The Golden Ratio and Fibonacci Numbers

Front Cover
World Scientific, 1997 - Mathematics - 162 pages
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 for 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.
 

Contents

INTRODUCTION
1
BASIC PROPERTIES OF THE GOLDEN RATIO
7
GEOMETRIC PROBLEMS IN TWO DIMENSIONS
15
GEOMETRIC PROBLEMS IN THREE DIMENSIONS
23
FIBONACCI NUMBERS
35
LUCAS NUMBERS AND GENERALIZED FIBONACCI NUMBERS
51
CONTINUED FRACTIONS AND RATIONAL APPROXIMANTS
63
GENERALIZED FIBONACCI REPRESENTATION THEOREMS
71
PENROSE TILINGS
97
QUASICRYSTALLOGRAPHY
111
BIOLOGICAL APPLICATIONS
123
CONSTRUCTION OF THE REGULAR PENTAGON
137
THE FIRST 100 FIBONACCI AND LUCAS NUMBERS
139
RELATIONSHIPS INVOLVING THE GOLDEN RATIO AND GENERALIZED FIBONACCI NUMBERS
143
REFERENCES
153
INDEX
157

OPTIMAL SPACING AND SEARCH ALGORITHMS
79
COMMENSURATE AND INCOMMENSURATE PROJECTIONS
87

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