As far as my memory serves my I do though think that the article is making one claim that is wrong.
It was Bach and not Mozart who was obsessed with numbers and not phi or fibonacci.
Harmony in western music is based on phi subdivisions of the octave.
Fibonacci is used in design and art quite a lot. the A paper format (A0, A1, A2, A3, A4 etc.) is based on it.
When you see a website that is "pleasing to the eye" that is often (but far from always) based on some interpretation of Fibonacci sequence.
Most probably it's the consistency in proportional difference that is pleasing and not phi in it self.
Phi and f are retrospectively pleasing because they are culturally imposed on us. If you divide an octave in ten (instead of 12) then you get a different division of the octave but it wont feel pleasing to the ear.
Just as the Arabic division of the octave is quite different and doesn't really allow for harmony.
So I think it's premature to just throw it all out and say nonsense. Neither Fiboncci nor Phi are some natural constant but they might be a cultural constant for the west.
Or as Wittgenstein said:
"The faculty of taste cannot create a new structure, it can only make adjustments to one that already exists"
I've not voted on your comment, but here are a couple of reasons I'm leaving it at zero:
* Grammar in the first sentence, the "As far as my" line, is quite garbled for an into to a reasonably long comment.
* I can't verify your other claims, but I know A series paper is based on the 1:sqrt(2) ratio, not 1:phi, so I suspect you're falling for the same trap that the article warns against, 0.62 is almost 0.71 right?
Speaking of Fibionacci, I was browsing through my old Data Structures and Algorithms lecture notes the other day for nostalgia and discovered that Fibionacci(n) can be computed in O(log N) time, at least in theory. I have completely forgotten about that fact.
Only if you ignore the cost of multiplication or the size of the numbers involved. The length of the answer is proportional to N, so you end up with N log N time with the typical exponentiating algorithm, and there's no way you'll beat linear time. The regular old additive for loop takes N^2 time, though.
A closed-form solution exists, so if you pretend that arithmetic is free, it's O(1)!
Bignum multiplication is significantly slower than bignum addition; if memory serves me it ends up coming out that you're essentially stuck with O(n^2) no matter what you do.
Fun fact: The naive, unmemoized recursive Fibonacci function has time complexity of precisely O(fib(n)). Scary!
Oh, I'm sorry, I was under the impression that there existed O(N log N) multiplication algorithms, but apparently that's just conjectured. Right now the best seems to be O(N log(N) 2^(log*(N))) ( http://en.wikipedia.org/wiki/Fürers_algorithm ), which is slightly better than O(N log N log log N). Okay then.
Also, the closed form solution would still be O(log N), since I don't think anybody yet considers exponentiation to be free :-)
I do wonder why phi gets so much attention (pi as well), but e doesn't? My suspicion is that it's because logarithms are "harder" to understand...