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### Summary

### Summary

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

Preface | p. xi |

Introduction Gnomons | p. 3 |

Of Gnomons and Sundials | p. 6 |

On Geometric Similarity | p. 9 |

Geometry and Number | p. 10 |

Of Gnomons and Obelisks | p. 13 |

Chapter I Figurate and m-adic Numbers | p. 15 |

Figurate Numbers | p. 15 |

Property of Triangular Numbers | p. 17 |

Property of Square Numbers | p. 20 |

M-adic Numbers | p. 21 |

Powers of Dyadic Numbers | p. 22 |

The Dyadic Hamiltonian Path | p. 25 |

Powers of Triadic Numbers | p. 29 |

Chapter II Continued Fractions | p. 31 |

Euclid's Algorithm | p. 31 |

Continued Fractions | p. 33 |

Simple Continued Fractions | p. 34 |

Convergents | p. 35 |

Terminating Regular Continued Fractions | p. 37 |

Periodic Regular Continued Fractions | p. 38 |

Spectra of Surds | p. 40 |

Nonperiodic Nonterminating Regular Continued Fractions | p. 42 |

Retrovergents | p. 43 |

Appendix | p. 44 |

Summary of Formulae | p. 45 |

Chapter III Fibonacci Sequences | p. 49 |

Recursive Definition | p. 50 |

The Seed and Gnomonic Numbers so | |

Explicit Formulation of Fm,n | p. 52 |

Alternative Explicit Formulation | p. 56 |

The Monognomonic Simple Periodic Fraction | p. 58 |

The Dignomonic Simple Periodic Fraction | p. 61 |

Arbitrarily Terminated Simple Periodic Fractions | p. 63 |

m Is Very Small: From Fibonacci to Hyperbolic and Trigonometric Functions | p. 66 |

Appendix: The Polygnomonic SPF | p. 67 |

Summary of Formulae | p. 69 |

Chapter IV Ladders: From Fibonacci to Wave Propagation | p. 74 |

The Transducer Ladder | p. 74 |

The Electrical Ladder | p. 76 |

Resistance Ladders | p. 77 |

Iterative Ladders | p. 79 |

Imaginary Components | p. 83 |

The Transmission Line | p. 85 |

The Mismatched Transmission Line | p. 86 |

Wave Propagation Along a Transmission Line | p. 88 |

Pulley Ladder Networks | p. 91 |

Marginalia | p. 95 |

A Topological Similarity | p. 95 |

Chapter V Whorled Figures | p. 96 |

Whorled Rectangles | p. 96 |

Euclid's Algorithm | p. 96 |

Monognomonic Whorled Rectangles | p. 99 |

Dignomonic Whorled Rectangles | p. 102 |

Self-Similarity | p. 108 |

Improperly Seeded Whorled Rectangles | p. 109 |

Two Whorled Triangles | p. III I |

Marginalia | p. 113 |

Transmission Lines Revisited | p. 113 |

Chapter VI The Golden Number | p. 114 |

From Number to Geometry | p. 117 |

The Whorled Golden Rectangle | p. 118 |

The Fibonacci Whorl | p. 120 |

The Whorled Golden Triangle | p. 121 |

The Whorled Pentagon | p. 121 |

The Golden Section: From Antiquity to the Renaissance | p. 123 |

Marginalia | p. 132 |

The Sneezewort | p. 132 |

A Golden Trick | p. 134 |

The Golden Knot | p. 134 |

Chapter VII The Silver Number | p. 135 |

From Number to Geometry | p. 137 |

The Silver Pentagon | p. 138 |

The Silver Spiral | p. 139 |

The Winkle | p. 142 |

Marginalia | p. 143 |

Golomb's Rep-Tiles | p. 143 |

A Commedia dell'Arte | p. 146 |

Repeated Radicals | p. 148 |

Chapter VIII Spirals | p. 151 |

The Rotation Matrix | p. 151 |

The Monognomonic Spiral | p. 153 |

Self-similarity | p. 158 |

Equiangularity | p. 159 |

Perimeter of the Spiral | p. 161 |

The Rectangular Dignomonic Spiral | p. 165 |

The Archimedean Spiral | p. 168 |

Damped Oscillations | p. 171 |

The Simple Pendulum | p. 174 |

The RLC Circuit | p. 177 |

The Resistor | p. 178 |

The Capacitor | p. 179 |

The Inductor | p. 180 |

The Series RLC Circuit | p. 180 |

Appendix: Finite Difference Equations | p. 183 |

Chapter IX Positional Number Systems | p. 187 |

Division | p. 187 |

Mixed Base Positional Systems | p. 191 |

Finding the Digits of an integer | p. 195 |

Chapter X Fractals | p. 198 |

The Kronecker Product Revisited | p. 198 |

Associativity of the Kronecker Product | p. 201 |

Matrix Order | p. 205 |

Commutativity of the Kronecker Product | p. 206 |

Vectors | p. 208 |

Fractal Lattices | p. 209 |

Pascal's Triangle and Lucas's Theorem | p. 211 |

The Sierpinky Gasket and Carpet | p. 215 |

The Cantor Dust | p. 219 |

The Thue-Morse Sequence and Tiling | p. 223 |

Higher-Dimensional Lattices | p. 225 |

Commutativity and Higher Dimensions | p. 227 |

The Three Dimensional Sierpinsky Pyramid and Menger Sponge | p. 227 |

The Kronecker Product with Respect to Other | |

Operations | p. 231 |

Fractal Linkages | p. 233 |

The Koch Curve | p. 234 |

The Peano Space-Filling Curve | p. 237 |

A Collection of Regular Fractal Linkages | p. 238 |

Mixed Regular Linkages and Corresponding Tesselations | p. 244 |

An Irregular Fractal Linkage: The pentagonal "Eiffel Tower" | p. 246 |

Appendix: Simplifying Symbols | p. 248 |

Index | p. 253 |