R =Φ
^{±P/n} R =φ
^{±P/n}
the spring of Creation The complete solution of Golden SpiralsAt the end of this non-exiguous research work, made up of in-depth study, improvisation and more or less exciting surprises, convinced that I had exhausted every useful or necessary argument, I was destined to discover precisely by virtue of the insistent checks, that I could have directly used the formula adopted for accuracy comparisons, as the only actual realization. the perfect stormA storm due to my distance from the school grounds, made up of tensions protracted day after day, which perhaps transpires from these pages, due to the insecurity deriving from the vortex in which I had trapped myself with such ease, has transformed inspiration as in a flash of bubbly that suddenly recompose in the calm of the resolution.This is how I define the unique parametric formula, the simplest and most efficient – which some authors announced rather complicated than that of polar coordinates – without limits of integer or fractional degree, nor of approximations other than those of calculators. In short, a flat equation, extremely concise and immediately applicable, which robs my mind of the courage to wonder why I never thought of it before: R =
R = φ^{±450/90} = 11, 090169943749474234011639053357
at 451° we will have the Radius: 11.149625650030747658140248563255 e così via. No special constant is needed, being able to modulate with 360°, 180°, 120°, 90° etc. whatever step and degree you want; and the result will be what it must be to trace each golden spiral exactly. To obtain the Cartesian coordinates of the intersection points with the axes it is obviously enough to apply the desired angle to the formulas, and the radius will correspond to x (with sin() = 1) or y (with cos() = 1). Such essential data can be read out with the [quad#] button for each curve type. Is this perhaps the alternative? or this, in its entirety? among the most reliable.
I'm not deleting anything of the first way from the script, even if only as [my] memory of the effort made, since having been programmed with a precise intention, in addition to documenting an elementary and non-academic logical-intuitive path, JavaScript is very versatile in offering cross-checks, and could be useful for those who want to conveniently reproduce real golden spirals in an operating environment that does not have advanced computing functions.
It is the complete ~volute circumference at 360° in fact that in this mode involves a total increase φ at each turn, according to the same diameters of the concentric golden circles. In the background, partial profiles of the concentric golden circles appear in the dashed line and, only with the extended zoom from the PDF, can we read some values of the radii at the angular intersections of the x and y axes (perhaps the browser zoom will not be enough), as well as the grid of the rectangles. Since the increment constant adopted here, for the most general consensus, applied directly to the 3rd type mentioned (as 4. at the console; φ as we shall see), for this script it is subject to the fixed reduction of ^{4}√α .
It is a spiral conformation, like the following one with a double step, perhaps easily recognizable in certain physical-chemical or biological processes, although it does not replace the expected golden spiral. Posthumous insert 2023 - 1^{st} trial
At the last stages of this research – intended to demonstrate the complete independence of the golden spiral from the one commonly defined as logarithmic, as well as of the natural golden ratio from Nepero's numerical artifice – I will have implemented an interactive dashboard, first to be able to experiment with the golden spiral models defined here, with various stages of equipment, and subsequently a more advanced one, which would allow for uploading images on which to experiment with each type of spiral proposed.
Once the work has been completed and tested, this is the right showcase for some results that prove the validity of the statements.
^{2}√α to double the pace of the first.
It is a mediation between Φ and φ, which reflects on each horizon, or axis, a golden ratio balanced in alternating symmetry between the various points of intersection: 0.618 · 1 · 1.618 and after 180°: 1.618 · 1 · 0.618. In the PDF it is easy to compare at the beginning of the spiral the mirroring positions of the modules φ with the previous ones.
It was also the result of a linear increase in the radius, which from 1 will be φ at 120°, multiplied at each degree by a constant in this case a
I don't know how significant it could be, since the priority ratio is 1=>2=>4 doubling, not 1, 2, 3, 4; but it is worth highlighting the flexibility of the formula and perhaps of the research.
(^{4}√ .It can be considered a proportion worthy of carrying over, integrating the golden series which evidently is not limited to a single expression of φ, however supported for its more immediate discovery. Posthumous insert 2023 - 2^{nd} trial
The 3. and the 6., thus marked in button order, seemed to have no peculiarity; but articulating on multiples of φ, we might as well make them visible in order to be able to compare them in nature.
But from a search among the alternative photos to the many Nautiluses, and immediately available, here is a confirmation of the 6^{th}, older than our geometry, which can indeed replace the graph.
Here again is a snail with a compatible spiral, but which extends on the outer side probably due to the three-dimensional perspective which hides the greater development in depth of the greater volute, which is less and less visible in the photo, indeed it is almost squashed.
What an admirable performance this guy! again φ at each 3^{rd} golden circle with four revolutions.
Its conformation, initially taken as secondary or accessory, turns out to be one of the most recurrent, at least among some animal forms, while the vegetal ones tend to expand much more widely, probably in accordance with a more rapid and extensive development.
the Golden Spiral par excellencethe Divine Proportion in the golden spiral |