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Are there any connect between the golden radio( or any well know constant like the fibonacci number) and physics? I ask this question because there are alot these occurence in math.

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"the fibonacci number" ? Fibonacci numbers form an infinite sequence; there's no special constant among the infinite terms (? or is there??)kant said:( or any well know constant like the fibonacci number)

Anyway, Fibonacci numbers can describe the growth of plants and trees (their heights...I believe, see http://www.cs.rit.edu/~pga/Fibo/fact_sheet.html).

They are important within the

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Regards,

Nenad

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Galileo

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I've also heard it's related to the patterns (positions) of flower petals, or fir cones. I think it's called Fibonacci phyllotaxis. Here's a good site:

http://www.swintons.net/jonathan/Turing/fibonacci.htm [Broken]

http://www.swintons.net/jonathan/Turing/fibonacci.htm [Broken]

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Nenad said:

Regards,

Nenad

reference?

there is another golden ratio called [tex]\pi[/tex] that occurs a lot in nature...

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matt.o said:reference?

there is another golden ratio called [tex]\pi[/tex] that occurs a lot in nature...

http://www.space.com/scienceastronomy/perfect_spirals_030917.html

Regards,

Nenad

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AKG

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lim

We can see this since φ is the ratio of the long side to the short side of a rectangle which, if you take out the largest possible square from it, you're left with a rectangle similar to the original.

Code:

```
a + b
----------------------
| | |
| | |
| |a |
| | |
| a | b |
----------------------
```

So a/b = (a+b)/a. Since we're looking for the ratio a/b, we may as well let b = 1 and then a is the number we're looking for:

a = (a + 1)/a

a² - a - 1 = 0

The quadratic equation gives a positive and a negative solution, and clearly we want the positive solution, which we will denote as φ.

On the other hand, if we look at the Fibonacci Sequence, letting L denote the limit we are interested in, we note that:

L = lim

= lim

= lim

= 1 + lim

= 1 + lim

= 1 + lim

= 1 + [lim

= 1 + 1/L

so L = 1 + 1/L

note that this equation is precisely the same equation that defines φ, i.e. L = φ.

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Ratio of the two lowest frequencies of magnetic spin waves found to be very close to 1.618. Predicted to be exactly PHI because of E8 symmetry.

Also the the mass ratio of the two lightest particles in the E8XE8 superstring is exactly PHI

but no observational example.

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Kepler was a warlock who believed in odd creatures that live on the moon and spouted the theory of universal gravitation before newton, but he did develop. A cracker of a theory re the music of the planets, helping to ensure the success of the copernican model. Ask why Venus traces a pentagram in the sky, and the phi ratio in the relative distances of celestial bodies (you need to accept nasa's distance measurements).

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The first rigorous results for KAM -golden mean were by Mather, but some physicists saw that happening using simulation. On the interval, it started with Milnor and Lyubish.

Now it is studied for dissipative embeddings of the disk that are perturbation of interval maps by Lyubish and Martens. The connection with physiscs is obvious but not trivial (the math cited are quite delicate). Plants and in particular flower stories are serious as other problems of morphogenesis. Some other examples are more fantasies: the precise lists of good and mad eludes me. Work is still progressing: see work by Lefever.

The golden mean is the limit of the ratios of consecutive Fibonacci numbers. It is an irrational number and more precisely a Diophantine number, with a particularly simple expression as a continued fraction.

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