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The Fibonacci Series and the Golden RatioEdit

One of the interesting features of the Fibonacci series is the proportion of each term to those before and after it. Look at the following table:

2 / 1

3 / 2

5 / 3

8 / 5

13 / 8

21 / 13

34 / 21

55 / 34

89 / 55

144 / 89

233 / 144

377 / 233

610 / 377

987 / 610

1597 / 987

= 2

= 1.5

= 1.666666666667

= 1.6

= 1.625

= 1.615384615385

= 1.619047619048

= 1.617647058824

= 1.618181818182

= 1.617977528090

= 1.618055555556

= 1.618025751073

= 1.618037135279

= 1.618032786885

= 1.618034447822

1 / 2

2 / 3

3 / 5

5 / 8

8 / 13

13 / 21

21 / 34

34 / 55

55 / 89

89 / 144

144 / 233

233 / 377

377 / 610

610 / 987

987 / 1597

= 0.5

= 0.666666666667

= 0.6

= 0.625

= 0.615384615385

= 0.619047619048

= 0.617647058824

= 0.618181818182

= 0.617977528090

= 0.618055555556

= 0.618025751073

= 0.618037135279

= 0.618032786885

= 0.618034447822

= 0.618033813400


Two things can be seen from the table:

1 The values in the second column converge on a value that begins 1.61803...

2 The reciprocals of these values (in the right-hand column) converge on a value that is one less than that in the second column.

At infinity, the value in the second column is one more than its reciprocal in the right-hand column. The number that meets this requirement is the constant: (√ 5 + 1) / 2 = 1.618033988749894848204586834365638. This number is called phi, and its symbol is φ.

The reciprocal of phi is: (√ 5 - 1) / 2 = 0.618033988749894848204586834365638. Phi is also known as the Golden Ratio. A rectangle is in Golden Ratio if the proportion of its width to its length is equal to the proportion of its length to the sum of its length and width. If its width = a and its length = b, then a / b = b / (a + b). The reciprocal of this amount is phi. b / a = phi. (a + b) / b = phi. A fuller version of phi (50,000 decimal places!) can be found here . This ratio features frequently in art and architecture in the proportions of figures and buildings, and has done so since at least the Greek classical period, and probably earlier. It is the proportion which the human eye finds most pleasing. For illustrations of this, see The Golden Ratio , by Steve Blacker, Jeanette Polanski, and Marc Schwach. Next 

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