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Inherited from the Egyptian school and introduced to Mediterranean geometry by the Pythagoreans, the illustrious mathematician could not fail to know the golden section.
So any claim to deploy `(√5±1)/2` starting from the Fibonacci series is only a senseless forcing. The calculation algorithm is simple: given `f1=0` and `f2=1`, it is repeated at will `f3=f1+f2: f1=f2: f2=f3`, and thence print `f3`, with the only issue that from a standard calculation system only the first decimal digits will be ob­tained, which could make a constant ratio to appear in the Fibonacci sequence, whilst due to the integers this is not so even at higher digits, which, more­o­ver, are the least suitable as parameters of specific phenomena.
That said, the Fibonacci modality does not boast definitive relevance to bi­o­log­i­cal or physical formations, due to the golden proportion, of which it par­tic­i­pates with a defect and asynchronously, given its initial pairs 1, 1 and then 2, 3, 5, 8, 13, far enough away from the φ succession to compromise its general adaptation right from the root. It must be said even if for some cas­es of natural correspondence, among the numbers that populate the se­quence it is certainly not difficult to associate some of them with de­vel­opments of the vital structures (*.
In any case, it would not be technically founded to build a geometric 'spi­ral' on an ungainly base – with circular arcs whose radius is an integer con­stant that cannot be recalculated – so no spiral pattern; it is even less so if it starts from two equal squares, and then associates it tout court with the 'gold­en' spiral, which has no beginning or end, simulating its graph­ic meth­od; since in any case they can not collimate.
It may be strikingly representative, but it's a sad graphic trick.

Basically, it is good to take into account some fundamental aspects.
1. First of all, if there is a graphic function that cannot be the object of rul­er and com­pass, it is the spiral, for the simple fact that each of its points, or de­grees, deviates more and more from the center, in contrast to the op­er­a­tion of a compass.
2. Secondly, said spiral cannot originate from, or simulate between sides of squares or rectangles; the simple passages widely demonstrative, draw noth­ing but quarter circles, non-spiral arcs of circumference and suit­a­ble yes and no to elementary schools, certainly not for academic vanity.

To be honest, I remember having accused the flaw ever since it was first taught to me in school. I came to terms with it, but I still I won­der which ones.
 fig. a – the naked and raw look of the curve de­nounces a sort of rec­tan­gu­lar squashing, a serious de­for­ma­tion that will per­sist with each volute boxed in this cage.
Nothing that can praise its functionality, al­though the eye does not easily grasp the dis­con­tin­u­ous hump between the end of one and the beginning of the next, seeing the mind what it wants to see and, in this case, sup­ported by the scaffolding of squares and rec­tan­gles : nor will it be possible to find a suit­a­ble sam­ple of them in the growth or de­vel­op­ment pro­cess­es in biology or botany, despite attempts at superimposition, often traced even if they do not correspond.

*) By the way, the so much-repeated profile of the Nautilus shell does not reflect the gold­en spi­ral (even if it is sometimes graphically resized ad hoc), but only a log­a­rith­mic, visibly less wide.