@golden spiral,
the spring of Creation

The complete solution of Golden Spirals

At 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 storm
A 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 = Φ ±P/n       R = φ ±P/n
  • ±P represents a Point, or Position or Process or Plot of the spiral, a real, integer or fractional, positive or negative number, ideally the de­gree, but of an infinite scale beyond ±360°, since as I said before it is the strong point of the formula: represents an angle in degrees, of the se­ries:
    for (var P = 1; P<= SpirLen; P += 1) converted by the calculator into module of 360 for the computation of < CODE class=baseMd>sin() and cos(), but with the absolute privilege of keeping the length reached by the radius for the number of rotations involved .
    In this operating context the π does not intervene, if only for the fact that in no point the radius is equal to another; and if it is present in formulas that define the golden spiral – except serve for the conversion of radians to de­grees – it is an inappropriate or unnecessary use.
    Naturally P, always positive, applied to Φ will render one spiral in con­trac­tion, and one in expansion at φ. In the first case it tends to the Start point, that single point, according to the Standard Cosmological Model, in which the Universe was so small as to remain unimaginably con­cen­trat­ed.
    Called a “Singularity,” it contained all the mass and energy that make up the known Universe, in inconceivable density. It would be the Big Bounce, which would have preceded the inflation of the Big Bang, the se­cond case in our showcase, vector of a boundless expansion.
    Just like at the 'extremes' of the spiral.
    At the minimum center of the PDF (max. zoom) a portion of the spiral add­ed in a thin line is visible using the first of the two partitions (Φ - its num­bers will be visible in the PDF), while the rest is the result of the se­cond (φ).
  • n represents the number of golden circles that, according to my no­ta­tion, a complete spire/revolution must cover, or if you want to cross; let's say its 'golden step', resulting from the form: 360/1, 360/2, 360/3, 360/4.
In the most classic case practiced, starting from a point of the spiral whose radius is R = 1, at 450° we will have the Radius:
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.
The source is available under the MIT license to anyone who wishes to use and/or reproduce it, provided that the source is cited in legible text:
https://pi-day.eye-of-revelation.org – © The Watch Publisher, 2022
I actually added the alpha option to the finished calculation board; to retry the first mode as well and be able to compare it, even by opening the same page in two separate windows and launching the procedures until viewing the two different lists (with the same starting range).
By clicking on the symbol, it turns red and activates the summation mode, up to the control of the filtered or total list; after which it is disabled, until fur­ther request.
If the option is activated, at the [full#] command the measurement of the ra­di­us at the 1st degree in the list is naturally equivalent to the increase con­stant suitable for that type of coil.

Let's examine in detail this distinction of types of golden spiral, one more har­mo­ni­ous and expressive than the other, a distinction raised since the first dis­cov­ery and disclosure of the four circles delimiting the large triangle (year 2002), with diameters scaled φ expanding, or Φ contracting, (see also pag­es 9 and 13 of the Treaty) which, concentric, highlighted certain char­ac­ter­is­tics of the golden evolution, especially in the architecture of the Śrī\ Chak­ra yantra, al­ready mentioned (and downloadable); so:
  • the most compact, whose coils intersect at zero degree each successive golden circle every 360°, amplifying in circular length in the ratio φ.
    The angle of intersections with the background golden circles can vary ac­cord­ing to the radius and scaling of the background, but always re­sponds in step.

    Golden spiral at the step
    of one golden circle
The figure gives an ex­am­ple of the gold­en in­cre­ment ap­plied to each full 360° ro­ta­tion, which is e­quiv­a­lent to ex­pand­ing the cir­cum­fer­ence from the ra­di­us of one con­cen­tric cir­cle to the next (or pre­vi­ous) one stand­ing in the gold­en pro­por­tion.
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 of4α.

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 - 1st trial
the most compact golden spiral is
for the slowest land invertebrate
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.

  • The second, intermediate, whose coils are amplified in circular length in the ratio φ at every 180°, thus jumping from a golden circle to intersect each 2nd successive circle in golden ratio at the same starting degree.
Golden spiral at the step
of two golden circles
It was also obtainable with a lin­e­ar increase of the radius, mul­ti­plied at each de­gree by the a­bove con­stant, in this case re­duced to 2α 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.

  • The n°3. (fourth in table order), whose coils are amplified by circular length in the ratio φ at every 120°, thus jumping from a circle to in­ter­sect each 3rd successive circle in the golden ratio at the same degree, it does not present particular outstanding motifs.
    The same can be observed with the 270° divider [5.*], where the spiral reaches the next 4th golden circle with three revolutions. Of this I leave the vision to the interactive test.
Golden spiral at the step
of three golden circles
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 (4α)3.
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.
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 - 2nd trial
Fossil with golden spiral, with
φ rhythm of 3 golden circles/4
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 6th, 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.

the Golden Spiral par excellence

the Divine Proportion in the golden spiral