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. 

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) 


B 
0 
2 
2 
1 
1.5 
1.7320... 
2 
1.6666666... 
1.6528... 
3 
1.6 
1.6287... 
4 
1.625 
1.6213... 
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} 
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 
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 
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}.






