geometric spirals are transcendent curves that are distinguished by the law that links the angle of rotation around the pole to the distance of the points of the curve from the pole itself. In particular:
s. of Archimedes, in which this distance is proportional to the angle of rotation …

fig. b – spiral geometrically veritable, calculated with a single center; even if repeated tests have convinced me that this is not the only case of the golden spiral, as it could have a probably as­ymp­tot­ic pole (not so coarse, however). This doesn't matter anymore; there is much more since 2023.
3. Finally and above all, the demonstrative curves inscribed in pieces within each square of the golden rectangle can only ex­haust the passage from one scale value to the next at a low level, and not only be­cause the departure and arrival radius is unchanged [already in itself a con­tra­dic­tion], but because the spiral must have its own center, or pole which, even if it is not stationary (logarithmic and hy­per­bol­ic spi­rals), certainly can­not dis­mem­ber the curve with centers at the fixed vertices of the scaled squares [fig. a];

I do not set myself to address this issue but, given the extreme ostentatious critical for the nth-th accuracy of the π, I wonder why no mathematician lifting a finger to call up a display adjustment. So much laxity in the face of so much ri­gor? It is worth the interlude to reiterate the absolute and ir­re­place­a­ble im­por­tance of the Golden Section (of which in the past I happened to read at Wiki­pedia[it] that it is just a number like any other!). 1-1-2 cannot represent Φ-1-φ!

- new insert 2023
Well, I couldn't overlook it! just hereafter I investigated at its root the correct way of drawing a spiral, resolved with a well-explained algorithm, then disclosed at least two ‘secrets’ pertaining to its geometric as well as creating power.
Now you can draw the exact spiral yourself, HERE or in your own website (free Ja­vaS­cript), with a choice of golden ratios over 5, and something more you never supected existed.…

- end 2023

In addition to the approximation of those arcs in the presumed spiral de­vel­op­ment, it would be good to try to explain why the so-called golden spiral curves thus ob­tained are at each turn more and more divergent and not su­per­im­posable for extremes if obtained from the golden rectangle, or from the golden triangle got from the star pentagon ...

Although as a circular structure it cannot be discretized by a scaffolding of rec­tan­gles and squares – except to challenge a problem even worse than squar­ing the circle – the attention remains firm on this specific development, as a pattern to panes, since they introduce the concept of a principle that reg­u­lates the pro­gres­sion of the Φ, implicitly attributable to the circular wind­ing quar­ters, as phas­es of a wavelength; we will see later the quadrangular ro­ta­tion scheme also recur in the quadrants of the circle, albeit in an un­dif­fer­en­ti­at­ed way.

Actually, the simulated spiral does nothing but reproduce the sequence of gold­en circles outlined here!
In that essay I worked out their series up to 8 in decreasing golden pro­gres­sion, then distributing them along the vertical axis up to the apex of the pyr­a­mid (which presumably would have housed a mysterious sphere with the size of the 8th), maintaining its tangentiality with the sides of the triangular profile, which revealed all the extraordinary tuning, with each center self­-dis­posed ex­act­ly on the pre­vi­ous [major] 2nd circumference as well as its tan­gen­tiality to the next 3rd; allowing us to hypothesize a set of vibratory fields with precise res­o­nance inside the struc­ture; which about seven years later seems to have found some confirmation at least from thermal scanning, thanks to the in­stal­la­tion of scientific detection in­stru­ments (muon detectors).

From this page a schematic view and a PDF excerpt from the cited essay.