insight always precedes
a theoretical formulation

The sought-after solution of the golden spiral

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 φ4 every 360°…

golden spiral with repeated increment of radius for φ/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:

D[egree] += 1  (or π/180 for radiant)
R[adius] ×= α  [alpha Constant]
to then arrive, ultimately, even more directly to:
 R = αD
lineto sin(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

φ 1.618
Therefore it is not 1.618: φ the 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 φ4 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:
1080°	1,00536111768685835835952555755714^1080 =
WIN:	321,99689295855895977188087280243  12	321,99689437998485765289480710049] 

JS:	321.99689295858394
EU:	321.9968943800276
PS:	321.983

1440°	1,00536111768685835835952555755714 ^1440 =

WIN:	2206,9995339060154982110049085301
JS: (const. 1.0053611176868584)
EU: (const. 1.005361118)
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:1080φ12 that is:(90×12)φ12. with a result from Windows® calculator [scientific] actually greater than
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:
Radius Lenght Rotation Δ ± α

advanced view:  

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#], 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 use the arrow or the "Back" command of 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 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 α 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.
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.
In case of comparisons with some biological species, to avoid deformations of the theoretical layout inherent in the early stages of growth, starting 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.

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°.
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 11 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.

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…
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 8th 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.
mytilus_bivalve-144° 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.

The background golden outline is an independent image, exact as a graphic but scaled locally, i.e. adapted by the browser (differently in the blackboard in an in­de­pend­ent page), unlike the real-time mathematical layout of JavaScript which, however, is superimposed on it in bitmap mode, that is, stone upon stone.
It is produced in SVG and remains vector; on the other hand, since the spirals drawn by the script cannot be, the automatic service zoom via mouse coarse­ly enlarges only the pixels already present on the display, and the same goes for any printout (not PDF or SVG).
This attests that the frame of golden squares and rectangles is not necessarily compatible with tracing the true spiral; but a surprise awaits us:

Perfect solution of the golden spirals

back to: Triangles, golden section and false spirals