Uses of the 'Golden Ratio'

Question

I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887.

Are there any actual mathematical uses that have been found of this number? What are its advantages? Just curiosity, really.

Answer 1

I'm not sure if you mean "man-made" as in real world applications or as in mathematical applications, but one of my favorite things about the golden ratio is that it is somehow the hardest number to approximate by rationals.

For one thing, it has the slowest converging simple continued fraction expansion. This is reflected in the fact that if you apply Euclid's algorithm to two successive Fibonacci numbers, you get a quotient of 1 at every step.

There is also Hurwitz's theorem, which says that for any irrational number $\xi$, there are infinitely rationals $\frac{n}{m}$ such that$$\left| \xi - \frac{n}{m} \right| < \frac{1}{\sqrt{5}m^2}.$$ The constant on the RHS can't be improved since, if we replace $\sqrt{5}$ with some larger number, then the statement of the theorem doesn't work when we let $\xi$ be the golden ratio. (Informally, we can't get too close, in a number theoretic way, to the golden ratio.)

Answer 2

This is one realm in which Wikipedia can be amazingly helpful. http://en.wikipedia.org/wiki/Golden_ratio lists all sorts of applications, with citations if you're picky (which is commendable).

Answer 3

One part that I can think of mathematically would be the use of golden ratio in fibonacci numbers.

The golden ratio is the part of series of well known fibonacci number series .

Here is the link since I am unable to write the actual series for the fibonacci here...... fibonacci

Answer 4

In physics (and maths...) Phi arises when we wish to maximize fluctuations in a system where an exponential random process P(x) = exp(-ax) is sensed by a device which responds exponentially to an input (e.g. out = 1-exp(-bx))

The ratio a/b=Phi maximizes the dynamics (in the stddev or variance sense) of the output. One example would be an exponential-law light detector sensing the intensity of a blinking star.

I found this some 5 years ago https://doi.org/10.1083/jcb.201506128 when trying to find the optimal (average) laser power in a optics perturbation method in cell biology. I was caught by surprise.

The reason I found this interesting was because it is an 'exponential-on-exponential' situation, not just some very particular experimental setting.

I would be grateful if anyone could point me to any reference where this fact is noticed. I was not successful in some basic googling attempts .