The Golden Ratio and Fibonacci Numbers (Google eBook)

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World Scientific, Dec 1, 1997 - Mathematics - 162 pages
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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 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.
  

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Contents

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
51
CONTINUED FRACTIONS AND RATIONAL
63
GENERALIZED FIBONACCI REPRESENTATION THEOREMS
71
OPTIMAL SPACING AND SEARCH ALGORITHMS
79
COMMENSURATE AND INCOMMENSURATE
87
PENROSE TILINGS
97
QUASICRYSTALLOGRAPHY
111
BIOLOGICAL APPLICATIONS
123
CONSTRUCTION OF THE REGULAR PENTAGON
137
RELATIONSHIPS INVOLVING THE GOLDEN RATIO
143
REFERENCES
153
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Page 11 - the ratio of the lengths of the two segments is the same as the ratio of the length of the longer segment to the
Page 1 - has been called the golden mean, the golden section, the golden cut, the divine proportion, the Fibonacci number and the mean of Phidias
Page 1 - K (the ratio of the circumference to the diameter of a circle) and e
Page 5 - the ratio of the circumference, c, to the diameter, d, of a circle;
Page 2 - the longer segment is the same as the ratio of the length of the longer segment to the length of the
Page 1 - An irrational number is one which cannot be expressed as a ratio of

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