Cover image for Gnomon : from pharaohs to fractals
Gnomon : from pharaohs to fractals
Gazalé, Midhat J., 1929-
Personal Author:
Publication Information:
Princeton, N.J. : Princeton University Press, [1999]

Physical Description:
xiv, 259 pages, 16 unnumbered pages of plates : illustrations (some color) ; 24 cm
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QA447 .G39 1999 Adult Non-Fiction Central Closed Stacks

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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.

Author Notes

Midhat J. Gazalé is an international telecommunications and space consultant and Visiting Professor of Telecommunications and Computer Management at the University of Paris IX. He has served as President of AT&T-France, as Chairman of the Board for Sperry-France and for International Computers-France, and as an executive and research scientist for other major companies. He was made Chevalier dans l'Ordre National du Mérite in 1981.

Reviews 1

Choice Review

The perennial task of bringing mathematics before the general public attracts expositors wielding a diversity of strategies who pursue goals that range from enticing further study and inducing appreciation to merely diminishing fear. Why Do Buses Come in Threes? purports to show us how mathematics, in particular probability theory, may enrich our daily experience of the world. Authors Eastaway and Wyndham strive to keep the mathematics so simple that often they stop short of delivering a satisfying explanation for this or that phenomenon, leaving the reader to settle for a mere kernel of insight. In these moments they fail to communicate precisely the explanatory force and predictive power of mathematics. Consider the question of the title--if one turns to that chapter one immediately finds the underlying supposition summarily dismissed as a myth; the chapter then mostly discusses hypothetical waiting times if buses did come in threes. Clawson's Mathematical Sorcery basically provides general information about the topics an undergraduate mathematics major usually meets in the first year or two of college: calculus, linear algebra, logical deduction, and proof. Demonstrating the fun of mathematics constitutes the author's stated purpose, but most likely he will convince readers already predisposed to think so. Clawson dangles many fascinating formulae before the reader, but frustration awaits neophytes who want the explanations and find they face years of study to get them. On the other hand, combinatorics, never mentioned here, offers many equally startling phenomena that nevertheless admit clever but fully elementary and self-contained explanations. By "gnomon" Gazale (Univ. of Paris) means a geometrical form whose addition to some other form leaves the shape of the latter form invariant, changing only the size. For example, each new chamber of a nautilus shell constitutes a gnomon. A form generated by the accretion of gnomons displays self-similarity, the same property characteristic of those forms known as fractals. Gazale's meditation on gnomons propels him through a suite of topics familiar to readers of popular mathematics: continued fractions, Fibonacci sequences, the golden number, spirals, and finally, fractals. In each case he manages to offer either fresh insights or a distinctive viewpoint. Less familiar topics include the silver number and electrical ladder. Though this book demands more from the reader than the previous two and parts should interest an undergraduate mathematics major, nevertheless in the main it falls in the category of popular exposition. Martin Gardner's "Mathematical Games" column in Scientific American inspired several generations of budding research mathematicians who first cut their teeth on the problems and puzzles published therein. The Mathemagician and Pied Puzzler consists of papers about recreational mathematics and puzzles in his honor. Since the ranks of Gardner's admirers do include some of the world's strongest mathematicians, recreational mathematics here hardly means frivolous and this volume certainly contains much to stimulate an undergraduate mathematics major, as for example J. Lagarias's contribution concerning the famous 3x+1 problem. Popular mathematical writing often returns repeatedly to the same, few, well-trodden topics, the ones sufficiently important to interest a wide audience but still simple enough to describe in a nontechnical way. Cipra's What's Happening in the Mathematical Sciences surveys late-breaking mathematical news. Though he includes material on such familiar topics as computer chess, chaos, Escher, and cryptosystems, he also discusses less familiar territory such as quantum computers, automated theorem provers, and algorithmic algebraic geometry. Here undergraduates might easily make their first acquaintance with a topic that could shape the course of their future studies and, beyond that, their professional lives. An essential acquisition. D. V. Feldman University of New Hampshire

Table of Contents

Prefacep. xi
Introduction Gnomonsp. 3
Of Gnomons and Sundialsp. 6
On Geometric Similarityp. 9
Geometry and Numberp. 10
Of Gnomons and Obelisksp. 13
Chapter I Figurate and m-adic Numbersp. 15
Figurate Numbersp. 15
Property of Triangular Numbersp. 17
Property of Square Numbersp. 20
M-adic Numbersp. 21
Powers of Dyadic Numbersp. 22
The Dyadic Hamiltonian Pathp. 25
Powers of Triadic Numbersp. 29
Chapter II Continued Fractionsp. 31
Euclid's Algorithmp. 31
Continued Fractionsp. 33
Simple Continued Fractionsp. 34
Convergentsp. 35
Terminating Regular Continued Fractionsp. 37
Periodic Regular Continued Fractionsp. 38
Spectra of Surdsp. 40
Nonperiodic Nonterminating Regular Continued Fractionsp. 42
Retrovergentsp. 43
Appendixp. 44
Summary of Formulaep. 45
Chapter III Fibonacci Sequencesp. 49
Recursive Definitionp. 50
The Seed and Gnomonic Numbers so
Explicit Formulation of Fm,np. 52
Alternative Explicit Formulationp. 56
The Monognomonic Simple Periodic Fractionp. 58
The Dignomonic Simple Periodic Fractionp. 61
Arbitrarily Terminated Simple Periodic Fractionsp. 63
m Is Very Small: From Fibonacci to Hyperbolic and Trigonometric Functionsp. 66
Appendix: The Polygnomonic SPFp. 67
Summary of Formulaep. 69
Chapter IV Ladders: From Fibonacci to Wave Propagationp. 74
The Transducer Ladderp. 74
The Electrical Ladderp. 76
Resistance Laddersp. 77
Iterative Laddersp. 79
Imaginary Componentsp. 83
The Transmission Linep. 85
The Mismatched Transmission Linep. 86
Wave Propagation Along a Transmission Linep. 88
Pulley Ladder Networksp. 91
Marginaliap. 95
A Topological Similarityp. 95
Chapter V Whorled Figuresp. 96
Whorled Rectanglesp. 96
Euclid's Algorithmp. 96
Monognomonic Whorled Rectanglesp. 99
Dignomonic Whorled Rectanglesp. 102
Self-Similarityp. 108
Improperly Seeded Whorled Rectanglesp. 109
Two Whorled Trianglesp. III I
Marginaliap. 113
Transmission Lines Revisitedp. 113
Chapter VI The Golden Numberp. 114
From Number to Geometryp. 117
The Whorled Golden Rectanglep. 118
The Fibonacci Whorlp. 120
The Whorled Golden Trianglep. 121
The Whorled Pentagonp. 121
The Golden Section: From Antiquity to the Renaissancep. 123
Marginaliap. 132
The Sneezewortp. 132
A Golden Trickp. 134
The Golden Knotp. 134
Chapter VII The Silver Numberp. 135
From Number to Geometryp. 137
The Silver Pentagonp. 138
The Silver Spiralp. 139
The Winklep. 142
Marginaliap. 143
Golomb's Rep-Tilesp. 143
A Commedia dell'Artep. 146
Repeated Radicalsp. 148
Chapter VIII Spiralsp. 151
The Rotation Matrixp. 151
The Monognomonic Spiralp. 153
Self-similarityp. 158
Equiangularityp. 159
Perimeter of the Spiralp. 161
The Rectangular Dignomonic Spiralp. 165
The Archimedean Spiralp. 168
Damped Oscillationsp. 171
The Simple Pendulump. 174
The RLC Circuitp. 177
The Resistorp. 178
The Capacitorp. 179
The Inductorp. 180
The Series RLC Circuitp. 180
Appendix: Finite Difference Equationsp. 183
Chapter IX Positional Number Systemsp. 187
Divisionp. 187
Mixed Base Positional Systemsp. 191
Finding the Digits of an integerp. 195
Chapter X Fractalsp. 198
The Kronecker Product Revisitedp. 198
Associativity of the Kronecker Productp. 201
Matrix Orderp. 205
Commutativity of the Kronecker Productp. 206
Vectorsp. 208
Fractal Latticesp. 209
Pascal's Triangle and Lucas's Theoremp. 211
The Sierpinky Gasket and Carpetp. 215
The Cantor Dustp. 219
The Thue-Morse Sequence and Tilingp. 223
Higher-Dimensional Latticesp. 225
Commutativity and Higher Dimensionsp. 227
The Three Dimensional Sierpinsky Pyramid and Menger Spongep. 227
The Kronecker Product with Respect to Other
Operationsp. 231
Fractal Linkagesp. 233
The Koch Curvep. 234
The Peano Space-Filling Curvep. 237
A Collection of Regular Fractal Linkagesp. 238
Mixed Regular Linkages and Corresponding Tesselationsp. 244
An Irregular Fractal Linkage: The pentagonal "Eiffel Tower"p. 246
Appendix: Simplifying Symbolsp. 248
Indexp. 253