Attractors and the Golden Mean

The golden mean is a mathematical constant that was discovered by the ancient Greeks. Like other basic constants (p, i, e, sqrt(2) ) it shows up in unexpected situations.

Written 2000

Formatted 2010

 

F= (1 + sqrt(5) )/2 = 1.618033989...

An attractor is the result of a recursive equation. Some approximation methods were developed by Sir Isaac Newton which use attractors. To attract the golden mean use one of the following equations recursively:

A <= 1 +1/A

or

B <= sqrt(B+1)

For example, start either recursive equation with 2 and watch what you get after each iteration:

Iteration

A

B

0

2

2

1

1.5

1.7320...

2

1.6666666...

1.6528...

3

1.6

1.6287...

4

1.625

1.6213...

We can see that both of these patterns are approaching the golden mean: F= 1.618033989...

 

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These two methods hint at some interesting properties of the golden mean. First compare the inverse of the golden mean to the golden mean itself:

F

1.618033989...

1/F

0.618033989...

F - 1/F

1

This pattern off matching decimals actually repeats for all odd powers of the golden mean:

F1

1.618033989...

1/F1

0.618033989...

F1 - 1/F1

1

F3

4.236067977...

1/F3

0.236067977...

F3 - 1/F3

4

F5

11.09016994...

1/F5

0.09016994...

F5 - 1/F5

11

F7

29.03444185...

1/F7

0 .03444185...

F7 - 1/F7

29

A similar thing happens for all even multiples of the golden mean if we subtract the inverse from one.

F2

2.618033989...

1-1/F2

0.618033989...

F2 + 1/F2

3

F4

6.854101...

1-1/F4

0.854101...

F4 + 1/F4

7

F6

17.94427...

1-1/F6

0.94427...

F6 + 1/F6

18

F8

46.97871...

1-1/F8

0. 97871...

F8 + 1/F8

47

We can subtract the numbers in the inverse column from the numbers in the powers column for the odd powers (Fn - F-n), and add for the even powers (Fn+ F-n) to create the Lucas Series: 1,3,4,7,11,18,29,47,... Notice how this parallels the common form used in engineering of ex + e-x. Thus, the Lucas Series seems to occur more naturally than the more popular Fibonacci series.

Still more may be noticed in this list. As in both the Fibonacci Series, and the Lucas Series, two successive numbers may be added together to generate the next number, the same is true for powers of the golden mean: Fn + Fn+1= Fn+2.

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