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| The Golden Ratio and Phi |
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| Picture Credits: Figures 1&2 http://tsc.k12.in.us/training/CURR/MATH/FIBN/fibnat.htm |
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| An interesting pattern evolves from the ratios of consecutive Fibonacci numbers. The graph below plots the quotients of the first 13 Fibonacci numbers and their antecedents. |
| Fig.2 |
| The ratios of successive Fibonacci numbers approach a value of 1.61803... The limit of the equation as n approaches infinity, is the irrational number, Phi. This value is known as the Golden Ratio. Like the irrational number Pi, the digits of Phi, go on forever without repeating. |
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| The line segment, AB, is divided by a point, C, such that: |
| Fig. 1 |
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| Since the decimal continues to infinity, the most accurate way to represent Phi is to calculate it algebraically and express it as the root of a quadratic equation. Consider the line segment below: |
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| Now, we can simplify and substitute to form a quadratic equation that can be solved using the quadratic formula. |
| We now see that Phi is actually the positive root of the quadratic equation. The negative root is -phi, such that phi = 0.6180339887499... |
| From the quadratic formula, the two roots are: |
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| References: Hoggatt, V.E. 1969. Fibonacci and Lucas Numbers. Houghton Mifflin Company, New York. 92 pp. Runion, Garth E. 1972. The Golden Section and Related Curiosa. Scott, Foresman and Company, Glenview. 150 pp. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html http://pass.maths.org/issue3/fibonacci/ http://www.friesian.com/golden.htm |
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