Some precisations about the "gold number"

PHI in electronics is still triggering a lot of people. Please find Rafaels precisations on the 'Gold Number'

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 X²-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

MORE BALLS PLEASE

Thanks for all your replies on the question of the Electronic Ball. There are more answers then I am able to publish on this blog. Below you will find an answer of Ron from New Zealand and Helmut from France. Ron uses LTSpice to get to the answer - Helmut the 'poor mans approach' Thanks guys!

wh

Hi Guys,

I have a solution to your resistor ball problem:

I flattened the ball, drew the schematic and loaded it into LTSpice circuit simulator, running on a laptop. I added a 1000 volt battery to the schematic and connected it to two diametrically opposite points. I ran the simulator in DC mode, and it produced a list of all the nodes with their voltages, and all the components (battery and resistors) with the currents through them. Knowing the applied voltage and the battery current, I calculated the resistance for question one.

Helmut from France wrote

Hi,

Here are some thoughts on the calculation of the equivalent resistance of two adjacent nodes, a subject that attracted less attention than the equivalent resistance of the two opposing nodes. Being more a practical kind of person I am not going to do a calculation that belongs in a theoretical treatise on electronics, instead I will try to find some poor man’s approach that will hopefully result in an acceptable approximation.

Some years ago a similar question was raised in Elektor about an infinite flat resistor grid and I was thinking that maybe it would be possible to reuse some of that. My starting point was (and still is) that the further a resistor is away from the two adjacent nodes, the lesser its contribution to the equivalent resistance will be. Intuitively, this value should be somewhere in between two limiting values, depending on the value 0 (short) or infinite (open) attributed to these far away resistors. So why not take the arithmetic average of the two? This approximation should get better when the number of jumps needed to get to the border nodes of the grid increases. These nodes will be given a value of open or shorted. The easiest case is the one with 0 jumps: short equals 0 ohm, open equals 1 Mohm, average equals 0.5 ohm. Averaging several approximate calculations of the equivalent resistance when the circuit gets more complicated gives this table (for the side of a pentagon):

Jumps	0	1	2
Short circuits	0.00	0.27 (3/11)	0.602
Open circuits	1.00	1.00	0.696
Average	0.50	0.64 (8/11)	0.649
Elektor measurement	0.65	0.65	0.65

With more jumps the calculation becomes complicated! But, with resistors of 2%, it seems that even with just one jump (still within reach of my mental calculation skills) we arrive at a good approximation.

Regards,

Helmut Müller