Nautilus and the Golden Number

The Golden Ratio exists, but does it really appears in natural patterns as often as we think ?

Nautilus and the Golden Ratio / Le nautile et le nombre d'or - Drawing by Okhaen

It is the proportion by which the ratio between two parts is equal to the ratio between the greater of these parts and the whole. Thus 1,618039887 ... and an infinite number of decimals. It is linked to the Fibonacci sequence, which is made up of whole numbers corresponding to many natural growth patterns and which tends towards the golden ratio.
But, notwithstanding the frequent assertion to the contrary, the shell of the nautilus is not a spiral of gold. In most books dedicated to the subject, and even more so on the Internet, there is information that the logarithmic spiral of Nautilus pompilius is developed according to the divine proportion.
The mathematician Marius Cleyet-Michaud explains: "The golden ratio has its devotees, and this devotion can manifest itself in very diverse forms, from the ecstatic contemplation to the fanaticism and even the violence. This provide serious difficulties for those who wish to initiate themselves sincerely, without prejudice, to the phenomenon "Golden Ratio". Most of the books written on this subject mix real mathematical facts and assertions that it is not known whether they are mere hypotheses. " in Que-Sais-Je, Le nombre d'or, PUF, 2009