Calculating the golden ratio (sqrt(5)+1)/2, or equivalently, the x where x^2 = x+1, by calculating the square root of 5 with the Babylonian method. The Babylonian method is represented by the algorithm: g becomes (g + n/g)/2, where n is the root (2 for square, 3 for cube, and so on). INTERESTINGLY: Despite the fact that this method does NOT rely on the Fibonacci numbers, all results that it produces are actually ratios of two consecutive Fibonacci numbers. Calculating this with WolframAlpha, these are the first few ratios that it produces to approximate phi: 2/1 (2) 5/3 (1.666...) 34/21 (1.619047619...) 1597/987 (1.618034448...) 3524578/2178309 17167680177565/10610209857723 407305795904080553832073954/251728825683549488150424261 You can check for yourself at https://oeis.org/A000045 that these are all numbers of the Fibonacci sequence. QUESTION: Why is it that these numbers are spaced apart as they are? The regular approximations of phi from the Fibonacci numbers are 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, and so on; these grow much quicker, even though they are still Fibonacci ratios. Phi can, in theory, be represented by non-Fibonacci ratios; I can write 1.62 = 81/50, approximating phi arbitrarily even though neither are Fibonacci numbers. So, if this does generate Fibonacci ratios, why only some? Let's look at their distances. We have the first ratio a/b, and the second, next calculated ratio, c/d, and we will then count how many Fibonacci numbers are between the Fibonacci numbers a and c: 1 (between 2 and 5) 3 (between 5 and 34) 7 (between 34 and 1597) 15 (between 1597 and 3524578) 30 (between 3524578 and 17167680177565) 64 (between 17167680177565 and 407305795904080553832073954) I am not smart enough to understand why, but essentially this is related to the behaviour of the algorithm, which is like Newton's method, when applied to a quadratic whose root is the golden ratio. The spacings will roughly follow the powers of 2.
