
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 

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:
F^{1} 
1.618033989... 
1/F^{1} 
0.618033989... 
F^{1} 
1/F^{1} 
1

F^{3} 
4.236067977...

1/F^{3} 
0.236067977...

F^{3} 
1/F^{3} 
4

F^{5} 
11.09016994...

1/F^{5} 
0.09016994...

F^{5} 
1/F^{5} 
11

F^{7} 
29.03444185...

1/F^{7} 
0 .03444185...

F^{7} 
1/F^{7} 
29

A similar thing happens for all even
multiples of the golden mean if we subtract the inverse from
one.
F^{2} 
2.618033989... 
11/F^{2} 
0.618033989... 
F^{2} + 1/F^{2} 
3

F^{4} 
6.854101...

11/F^{4} 
0.854101...

F^{4} + 1/F^{4} 
7

F^{6} 
17.94427...

11/F^{6} 
0.94427...

F^{6} + 1/F^{6} 
18

F^{8} 
46.97871...

11/F^{8} 
0. 97871...

F^{8} + 1/F^{8} 
47

We can subtract the numbers in the inverse
column from the numbers in the powers column for the odd powers
(F^{n } F^{n}), and
add for the even powers (F^{n}+ 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 e^{x} + 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: F^{n }+ F^{n+1}= F^{n+2}. 
Links and References
