Gnomon: From Pharaohs to Fractals

Overview

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 ...

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Overview

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.

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Editorial Reviews

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

"[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

Cait Anthony
By definition, a gnomon is a self-repeating shape such as those found in spiral seashells and defined by Fibonacci sequences of numbers. Gazale fully considers gnomons by virtue of their properties and application as well as their historical heritage, founded in ancient Egyptian and Greek timekeeping. Not for readers who eschew math, this book is loaded with formulas exemplifying "gnomonicity" in fractions, logarithms, spirals, fractals, and "the famous golden number.
Science News
Susan Duhig
Gazale, a mathematician and engineer who teaches at the University of Paris, defines a gnomon as a figure which, when added to another figure, results in a figure similar to the original--whorled figures, logarithmic spirals, fractals and others. Apart from the crisp introduction, the text of the book reaches beyond the nonspecialist, but the general reader can browse 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. My favorite is the one in which Vladimir Nabokov describes a spiral as a circle that has been set free.Chicago Tribune
Donald W. Good
"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.
—Donald W. Good, Mathematics Teacher
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Product Details

  • ISBN-13: 9780691005140
  • Publisher: Princeton University Press
  • Publication date: 4/19/1999
  • Pages: 280
  • Product dimensions: 6.37 (w) x 9.54 (h) x 0.93 (d)

Table of Contents

Preface xi
INTRODUCTION Gnomons 3
Of Gnomons and Sundials 6
On Geometric Similarity 9
Geometry and Number 10
Of Gnomons and Obelisks 13
CHAPTER I Figurate and m-adic Numbers 15
Figurate Numbers 15
Property of Triangular Numbers 17
Property of Square Numbers 20
m-adic Numbers 21
Powers of Dyadic Numbers 22
The Dyadic Hamiltonian Path 25
Powers of Triadic Numbers 29
CHAPTER II Continued Fractions 31
Euclid's Algorithm 31
Continued Fractions 33
Simple Continued Fractions 34
Convergents 35
Terminating Regular Continued Fractions 37
Periodic Regular Continued Fractions 38
Spectra of Surds 40
Nonperiodic Nonterminating Regular Continued Fractions 42
Retrovergents 43
Appendix 44
Summary of Formulae 45
CHAPTER III Fibonacci Sequences 49
Recursive Definition 50
The Seed and Gnomonic Numbers so
Explicit Formulation of Fm,n 52
Alternative Explicit Formulation 56
The Monognomonic Simple Periodic Fraction 58
The Dignomonic Simple Periodic Fraction 61
Arbitrarily Terminated Simple Periodic Fractions 63
m Is Very Small: From Fibonacci to Hyperbolic and Trigonometric Functions 66
Appendix: The Polygnomonic SPF 67
Summary of Formulae 69
CHAPTER IV Ladders: From Fibonacci to Wave Propagation 74
The Transducer Ladder 74
The Electrical Ladder 76
Resistance Ladders 77
Iterative Ladders 79
Imaginary Components 83
The Transmission Line 85
The Mismatched Transmission Line 86
Wave Propagation Along a Transmission Line 88
Pulley Ladder Networks 91
Marginalia 95
A Topological Similarity 95
CHAPTER V Whorled Figures 96
Whorled Rectangles 96
Euclid's Algorithm 96
Monognomonic Whorled Rectangles 99
Dignomonic Whorled Rectangles 102
Self-Similarity 108
Improperly Seeded Whorled Rectangles 109
Two Whorled Triangles III I
Marginalia 113
Transmission Lines Revisited 1 13
CHAPTER VI The Golden Number 114
From Number to Geometry 117
The Whorled Golden Rectangle 118
The Fibonacci Whorl 120
The Whorled Golden Triangle 121
The Whorled Pentagon 121
The Golden Section: From Antiquity to the Renaissance 123
Marginalia 132
The Sneezewort 132
A Golden Trick 134
The Golden Knot 134
CHAPTER VII The Silver Number 135
From Number to Geometry 137
The Silver Pentagon 138
The Silver Spiral 139
The Winkle 142
Marginalia 143
Golomb's Rep-Tiles 143
A Commedia dell'Arte 146
Repeated Radicals 148
CHAPTER VIII Spirals 151
The Rotation Matrix 151
The Monognomonic Spiral 153
Self-similarity 158
Equiangularity 159
Perimeter of the Spiral 161
The Rectangular Dignomonic Spiral 165
The Archimedean Spiral 168
Damped Oscillations 171
The Simple Pendulum 174
The RLC Circuit 177
The Resistor 178
The Capacitor 179
The Inductor 180
The Series RLC Circuit 180
Appendix: Finite Difference Equations 183
CHAPTER IX Positional Number Systems 187
Division 187
Mixed Base Positional Systems 191
Finding the Digits of an integer 195
CHAPTER X Fractals 198
The Kronecker Product Revisited 198
Associativity of the Kronecker Product 201
Matrix Order 205
Commutativity of the Kronecker Product 206
Vectors 208
Fractal Lattices 209
Pascal's Triangle and Lucas's Theorem 211
The Sierpinky Gasket and Carpet 215
The Cantor Dust 219
The Thue-Morse Sequence and Tiling 223
Higher-Dimensional Lattices 225
Commutativity and Higher Dimensions 227
The Three Dimensional Sierpinsky Pyramid and Menger Sponge 227
The Kronecker Product with Respect to Other
Operations 231
Fractal Linkages 233
The Koch Curve 234
The Peano Space-Filling Curve 237
A Collection of Regular Fractal Linkages 238
Mixed Regular Linkages and Corresponding Tesselations 244
An Irregular Fractal Linkage: The pentagonal "Eiffel Tower" 246
Appendix: Simplifying Symbols 248
Index 253

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