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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:

2 = + 1

Which is the same as:

2 = 1 + 0

And this leads to the fact that for any n:

n+2 = n+1 + n

Thus each two successive powers of phi add to the next one!

n

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:

n  +  1/n = a whole number

For any odd integer n:

n  -  1/n = a whole number

Examples are shown in the tables below:

for n = even integers

n

n 1/n n + 1/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

n 1/ n n - 1/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
 

Phi - The Golden Number
A source to some of Net's "phi-nest" information on the
Golden Section / Mean / Proportion / Ratio / Number,
Divine Proportion, Fibonacci Series and Phi (1.6180339887...)

The Evolution of Truth, 1999-2001

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