Is There a Relationship Between the Golden Ratio and Physics?

[kant]

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 a lot these occurence in math.

 

[bomba923]

( or any well know constant like the fibonacci number)

"the fibonacci number" ? Fibonacci numbers form an infinite sequence; there's no special constant among the infinite terms (? or is there??)

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 PHYSICS of plant growth

 

[Nenad]

Im firly sure the golden ratio is seen in the structure of galacies and spiral galaxies in the universe.

Regards,

Nenad

 

[Galileo]

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

 

[matt.o]

Im firly sure the golden ratio is seen in the structure of galacies and spiral galaxies in the universe.

Regards,

Nenad

reference?

there is another golden ratio called $$\pi$$ that occurs a lot in nature...

 

[Nenad]

reference?

there is another golden ratio called $$\pi$$ that occurs a lot in nature...

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

Regards,

Nenad

 

[AKG]

The Golden Ratio and the Fibonacci Sequence have a very interesting relationship. If we let φ(n) denote the n^th Fibonacci number, and φ denote the Golden Ratio then:

lim_{n → ∞}φ(n+1)/φ(n) = φ

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.

           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_{n → ∞}φ(n+1)/φ(n)
= lim_{n → ∞}[φ(n) + φ(n-1)]/φ(n) by definition of the Fibonacci Sequence
= lim_{n → ∞}1 + φ(n-1)/φ(n)
= 1 + lim_{n → ∞}φ(n-1)/φ(n)
= 1 + lim_{n → ∞}φ(n)/φ(n+1)
= 1 + lim_{n → ∞}[φ(n+1)/φ(n)]^-1
= 1 + [lim_{n → ∞}φ(n+1)/φ(n)]^-1
= 1 + 1/L

so L = 1 + 1/L

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

 

[Renge Ishyo]

You see a LOT of the golden ratio in advanced studies in art as well. Everything from classical vase design, to the dimensions of the movie screen you see at theaters, and even the arrangements of our facial features can be drawn in terms of golden rectangles that conform quite precisely to the divine proportion. There is a really great book on the Golden Ratio called "The Power Of Limits" floating around if you are interested (it explores both geometrically and visually the places where this "growth ratio" appears in nature and in art).

 

[liam1618]

8th January 2010:first experimental evidence of PHI in the Quantum world.
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.

 

[skeptic2]

It has always annoyed me a little that the 3-way light bulbs typically have the lowest level at 50 watts, the second at 100 watts and the third as the sum of the first two or 150 watts. Going from 50 watts to 100 watts is a much greater change in light level than going form 100 watts to 150. If they made the first two light levels at the golden ratio, the third to the second would be also.

 

[BenC]

Mate, check out kepler's 3rd law, i think it is, re harmony. The golden mean is all about the proportions as expressed in integers by n/ n-1 of the Fibonacci sequence.
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).

 

[charlylebeaugosse]

The Golden mean is associated to the destruction of KAM tori. (Kolmogorov, Arnold, Moser). Hence some relation to issues in dynamics. It is also attached to a intertesting type of recurrence for interval maps (the other case, KAM, being rather related to circle maps.
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.