I’ve been playing with numbers with Python. I wrote a function that returns the nth Fibonacci number. Fibonacci numbers are generated by taking zero and one as the first two numbers, then adding the previous two Fibonacci numbers together to yield the next one. The next Fibonacci number after 0 and 1 is also 1 since 0+1=1. The next one is 2. The next one is 3. The next one is 5. And so on. Strictly speaking, the first Fibonacci number is 1 since there is another trait that Fibonacci numbers exhibit and 0 can play no part in that. Given two successive Fibonacci numbers, the ratio of the greater to the lesser of the two will tend toward the golden mean or phi.<\/p>\n
The golden mean is approximately\u00a01.61803398875 although, like pi, it is an irrational quantity that has no exact expression. It can be written as the expression (1 + square_root(5))\/2. It is the square_root(5) term that makes the number irrational. But, if you will notice, 1\/1 is 1, 2\/1 is 2, 3\/2 is 1.5, 5\/3 is approximately 1.667 and as you continue to compute the ratio with larger and larger Fibonacci numbers, the result gets closer and closer to phi.<\/p>\n
Phi is an interesting quantity. It shows up in many places in nature. Some of the claims about it, for instance that it is evident in the logarithmic curve formed by a Nautilus shell, have been disproven by empirical study. Others, like the fact that the ratio of the length of a diagonal of a regular pentagon to the length of one of its sides is phi, are on sound footing. In any case, computing Fibonacci numbers and phi can be an interesting pastime.<\/p>\n
I found a simple function in Python that yields the nth Fibonacci number in relatively short order without having the unfortunate side effect of consuming the entire stack.<\/p>\n