The Golden Ratio and Fibonacci Numbers

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In this invaluable book, the basic mathematical properties of the golden ratio and its occurrence in the dimensions of two- and three-dimensional figures with fivefold symmetry are discussed. In addition, the generation of the Fibonacci series and generalized Fibonacci series and their relationship to the golden ratio are presented. These concepts are applied to algorithms fur searching and function minimization. The Fibonacci sequence is viewed as a one-dimensional aperiodic, lattice and these ideas are extended to two- and three-dimensional Penrose tilings and the concept of incommensurate projections. The structural properties of aperiodic crystals and the growth of certain biological organisms are described in terms of Fibonacci sequences.
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Product Details

  • ISBN-13: 9789810232641
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 12/28/1997
  • Pages: 172
  • Product dimensions: 5.94 (w) x 9.12 (h) x 0.39 (d)

Table of Contents

Ch. 1 Introduction 1
Ch. 2 Basic Properties of the Golden Ratio 7
Ch. 3 Geometric Problems in Two Dimensions 15
Ch. 4 Geometric Problems in Three Dimensions 23
Ch. 5 Fibonacci Numbers 35
Ch. 6 Lucas Numbers and Generalized Fibonacci Numbers 51
Ch. 7 Continued Fractions and Rational Approximants 63
Ch. 8 Generalized Fibonacci Representation Theorems 71
Ch. 9 Optimal Spacing and Search Algorithms 79
Ch. 10 Commensurate and Incommensurate Projections 87
Ch. 11 Penrose Tilings 97
Ch. 12 Quasicrystallography 111
Ch. 13 Biological Applications 123
App. I Construction of the Regular Pentagon 137
App. II The First 100 Fibonacci and Lucas Numbers 139
App. III Relationships Involving the Golden Ratio and Generalized Fibonacci Numbers 143
References 153
Index 157
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