Some of what you identify as non-Fibonacci ratios are in fact ratios of non-adjacent Fibonacci numbers. For example, 0.382 is the asymptotic ratio of two Fib numbers that are two positions apart in the sequence (e.g., 13/34 and 21/55). That these "skip one" ratios converge to the square of adjacent ratios is a consequence of the fact that the adjacent ratios converge to constant value. This would be true for any geometric (constant ratio) sequence, which the Fibonacci sequence approaches at the higher numbers.
Likewise, any ratio of non-adjacent Fib numbers of wider separation will converge to a power of the adjacent ratios. The elements of the sequence 1.000, 1.618, 2.618, 4.236, 6.854, etc, (i.e., the powers of the adjacent ratio 1.618 starting with power zero) are all ratios of non-adjacent Fib numbers with increasing separation. It's interesting to note that this sequence has the additional property that each element of the series beyond 1.618 is the sum of the two previous elements, just like the Fibonacci series itself.
The inverse series, 1.000, .618, .382, .236, .146, .090, .056, .034, where each element is a power of .618 are themselves ratios of non-adjacent Fib numbers, and have a similar addition property in that each element is the difference between the two preceding elements in the series.
If you look at the non-adjacent ratios of the lower Fibs, you can come up with several additional Fib ratios. The largest ratio less than 1 is the adjacent ratio 2/3, and the smallest ratio greater than 1 is 1.5, so ratios of these small Fib numbers do not produce numbers that fill in the big gaps near the ratio one. I don't know who thought up the idea of using roots of the ratios to do fill in, but I've always assumed it was done because it is the extension of the asymptotic geometric sequence to which the Fib numbers converge rather than some arbitrary scheme.
Your point is well taken that the powers and roots represent a much larger set of numbers than those usually included in the set of "important numbers". I'm a long way from being convinced that the market turns at or near these numbers because of some fundamental law of nature as opposed to the expectations of the large number of people who follow them. |
