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Unformatted text preview: Fibonacci Numbers and the Golden Ratio James Emery 4/30/2011 Contents 1 Fibonacci Numbers 2 2 Some Large Fibonacci Numbers 12 3 The Binet Formula 12 4 The Golden Section Search 15 5 Phyllotaxis, the Golden Rectangle, Numerology, and Pseudo science 18 6 The Pentagon and the Golden Ratio 19 7 Appendix: How the Figures in This Document Were Drawn 23 8 Appendix: Continued Fractions 42 8.1 The Continued Fraction Representation of a Rational Number 42 8.2 The Continued Fraction Representation of a Real Number . . 46 8.3 Computing the Convergents . . . . . . . . . . . . . . . . . . . 48 8.4 Properties of the Convergents . . . . . . . . . . . . . . . . . . 52 8.5 Continued Fractions Bibliography . . . . . . . . . . . . . . . . 55 9 Notes 56 10 Bibliography 56 1 1 Fibonacci Numbers The Fibonacci numbers are defined to be an infinite sequence of numbers f , f 1 , f 2 , .... . The first two numbers are f = 0, and f 1 = 1. The third number is the sum of the previous two, so is f 3 = 1. The k th number in the sequence is the sum of the previous two numbers. So f k = f k − 2 + f k − 1 . The first few numbers in the sequence are , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , ... A second sequence of ratios of Fibonacci numbers, { r n } ∞ 1 , is also of interest. These ratios are defined by r n = f n +1 f n . It turns out that this second sequence converges to the Golden Ratio. So let us define the Golden Ratio. Suppose a rectangle has height h and width w . Suppose the ratio of h to w is the same as the ratio of w to the sum of the sides h + w . Then h w = w h + w . Without loss of generality suppose h = 1, then 1 w = w 1 + w , or w 2 = 1 + w. w 2 − w − 1 = 0 . Thus w = 1 + √ 5 2 . 2 A rectangle with these proportions is called the golden rectangle and φ = w h = 1 + √ 5 2 , is called the golden ratio. This is approximately φ = 1 . 618033988749895 As a second way of defining φ , let us consider the division of a unit interval. Let the unit interval be divided into two intervals, one of length x and the other of length 1 − x . Let us assume that x ≥ x − 1 . The extreme ratio is defined as r e = 1 x The mean ratio is defined as r m = x 1 − x . If r = r e = r m , the interval is said to be divided in extreme and mean ratio. This is a concept that appears in Euclid’s Elements. Definition Euclid’s Elements, Book 6, Definition 2: A straight line said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser. In this case r = 1 x = x 1 − x . So x 2 + x − 1 = 0 . And so x = − 1 + √ 5 2 . Thus r = 1 x = 1 + √ 5 2 , 3 which again is the golden ratio φ . Here is another way of constructing a golden rectangle. Draw a square of side 1. Bisect the lower horizontal edge and take it as a circle center c . Let the radius of the circle be the distance from the center c to the upper left corner of the square. Then r = q (1 / 2) 2 + 1 2 = √ 5 2 Swing the circle down from the upper left corner of the square to the line...
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.
 Fall '11
 N/A
 Economics

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