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
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 SPAN >, 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.
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.
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.
*) 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.