WH

Salutations!

First I have to note that Phi is not the "Golden ratio", as you stated in your article. Exactly it is the "Gold-number", or "Number of Phydias", as it is present in the architecture of the Parthenon, in Athens.

Well, exactly "Phi" is the limit in the ratio of two consecutive numbers in the series of Fibonacci. It is a perfect "irrational" number, and as for "Pi" or "e", it only can be defined by a relation in a series of numbers. In this case it also represents the solution for the "Aurea proportio" in the division of a segment. It seems you think that Phi was firstly defined by Pacioli. In fact it was discussed by Euclid in his "Elements", as you can read in the following articles.

http://en.wikipedia.org/wiki/Euclid

http://www.pauloporta.com/Fotografia/Artigos/epropaurea1.htm

When you consider the ratio between three segments which satisfy that L=a+b, while L/a=a/b, the calculation brings us the formula X

^{2}-x-1=0, whose solution is x=(1+5

^{1/2})/2; but also x=(1-5

^{1/2})/2. As the second solution is not a real number, then we use the first for the operation in the physical world, and we call it "The Golden number".

Well, the fact is that the Fibonacci series is present in a lot of events, in the nature. It is the factor for the growth of a specimen, for the "Harmonische" (tempered) scale of music, for the aesthetic look of a sculpture... But you also find it in our own body. It is the ratio between the facial segments, between the length of the members... Considering this, and as electronics is only a matter of nature, it is normal than you'll find "Phi" in several effects. For example, this is the ratio in the growth of a Larsen signal, or the dumping factor in a natural attenuation. Actually, whenever you have any event where several (more than three) values of the same magnitude vary in the time, or in a ratio of parameters, in a re-entrant manner, you'll meet the Fibonacci series, hence, the Golden number in the progression.

As you can check, the value is not steady till you consider larger numbers in the series. For instance, as the series is caused by he addition of the precedent pair, the ratio between 2 and 1, 3 and 2, 5 and 3, 8 and 5, is far away from Phi. This is the cause that normal observations don't consider the fact, as the values are limited to a few observations, and the Maxwell theory provides an exact answer to the continuous environment (in the linear equations domain). But really the matter is that Phi and the Fractal geometry have a lot in common; thus, the "Golden ratio" is a simple approximation to the phenomena of "fractal growth".

Then don't ask for "applications" where "phi" is of use; better if you ask for the contrary, which phenomena don't fit the theory. Or if you like, ask for examples of how it is observable in the electronic field.

For instance, whenever you have to compute the impedance values for an attenuator, and you need match commercial values, sure than Fibonacci will come in your help.

Best regards

R. de Miguel