

Fibonacci Flim-Flam - mhansen
http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm

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camccann
That was a nice thrashing of some of the silly numerology that seems to
collect around the Fibonacci sequence/golden ratio.

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...

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rw
Chaitin's constant is also quite underappreciated.

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earthboundkid
Chaitin’s constant has the problem that unlike pi, no one can memorize its
first n digits.

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SandB0x
Nice as a continued fraction:

phi = 1 + 1/(1+1/(1+1/(1+...)))

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ThomPete
Great great article.

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"

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ThomPete
Why the downvote?

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DougBTX
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?

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ThomPete
Yeah you are right. My bad.

Perhaps the confusion comes from how you would normally try and structure your
grid.

I am not sure though if 0.71 is too far from 0.62 to be considered close
enough.

Regarding grammar I unfortunately I can't edit it sorry.

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DrJokepu
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.

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shrughes
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.

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camccann
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!

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shrughes
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 :-)

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akkartik
I remember being surprised by this back in '07:
<http://akkartik.name/blog/10476036>

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dazmax
Ah phi, the most irrational number.

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jfarmer
It's not even transcendental!

