Powers of Phi
Phi has a unique additive relationship
The powers of phi have unusual properties in that they are
related not only exponentially, but are additive as well. We
know that:
Ø^{2}
= Ø + 1
Which is the same as:
Ø^{2}
= Ø^{1} + Ø^{0}
And this leads to the fact that for any n:
Ø^{n+2}
= Ø^{n+1} + Ø^{n}
Thus each two successive powers of phi add to the next one!
n 
Ø^{n}

0 
1.000000 
1 
1.618034 
2 
2.618034 
3 
4.236068 
4 
6.854102 
5 
11.090170 
6 
17.944272 
Powers of Phi and its reciprocal
Another little curiosity involves taking phi to a
power and then adding or subtracting its reciprocal:
For any even integer n:
Ø^{n}
+ 1/Ø^{n} = a whole number
For any odd integer n:
Ø^{n}
 1/Ø^{n} = a whole number
Examples are shown in the tables below:
for n = even integers
n 
Ø^{n} 
1/Ø^{n} 
Ø^{n}
+ 1/Ø^{n} 
0 
1.000000000 
1.000000000 
2 
2 
2.618033989 
0.381966011 
3 
4 
6.854101966 
0.145898034 
7 
6 
17.944271910 
0.055728090 
18 
8 
46.978713764 
0.021286236 
47 
10 
122.991869381 
0.008130619 
123 
for n = odd integers
n 
Ø^{n} 
1/ Ø^{n} 
Ø^{n}
 1/Ø^{n} 
1 
1.618033989 
0.618033989 
1 
3 
4.236067977 
0.236067977 
4 
5 
11.090169944 
0.090169944 
11 
7 
29.034441854 
0.034441854 
29 
9 
76.013155617 
0.013155617 
76 
11 
199.005024999 
0.005024999 
199 
The whole numbers generated by this have a relationship
among themselves, creating an additive series, similar in structure to the Fibonacci
series, and which also converges on phi:
Exponent n 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
Result 
2 
1 
3 
4 
7 
11 
18 
29 
47 
76 
123 
199 
