To: FormerLurker
Fibonacci numbers are related to "phi", but they aren't "derived" from it. Thus, "phi" is not foundational (i.e., part of the underlying axiom system) for Fibonacci numbersDo you care to restate that Mr. Wizard?
You really don't understand, do you....
from YOUR link ("Binet's Formula for the nth Fibonacci number"):
the n-th Fibonacci number is the sum of the (n-1)th and the (n-2)th.
That's the definition of the numbers in the Fibonacci sequence. Nowhere is "phi" or "Sacred Geometry" including its "vibrational resonances" needed to define it! Fibonacci didn't use "phi" to define his sequence; the fact that it can be used to compute the nth term of the sequence is all very nice, but doesn't contradict a thing I wrote.
The fact that the Fibonacci numbers can be (and were) defined without "phi" means, by definition, that "phi" is not foundational or fundamental to the Fibonacci sequence.
You seem to be struggling very hard not to get this very simple point.
To: longshadow
Not to pop your bubble, but I do have other more important things to do right now, so let me just straighten you out on something. You have gone on record as saying that Phi is not important or relevant to modern Math and Physics.
Just take a peek at the following links as far as the Golden Ratio Phi is concerned.
Atomic Vortex Theorem of Energy Motion
Search for golden ratio in the following PDF file..
arXiv:physics/9810032 v1 16 Oct 1998
As far as the Fibonacci numbers, you seem to have a hard time grasping the fundamental concept of what they are. I'll get back to you soon on those, but for now, enjoy the links I've just provided...
To: longshadow
the n-th Fibonacci number is the sum of the (n-1)th and the (n-2)th. That's the definition of the numbers in the Fibonacci sequence.
No kidding. BUT, you appear to have failed to read the question, "Can we find a formula for F(n) which involves only n and does not need any other (earlier) Fibonacci values?"
The Binet formula which does in fact provide Fibonacci numbers without knowing any earlier value IS BASED UPON Phi, so you could safely and accurately say that it is DERIVED from Phi.
To: longshadow
Concerning your view that the Fibonacci numbers are just trivially related to Phi...
From Phi: That Golden Number
If we take ratios of the length we will see that the series of whirling rectangles will begin to estimate the Golden Ratio.
2/1 = 2 3/2= 1.5 5/3 = 1.666... 8/5 = 1.6 13/8 = 1.625 and so on.
Hence as we increase the number of squares we get a figure that begins to look more and more like the Golden Rectangle. It might also be noticed that there is something special about the sides of the squares. If we list them we have, 1, 2, 3, 5, 8, 13, ... This of course is the famous Fibonacci sequence. As will be shown in the rest of the essay, the Fibonacci sequence and the golden ratio are intertwined with each other.
To: longshadow; dennisw; rwfromkansas
Have you figured out where Fibonacci numbers come from yet? Do you still want to claim that Phi isn't a universal constant that is extremely important in modern science? Hello, are you still there?....
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