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.
Rezensionen / Stimmen
[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 Gazale describes clearly the concepts underlying gnomic patterns such as contained fractions, Fibonacci sequences, whorls, spirals, and fractals. Gazale 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
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Produkt-Hinweis
Illustrationen
24 farbige Abbildungen, 124 s/w Abbildungen, 16 Tabellen
24 color illus. 124 black and white illus. 16 tables
Maße
Höhe: 229 mm
Breite: 152 mm
Gewicht
ISBN-13
978-0-691-00514-0 (9780691005140)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Midhat J. Gazale 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 Merite in 1981.
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
