Before re-proposing the most celebrated and significant one in a new light, that spiral whose expansion/contraction ratio refers to the φ for each of the four phases of the circle, and then by φ^{4} every 360°…

I had even published a broader one, produced for the sole purpose of a comparison with the deformations of the alleged Fibonacci, with a first intuitive constant increase of the radius which, given a step of φ_{/360°}, multiplies itself at each degree by that step. A little rudimentary, but not too far from the more widespread images of several galaxies, and recently discovered stars 'drawing spirals' (NASA/ESA). (see pages 10~11 and 26 of the Treaty)

Having reached this phase of my exposition which involved the spiral, hoping to be able to suggest a valid and practical replacement for the touted petty solutions, I thought of providing a prototype that would complete those first observations along the lines of what was briefly introduced in the treatise on the π; but, despite having insistently searched the Web and attempted indications for an accessible formula, which would allow me to reproduce the spiral parametrized on a system of Cartesian axes in PostScript which, like most of the readers, is not familiar with polar coordinates, I have not found that disparate notations from elementary to complex, little or not clear enough and for me immediately recyclable, with laborious conversions, perhaps not always tested and worthy (it is rare that the mathematician and webmaster are the same person, or that the competence is bijective).
And on the sidelines, rivers of code by code to display the output of a homemade spiral.

To conclude tout court for the purpose of illustration, I first resolved to devise a new algorithm, which would allow the dynamics of the golden section and spiral to be exalted in a more truthful way, which without going through the formulas of logarithmic sequences, provided for a continuous path, based as previously on a linear increase of the radius multiplied step by step by a constant up to the desired expansion, making everything clear and applicable even by laymen, given that everyone makes school of it.
The initial formulation was conceptually and from the executive side the most linear, repeating from a starting point x,y [moveto] up to the established term and without interludes the following three instructions:

to then arrive, ultimately, even more directly to:

R =α^{D}linetosin(D) ×R cos(D) ×R

A not entirely orthodox development method, which allows, once found the constant, to appropriately institute a sort of new definition, such as that of "spiral of 360°, 720° or 1999°..." destined, as will emerge at the end of the stroke, to result in a valuable aid for a technically ideal solution.
But let's follow a first maturation of it for now.
While aware of inevitable compromises, such as the accumulation of approximations, I aimed in the first and isolated instance - I repeat - to override the most widespread erroneous figurations, and indeed to bring attention back to the golden aspects of new evidence, after having detected/revealed the four proportional circles at the base of my study on the great golden triangle; and we will see how this can be integrated into theory and practice.

Let's redefine Phi Φ on our page: Φ × (1 + Φ) = 1
It should be acknowledged that the term ‘section’ defines the specific part of a unit or unitary structure, which in this case operates according to the proportion, or 'ratio' indicated by the digit itself.

Φ .618

1

1

φ 1.618

Therefore it is not 1.618: φthe definition of the Golden Section: nor should it be considered as such, but only as 1 added to Φ, thence in itself not a primary argument but a derivative, say as cause: Φ and effect: 1 / Φ; it is an application of the golden section, and the opposite is not true, even if it will subsequently assume highlighted importance.
In fact Φ it would be suited in any direction, multiplied or divided, a priority that will become exclusive when the 7.8615… (its square root, whose square root is the side of the square with the same area as the circle) will somehow be identified with the actual π.
Interestingly, similarly, 1.618 squared: φ²= 2.618 and Φ² + φ²= 3.
Dulcis in fundo, Φ × 2 +1 that is Φ + φ = √5, the key to an upcoming interesting application.
Under the metaphysical profile, its symbolic transcendence seems to indicate the process of continuous expansion deriving from its sub-division, where multiplying its potential leads back to the Origin, the Source point from which existence proceeds.
Like the π, it is like a perennial tip of the balance, active or latent in any passage, from the infinitely microscopic to the entire universe, guaranteeing the balance and connexion of any development and the compatibility between the parts and the vastest conglomerates and distant; therefore it is called Divine Proportion; and each of its spiral embodiments represents the advanced and vital expression of this mathematical soul, if such can be defined as the module of a calculation system that is not decimal, nor binary, nor of any definable cipher, if not of a completely transcendent measure, celestial unit of measure, inherent and emanating from the Creative Unity.

the Developments of the Golden Spiral

To begin with, if the framework, which I will also call cage of squares and consequent rectangles is generally represented with the sides tangent to the spiral, or with intersections at the corner quarters, it is only due to the deviation occurred from the construction progressive of the golden rectangles, of strong inspiring value, then by the consequent [ab]use of the arcs of circles; not being able to touch the curve starting from its only center with radius eg. equal to 1, as in our figure, but only with a special radius or scale, keeping by default the line of the initial radius parallel to the x axis.
The gap becomes apparent when you thoroughly examine (ie with a good reader) the PDF diagram.
So in this case the spiral will be drawn in a corresponding scale of 1:7,663.
In spite of the critique of the use of the compass on the squares, I will expose on this page and for the first time two recondite aspects of the golden spiral, which will allow a new approach: the first computer-mathematical, the second geometrically revelatory, self-exalting by itself just this starting frame.

my first formulation: algebraic-graphic

From a pragmatic point of view, therefore, it was a question of defining that constant which fed the progressive expansion of the curve, from 1 to φ^{4} according to a pre-set step, giving rise to an algorithm which easily replaced, with the same computer tools that basically everyone uses, the use of straightedge and compass.
Chose the step of 1 degree and, with some forwarded research I arrived at the constant a= 1,00536111768685835835952555755714 which increases the radius, multiplying it iteratively in the form:
R *= a, to generate the desired golden spiral sequence in its development step by step, not only with precise evolution φ to every quarter of a circle, but also in its appropriate variants, as can be verified from the following graphs which I have made interactive on purpose; of which those who are practical enough can easily read the internal javascript code accessible under the MIT license.
I don't know how much precision the various development environments will be able to use of it.
The best value achieved by javascript should exceed 10 decimals, where the usual software begins to withdraw (but in astrophysics they seem to settle for 8).
In fact, this parameter produces a sequence rounded by the interpreter in terms of which I have reported a brief couple of points indicative, to an accumulation of 1080° also visible in the PDF:

What matters, the constant offers fairly reliable results with the same algorithm for all tested cases; JavaScript uses 16 decimal digits, unlike EUPHORIA [C] which limits itself to 9 and maintains precision at 8 digits - while PostScript stops much earlier.
Nor do I find it necessary to venture into the design adopted for the Śrī Chakra yantra, to solve which it was essential for me to program an indeterminate number of decimals, established by the user of the program; but in that case the perfection in exactly intersecting 9 triangles for a pattern definitive, and never resolved before, was a must…
However, the optimal accuracy for the executable script appears to be around 1.005359, for which the curves both in summation and exponential mode overlap without appreciable difference, while a minimum would be visible with the complete constant.
That minimum gap makes me think again about a fixed defect, implicit in the digital increase of the radius, not governed by the π… but this is a story already discussed on the π.
It is more than sufficient (and never requested or validated in the available demos, which do not exceed 3 decimals of units), superior to any need for current performance; not to mention that 5 decimals on a 3 or 4 digit integer remain valid, despite having to deal with repeated rounding in summation by multiplication, so PostScript and C eg: can produce slightly different results at microscopic levels; particularly in extending / contracting the diagrams with extreme zooms (up to 6400% on a 6.900 × 4.800 pt page), which challenge the calculation and resizing of micro- and macro-curves.
The constant, even with narrow gauge, maintains ample precision even for spirals that can be tested here and higher than 5700°; but to dispel any executive doubts about accumulation, I figured I could reset the radius value to the power of φ according to the desired interval, and continue counting the degrees without affecting the graph… and this paradox indirectly led me to the main road.
A lot of effort in fact, to realize that the golden spiral is summed up to an exponential function, eg with a base:^{1080}√φ^{12} that is:^{(90×12)}√φ^{12}.
with a result from Windows® calculator ^{[scientific]} actually greater than
0,000000000004…
[0,00000000444…only by power of 12, at 1080°, obvious software limitation]
compared to what I had calculated with my methods and tools.
In fact, this attests to the inevitable imperfection of even scientific calculations and up to 32 decimals, which cannot replace the actual Φ, and echoes the same that occurs with the repeated use of formulas containing a π artificial.
With my own JavaScripts I offer the way to check their compliance.
I wanted to call it α 'Alpha', like the first harmonic stone of the immense omnipresent building; but it's still a stone that might not hold a skyscraper.

from theory to practice

A graphical calculation procedure that has to deal with pixels, from tiny to very large strokes on the same screen, cannot fail to make compromises. We are already full of figurations with thick and hard to decipher curves, technically unreliable or useless; for my part I have tried to keep all the details clearly visible, so that getting an idea of any defects can lead to certain results in the development of appropriate concepts.
In this specific case of the spiral, as I said, I did not intend to reproduce formulas that were already known, but to venture into a conceptual study that could dismantle superficial disclosures and validate, perhaps rediscover and better understand, the intrinsic characteristics of this celestial development. In the end with more success than I supposed, but in the meantime I preferred to take the risk, and while his proposition - overcome on this same page - might not be of interest, if not for the purpose I set for myself, the graphical result on the Cartesian axes speaks for itself.
So here it is operational on this page, in its various conformations.
With the center in x=0.4472135954999579 y=0.7236067977499789, coordinates in the golden ratio between them and the translation that identifies with respect to the zero point of the Cartesian system (a white cross), the center of the spiral (a red circle at maximum zoom), from which the radius starts (see pdf diagram); that point also underlined by the writer and essayist Clifford A. Pickover as the "eye of God", as an infinitely unreachable center from the spiral of converging rectangles.
Each spiral represents an open process, with indeterminate extremes, which can only be referred to from afar as a circumference, since it does not enclose any area.
While attracting the most attention the one inspired by the succession of golden rectangles in scale, with pivots at the vertices of the squares from which derive, therefore with a φ² step, the multiple possibilities of development of the golden spiral, i.e. the ratio of its winding/ unfolding in the golden proportion, refer to the quaternary also transversally, that is with at least three other fundamental modalities, deriving from the subdivision of the 360° circle by 1, 2, 3, 4, and we will also see 5.
Mainly from the achievement of a vector radius, increasing or decreasing, of a successive ratio { 1_{× 2} = 2_{× 2} = 4 }, by φ or Φ with its starting measure, and you can experiment them directly on this special graphic board:

JavaScript – Drawing Golden Spirals

1. the one for which this value is reached after a complete rotation 2. the one for which this value is reached after half a rotation 4. the one for which this value is reached after a quarter of a rotation

to these primaries we will add with good reason:

3. the one for which this value is reached with a third of a rotation 5. absolutely the one that reaches the φ at one-fifth of a rotation:

the polygonal structure that best represents the golden section.

6. and not? the one for which it is reached each three quarters of turn
7. the one for which it is reached each two fifths of turn
8. with this the φ is achieved at every
multiple of 360°.

Click the buttons to plot them; below you can vary the following parameters:

Hold down the left mouse click on the image to enlarge it.
The [console] button will open an external [tab] screen of the graphics console, much larger and independent.

For the last activated mode you can also view the list of radius and radian values for every degree° [full#] (will become $), or filter them every occurrence of the modular sector [module#]; JavaScript will replace this page with the output of the list, and to return to reading you will have to refresh, or use the arrow or the "Back" command, depending on the browser.
Note that the total computation for the defined spiral length can exceed the graphical output limit of the frame, providing data for any list, as long as the base software permits (excessive lengths can be ignored).
At the same time, [module#] (will be #) will respond only in the presence of integer intervals corresponding to the latest model launched.
The [±] button will reverse the motion of the spiral for the given radius: towards the center if the length number is positive, and with the same rotational direction, eventually completing the same spiral already traced expanding (if) with a large starting radius.
Even if the visibility of the spiral should be practically zero, like a tiny colored circle in the center, calculations and tables will be performed.
Introducing instead a negative length number you will also obtain an inverse tracking, but in this case you will be asked to introduce a starting radius suitable for visibility, with a length that can even be greater than the window, and the rotation will be the opposite starting from the same point of start. Note that the two commands together compensate each other.
To enter fractional numbers, use a comma, not a point.
The 8 option will be explained later.
If it is not active, the increment option is available Δ, which allows you to enter a parametric value, even decimal, as the unit for jumping the track from one degree to another, and can be, for example: 0.5° or 24° or 48°…, obtaining curves made up of more or less visible segments, with the relative calculation lists. In the final version, with a negative value, a dashed curve with alternating segments will be obtained
The intersection angle will depend on the initial radius, which implicitly performs a zoom function, both graphic and numerical; it goes without saying that in any case the radius can never be equal to zero, equivalent to the metaphysical Origin! Lengthening the radius graphically extends the spiral even a lot, so the Len value could invisibly override the tolerance set by the javascript for this page, and produce no output; parameters may need to be scaled.
If the Len field is empty or 0, when a spiral is launched it will assume the default value, introduced to be included in the diagram, purely indicative but useful for evaluating proportionate variations, that don't block the script. But be careful, since if it is not reset, each spiral adjusts itself with the parameter it finds, so if, for example: the ^{1}/_{5} has been launched leaving length 935, then launching the ^{3}/_{4} with a starting radius even just higher than 5, you would not see anything. Testing all spiral buttons, with Rad=1 and Len=1000 everything will be clearer.
In case of comparisons with some biological species, to avoid deformations of the theoretical layout inherent in the early stages of growth, as we will see later, it will be advisable to start the spiral with a fairly consistent radius, for example: 20 or 30, will result in a more realistic rendering. The [±] button, by clicking the same option again, will allow you to view the skipped part.
From the first successive tests it will in fact be evident that in the spiral evolution of any organism the physiological barycentre may deviate from its virtual starting point, to depend more and more on the overall conformation, to which we will have to refer more than to the actual center of the geometric function.

In order not to leave anything to chance, following some philosophical reflections sketched at the end of the run, I wanted to introduce a custom option @, available for any test bank you can have in mind: it allows you to draw spirals with all sorts of steps, reducing or increasing the step using the 360° multiplier parameter (the comma reminds you that it accepts decimal fractions), which establishes the number of revolutions necessary for the spiral to intersect each successive golden circle, i.e. at the radius to resize in the ratio of Φ.
In fact, this new mode can simulate all the previous ones, from which we start and as such remain to be highlighted; for example: the multiple 0.5 would repeat option 2 for 180°; and while 0.25 reproduces 4, 0.3333, resulting in 119.988° will very closely simulate option 3 of 120.00°. We will only see at the end the probably most important one, for 144.00°.

A particular example is provided by this beautiful shell that populates many seas, from whose center more than one spiral seems to start, of which at least two have different parameters, one bearing along the container edge (classic red) with φ at every 90° ie 0,25, and at least one contained internal (green) with steps φ customized at 104.4° with 0.29.

This gave me the idea of assigning it with a color of your choice among those of the other seven: it will adopt the color of the last one drawn (even if only to set the color, and then cancelled).
It may prove convenient with certain background images, what we'll meet in the third stage of the project's development. For a parallel reason, it will also maintain the thickness of the last trace, to avoid thickening by clicking several times to try new parameters; thence for the first draw only.
Not enough, further tests suggest being able to vary the color on the basis of the background images; I have therefore arranged that the color of the customized curve is taken from the #HEX field (already used for the blackboard background) and take precedence here as well, if a valid one exists.
With this combination, clicking ie the 1^{1} button three times triples the thickness of the curve; to transfer this to @. it must be deleted with [×] and then click @. and its curve will keep the thickness reached.
A final refinement allows, by varying the color in the #HEX field, to superimpose a new curve on the thick one, without rotation and without changing the parameters, which will no longer have added thickness, so that it will appear framed inside it.
In any case, the color used will be transcribed in that field in order to be easily changed.
Thus any type of curve can be manipulated and reproduced at will, and with the desired color.
In my opinion, a no less interesting research will emerge in the contraction of the motion, although everything is limited in both directions by the graphic detail, or by the size that the system in use can support, until the track is reduced to a black dot at center (that can escape).
NB: a possible non-response, ie no output, does not mean that the script does not work, but that the entered parameters are inadequate, too large or small to be made visible or applicable to the space provided; which cannot exclude spreadsheets; checking them may explain.
For example: R:10 SpLn:999 and @:0,12, with SpLn:1999 no longer responds; then compare the curl to R:5… Pay special attention to the length, with parameters of @. less than 0.2!
Even if with the instructions given it shouldn't be difficult to reprogram it in a more satisfactory ambit, I believe that the present impact is what matters most, in itself already capable of stimulating major reflections on the subject, some freely arising at the end of the presentation, between a couple of pages.

February 2023 upgrade

After having discovered the power of the spiraloid, at the end of the work carried out here, capable of bringing irregular figures to the perfect pentagonal triangle, and having returned to deepen the properties of the triangle, square and pentagon at the site dedicated to π at the origin of all of which, I have come to realize the most obvious of resources right where it was most hidden.
It was about understanding that it is the value of the pentagon that gives impetus and progress to life and not only that, but that the parameter of ^{1}/_{5} of rotation that had already fascinated me while remaining out of any context, represents only one side of the coin.
The other, perhaps the most significant for the whole context of research in nature, is given by the concave or starry pentagon, called for centuries pentagram with the most dutiful respect of philosophers and occultists, and yet almost ignored, since it does not offer a elementary like that of the rectangle, and at the most crowned by a deformed spiral devoid of natural development.
I therefore proceed to insert an 8^{th} button on the console, which reports with ^{2}/_{5} [×360] the value of 144° necessary for the radius to increment by φ.
I have introduced two other options, made to project various types of spirals on images of molluscs, which get various usefulness in comparisons.

The first is transparent, and provides that each repeated click on the same button increases the thickness of the line by one pixel ([Rot] must be = 0).

The second [Rot] provides for a rotation of the diagram parameterised in degrees, which will add up to the current one, so that with each click, the chosen spiral will be drawn rotated according to the requested value.
If this is active, i.e. different from zero, the thickness of the line remains unchanged, therefore it can be increased (when stopped) and then maintained as it is in the rotation of the same curve.
The angle rotates clockwise and can also be negative; (if the browser doesn't accept it, for a counterclockwise [negative] rotation just add 360.
To apply the optimal quotient, found after a multiple application of rotations, just clear the previous spirals by clicking [×] and assign the total value to the [Rot] field, i.e. the sum of the previous rotations.

Warning: starting from 0 after a clear [×] without reloading the page, keeps the last linewidth and rotation setting. The clear [×] command after rotations may not clean the whole view of longer spirals; you can always refresh the page, but not all browsers will keep the current settings.
Here is an example applied to a typical shell of the Tyrrhenian Sea (mytilus edulis); the spiral dilates achieving the radius an increment φ every 36°, as parameter 0,1 [×360] exposes, and proves to be very valuable, although not included in the list. (click to enlarge):
This too is of course a golden spiral in all respects; it simply doubles the rhythm of the pentagonal, that of a decagon where all angles are 144º! as if to make that beat of ^{2}/_{5} resonate.
In fact, an ideal spiral for two shells with a symmetrical specular development designed to enclose the mollusk.
And let's not forget that it's the half of a bivalve shell.
Neither rectangles nor squares are part of it (the background template stays there to define the center), but its vector is only the 36° angle, which rules at the vertex of the star triangle to be reached at the end of this research, where we could even recognize it as the most sought after golden achievement.

march 2023 upgrade

After so much effort, the final realization could not be missing, the one for which each of us can put into practice how much it is possible for him to develop authentic golden spirals, applying them or better, obtaining them from living creatures, instead of so many superficial mystifications.
To this end, I have taken the first model tested on the mytilus edulis and demonstrated there with a first composition tool, to build a complete and versatile one, although still rudimentary, functional enough to allow a direct online approach.
The complete help for any new section is available below the console.