, spirals, and their closely related number sequences, especially the famous golden number. Gazalé's elegant explorations lead him into fractals and triangular spirals that generate another famous irrational that he calls the 'silver number.' You put down his book with a heightened sense of awe and wonder at the gold and silver of pure geometry and its astonishing applications to the material world."--Martin Gardner, author of many books, most recently The Night Is Large and The Last Recreations: Hydras, Eggs, and other Mathematical Mystifications

"Gnomon entices mathematically inclined readers to embark upon a highly rewarding voyage of discovery. . . . Dr. Gazalé 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

File created: 1/4/2017

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