‘'Sacred geometry' means living geometry,
living geometry is sacred geometry.

The most admired Golden Spiral


Here we are finally with the best-known one, whose 360° vaulting is worth φ4 as it reaches the fourth golden circle after the one from which the radius starts, which is why I have called it 'full' or complete.
So, finally, if we want to experience the charm of a quarter moon per­spec­tive, those even sinusoidal phases, whose value I have supported since the dec­la­ra­tion of the true π, since they echo the seasons of time, the wind rose which has always indicated the directions of nature, and the tetraktýs of the four El­e­ments constituting the tangible plane, we are at the easily prac­ti­ca­ble solution as I have already produced, in any operating environment and de­vel­op­ment language, which will shortly give rise to an analysis well de­served, suggestive and up to now unknown.
Once and for all, anyone who claims, or supposes to demonstrate with low­-qual­i­ty graphs, that the difference between a quarter of a circumference and a spiral is minimal, utters nonsense not to be disclosed, such as this en­large­ment shows with sufficient mathematical evidence: the information distortion to which we must refer is not so much in the non-overlapping of the two curves, but in the inversion of the motion trend, one fixed circular and the other in centrifugal expansion, which is why the two curves intersect in­di­cat­ing a before and a after non-random, which precisely meets the diagonal of the golden rectangle that delimits it, about 4 parts against 5 of the same quad­rant, more or less subtly distorting the correct mental perception, especially when the two curves are thick and overlapping.
Is it important? someone will ask…; but what is the use of reproducing this spiral with such emphasis, if we cannot then contemplate it in its true essence?

The constructions proposed here are instrumentally correct, not based on a collage of squares, or segments programmed one by one (a procedure that needs to be reviewed for each type of spiral), but on a route of total ho­mo­ge­ne­ous continuity, from and towards the infinite; they present the nu­mer­i­cal re­quire­ments of the golden progress, which you check with a click, ex­clud­ing the presumed usefulness of the one defined with a compass and right quar­ters, not at all educational… or worse on the basis of two equal squares! not with­out the in­vi­ta­tion to be cautious about the jumble of pro­po­sals that churn out disparate formulas on the net, with thick lines and a single couple of rotations that are not always verifiable (I superimposed two obtained with pa­ram­e­terized automatic tracking apps, and they did not correspond to each other) ; moreover, even the programming divided into the four sectors can reserve surprises on the homogeneity of the ring road, imperceptible to the eye.
Nonetheless, the compulsive use of boxes may have drawn instinctive in­spi­ra­tion from the fact that – I will demonstrate it shortly – these arches ac­tu­al­ly rotate on a similar carousel, but in this case responding to per­fec­tion, as it is made of authentic quarters of ellipses; and there's more …


I had intended to propose it as a new approach to a design that could be translated into any context, starting from javascript and HTML; but if that weren't enough, incredible but true... making it usable with precision even with only manual instruments, not only for the garden but also ar­chi­tec­tur­al­ly, as long as there is enough space to exhibit its extension.

the complete Spiral of the Golden Section,
maintained uniformly by the Ellipses
The rectangle with an orange dotted line and ivory field highlights first of all how each quarter of the curve (π /2) affects a portion or arc that is com­plete­ly unrelated to the square with which it tends to be represented, dis­torting its dynamics, since it is this rectangle that should rotate around its first vertex at the center of the spiral (not of the Cartesian system), am­pli­fy­ing itself by φ at every 90°, as can be followed by rotating the base around the yellow axes.
my second formulation: geometric-graphic
We will then attend the appearance of a second rectangular perimeter, with proportions much closer to the square, to be exact in scale 1:4 √φ al­ter­nate according to the same principle at every quarter, but around the ver­ti­ces of a golden rectangular profile, of which we will discover the very spe­cial one scope.
In order not to weigh down the figure, one of these is outlined in gold, which has three vertices at 1, B and 2; and we will get the sense of it shortly.
Both factors are clear evidence of the incompatibility between the de­vel­op­ment of the golden spiral and the sequence of golden rectangles usually adopted as a scaffolding, even in a Cartesian rather than a polar system.

From the spiral to the golden ellipses

A curious figurative effect, due to the inversion
of a couple of statements in the PostScript code,
which explodes and shows the ellipses separately.
It was this method of re­search that al­lowed me to high­light an ex­cep­tion­al and yet un­prec­e­dent­ed la­tent pro­cess in the de­vel­op­ment of the gold­en spi­ral, re­pre­sented by none oth­er than the con­cur­ring fig­ures of as many el­lipses for as many quad­rants as there are.
Can we call them ‘gold­en’?
not for the ra­tio of their axes (they are not rec­tan­gles to e­quate them to; if an­y­thing, an­oth­er cri­te­ri­on should be ap­plied) but for their func­tion­al­i­ty in this context.
However, I investigated for a virtual re­la­tion­s with the golden rectangle, and something emerged. Inscribed the rectangle in one of the ellipses, I traced its diagonals; these delimit two almost equilateral triangles, right and left in the fig., whose heights intersect in points very close to the foci of the ellipse (the left in the fig.), but in the fairly precise graphic construction they do not reach them exactly.
I had taken it as an invitation to an even more careful general reconstruction; I have already stated how delicate the management of the ellipses is, moreover with the manual tools used, but even without an extreme set-up it does not depend on an error, given the coherence of the various parameters; on the contrary, my imagination likes to suppose that the failure to achieve ideal parameters could be a reason for intelligent tension towards a continuous expansion of the form; but it's just a thought dedicated to the spiral...

A stimulating expression of semantic correspondence
If a golden rectangle can be considered an emblem of φ, it evolves and mul­ti­pli­es by pivoting in a spiral around its 'square root' (free symbolic ex­pres­sion of the square it derives from).
Although the scaffold of golden rectangles is not necessary to trace the spiral, here is how to restore new dignity, for the intimate participation.
By adopting any rectangular scheme among these, or in any case by tracing any golden rectangle from scratch, so that it is tangent to the spiral with at least three of its sides, and scaling it starting from the center of the spiral in the proportion √φ [1,272], its vertices will reach the centers of as many el­lipses, of which the quarter delimited by the extensions of the sides of the rec­tan­gle, which become their axes, will be part of the spiral!
Given a system that adopts radius = 1 as the reference unit at the 1st step, its height, or the shortest side, will be: φ12 [321,99689437998485765289480]
These axes, sides of that second rectangular perimeter, are easily meas­ur­a­ble for profiling the ellipses, since along the development of the cage they scan the spiral perpendicularly and exactly at each of its quarters (90°, 180° etc.) on the tangent cage, which the rectangles of the ordinary cage they an­tic­i­pate slightly.
From the axes it is easy to go back to the foci, from which to trace the ellipses, where the rectangular perimeter (e.g. outlined in gold with 1, B and 2 ) spirals around the vertices (in fig. A -->1, B -- >2, C -->3) of each rectangle scaled to √φ with respect to the progressively tangent ones.

If this description is too concise (or the translation is not optimal :(), the graphs should well replace the equations and lead to the intuition of this hidden har­mony.

This survey, which makes the parametric process much more expressive than the one with polar coordinates, and clearly distinguishes the golden spiral from a standard logarithmic one, aroses from having applied to the in­ter­me­di­ate spiral of rectangles inside the perimeter C,B,A,d – a choice as valid as any other, since the spiral is a repetition without beginning or end – a rec­tan­gu­lar projection [green] whose vertexes remain on the diagonals that identify the spiral center, with a proportion between the sides always 1 to φ, but a­sym­met­ri­cal with respect to the starting profile, naturally spaced progressively by virtual rectangles, one for each vertex in alternating ver­ti­cal / horizontal rotation, in such a way as to maintain a ratio Φ be­tween their dimensions in equal rotation.
We can thus observe at corner A a diagonal distance quoted Φ²: Φ, against Φ: 1 at corner B, which becomes 1: φ to rectangle C.
The latter highlights how the sum of the bases of the two: B and C dis­trib­utes the proportion of the additional rectangles with the inevitable ratio √5.

Well yes, the vertices A, B, C, in fact applied to the new external [green] rectangle, denote nothing less than the centers [in the same color] of as many ellipses, of which a quarter of each is equivalent to the quarter of the volute of the golden spiral under consideration.

Such a diagram probably insinuates the evocative image of a spiral path in which at the beginning of each virtual quarter the curve wraps itself in the ellipse that shapes it, covering it all, and then once again reaching 90° of the arc, repeat the circumnavigation on the next ellipse. A cadence like cycles and recurrences of growth, as is known an alternation of traumas; a rewinding on itself, an awareness of the experience before a new momentum forward.
Undoubtedly a whimsical vision, but which perhaps hides a precise truth in the wake of the showy continuous motion, made up of courses and recurrences, capable of giving a pulsation to the historical and vital reality. Arguments of connection between the circular principle and its expansion and contraction, capable of circumscribing the phenomenal world in single events.
I cannot help thinking how each ellipse virtually contains all the spiral that precedes it in growth, both in the natural direction and in the opposite one, thus almost constituting a stage of completion or transition.

Meditate to understand, on the fact that the spiral is not a closed and complete entity, defined as any regular polygon is.
On the contrary, it has no beginning or end but only movement, a continuous expression of centrifugal and centripetal energy. The spiral represents the dynamics of life, the way of contracting and expanding, transforming to e­volve but, by its very property, it is the mere vectorial reproduction of the trans­par­ent 'ratio aurea'; not of any finite object of life itself, whose essence and structure Creation crystallizes through polygons and polyhedrons, each with primary functional qualities of arrangement and development, naturally inter-dependent on numerical relationships and sacred geometry.

Let's focus for a moment on the dynamics of cyclones, rather than on the less showy (although no less expressive) dynamics of shells and plants.
The cyclone arises from a set of atmospheric turbulences determined by high equatorial temperatures, which by generating centers of minimum pressure cause aspiration. This causes the winds to converge according to a spiral mo­tion which combines into a vortex, similar to a gigantic funnel.
However it is connoted, we can more easily assume that said vortex satisfies a gravitational principle, conveying cyclones and anticyclones, forces of at­trac­tion and repulsion.
Its image is more easily traceable to the temporarily closed one of an ellipse, with its property of reflecting the spiral – of which it's a daughter – within its circumstantial perimeter, enveloping it consistently in both directions.

Just as the past and future manifest themselves in a spiral over time in con­stant though almost impalpable dilatation; to integrate thanks to the golden ratio what I said about the π fourth dimensional,
Established a coefficient of growth, it is unique, infinitely and eternally equal to itself.
If we were able to imagine a spiral whose radius expands at each coil by a single minute or a second, we would have conceived a magical symbiosis of time and spiral, an unusual way to visualize and geometrically represent the passage of time. After all, the hands of traditional clocks already do this, since the matter itself expands (and expansion, 'na­tu­ra docet', can only occur in a spiral); yet we do not notice it.
Try anyway the customized option in the console at your disposal. Apply eg: to @. 16,18, and 43 to the Radius, and imagine the passing of time, as if each 'sound' on the traces of a CDROM – maybe like me you prefer analog vinyl? – was a year, or a century or a millennium, at evolutionary cosmic rhythms secretly articulated by the Φ…
It is enough to follow what instinct inspires, from the spring of a watch which is extremely compact the more it is loaded, to the extreme of the latent wave frequency; try a Radius: 13 and length:49999 then overlay, without clearing the screen, a length:99999 (a parameter that only this option can handle) or at maximum expansion, which you can glimpse by trying with a Radius: 3 and a spiral length: 99 the steps from 0 .05 to 0.02. All truth is within you.
In fact, the golden section did not arise within logarithmic instrumentation or to satisfy mathematical notation. I've already called it “celestial unit of measure” and I won't add anything else; but I do not exclude a deepening of further unexpected properties of the cosmic soul of the spiral, perhaps having glimpsed something even more surprising to examine in depth.

The proportions on-the-fly in the current graphic layout of the ellipses are 1.075281212 * 0.9534 – eccentricity 0.4972479446 –, and the relative di­mensions of their axes and foci, represented by the ends of the respective col­or­ed segments, are obviously referable to the same golden ratio which reg­u­lates the diameters of the concentric circles supporting the great golden triangle.
It goes without saying that I traced them with instructions of circles in a suit­a­ble x, y scale, and that some specifications could undergo minor var­i­a­tions to an actual calculation, as the work of setting up this study is necessarily planned visually (or I would not have discovered none of that…) and is based on rounded Φ and multiples thereof, with axes transferred by blocks scaled in subsets at various levels.
The formal ratio achieved at the end of the page will make them more exact; but as I mentioned, more important work awaits…

A fourth ellipse Φ³:Φ² inserted afterwards with identical parameters from the center 'd' immediately integrated the curvature, placing itself for the due scale at π², at the diagonal intersection point [ B-d, orange] and rectangle [green], com­fort­ing the initial expectation.
And for even greater satisfaction and verification, in terms of project, I have grouped the complete four parts in a single function, subjecting them in the con­text to a very critical command for this kind of output: repeat it on a re­duced scale 1 /φ4; and the outcome [fig. below] is more than satisfactory, despite the computational boundaries of the system.
All this cannot fail to suggest a new way of constructing this spiral, proceeding in leaps of φ4 to reconstruct the quarter ellipse~spiral along the trajectory of the diagonal axes; definitely more laborious, but still manual and truthful.

Therefore, as I had promised, the position of the relative foci that can be de­rived for each ellipse is such as to allow tracing with any proportion and on any terrain: up until now, in fact, both a circle and an ellipse could be drawn in the open field, however difficult it may be to draw ellipsis, but direct and scientific access to the golden spiral was in fact precluded (which must have contributed to the success of the Fibonacci series).
It is virtually open at this point.

How the Ellipses delimit the full Golden Section Spiral
This also means that a new algorithm could take shape, to draw the spiral (or spirals, if someone wants to advance in the direction of this study), consisting in repeating not the single degrees or fractions, but properly the quarters of an ellipse, just like i[t wa]s used with circles.

Said and done, I planned it in the graphic above, at the very least to light­en the SVG code on thousands of in­struc­tions.
With a good PDF reader, zooming in on the junction of two ellipses reveals the difference between the gold spiral curve, segmented by thousands of calculations of sin and cos for each 0,5° of which it pres­ents the detachment, and the blue and orange curves of the ellipses, pro­grammed with a single instruction as con­tin­u­ous arcs of 90°, without fractures in the plotted.

How to finally build the
true Golden Section Spiral
Having thus reached the finish line, starting from scratch, it is interesting to evaluate that via software – which now effectively replaces the compass – the effort to draw an ellipse in this case is minimal, since it does not require the use of equations but only the use of primitive instructions, i.e. quarter-cir­cle arcs, framed in x scale: y of 1:1.12783849 and vice versa, as I have already explained above; note also that quadrilateral of aspect ratio 1:4√φ repositions itself with each 90° rotation in φ scale at its vertex at the top left (in counterclockwise mode) which does not require any external scaffolding or ref­er­ence, allowing any portion of the spi­ral to be directly and accurately re­pro­duced.
And here it is served, in its most elegant form and 
feasibility, for the better right of future generations.
Unless you definitely want to accept this as a real spiral, which who knows how and why is defined as logarithmic, even though it is made to fit the golden rec­tan­gle in the background, while it is neither one nor the other.

To make it even shorter and more exhaustive, I have elaborated the PS code of one of the various ways to reproduce it easily, with any tool capable of reading PostScript directly, such as Photoshop itself, or Gsview or other Post­Script Viewers available on the net.

It is a fairly simple RPN routine, of about ten lines, in addition to the greater vol­ume dedicated to the accessory parameters. It can be copied and pasted into a file like "GoldenSpiral4Ell.ps" to be compiled or used for this purpose: 
%!PS-Adobe-3.0 EPSF-3.0
%%Title: a 'new' Golden Spiral exact drawing formula, with ellipses
%%Creator: Antonio Alessi
%%Copyright: The WATCH Publisher © 2022 - https://golden-spiral.eye-of-revelation.org
%%BoundingBox: 0 0 6900 4800

%% MIT License
%% Golden Spiral construction - buildup or exact math formulas are described in website
%% Permission is hereby granted, free of charge, to any person obtaining a copy of this software,
%% including without limitation the rights to use, copy, modify, merge, publish, distribute,
%% provided that the above copyright notice and this permission notice is included
%%  in all copies or substantial portions of the Software.
%%EndComments
%%BeginProlog
/setpagedevice where { pop 1 dict dup /PageSize [ 6900 4800 ] put setpagedevice} if

/RGBco { {255 div} forall setrgbcolor }def
/gold {[206 190 108] RGBco }def
/silv {[207 213 233] RGBco }def

/phi { 5 sqrt 1 add 2 div } def
/SideRatio { phi  2 {sqrt} repeat } bind def
/phiH4 { 1 SideRatio div } bind def

%%EndProlog
3450 2400 translate

/baseline{ gsave silv .1 setlinewidth
 0 0 moveto 0 radius 0 lineto
 0 radius  rlineto radius neg 0 rlineto closepath
 stroke grestore } def

/newquadrant {SideRatio 1 scale
% baseline
} def

/radius 33 def
/lwidth radius 20 div def

1 setlinecap gold
%-------
14 { newquadrant
 0 0 radius 0 90 arc currentpoint
 stroke

90 rotate
 SideRatio phi mul radius mul sub 0 translate
 pop

 phi dup phiH4 mul scale

/lwidth lwidth phi sqrt div def
 lwidth setlinewidth

	} repeat





% (V5 + 1) /2
% 1.1278384855616822602648354797459



% center start







% unlock this  to draw the frames







% set the proportion for each new arc you want
% draws the 1st arc from any 0,0
% and gets the last point x, y, then:

% turn one quadrant clockwise, for any frames
% from the previous y, moves to the new center
% ( delete the remaining x from the stack )

% renews the proportions of the next phi ratio

% resizes the line width as to the new scale
% to fix a flat line, remove the 'sqrt' from above

showpage
Confident in the time invested so far in this contribution, I trust that some patient Mathematician will want to retrace it in the formulas resulting from the illustrative procedure, with exhaustive and impeccable formulations.

Since this figure is completely vectorial, for its greater visibility I recommend adopting the landscape screen and scaling it with the maximum zoom al­lowed by the browser; moreover, in order to lighten the loading of many complex SVG figures, I have adopted a data compaction that increases the ap­prox­i­ma­tion of almost all values; however I don't think that a few dec­i­mals less can affect the appreciable yield via browser.
In any case, the PDF of all the variants of the graphs is clearly legible, which will make it possible to overcome any gaps in the visibility of the lines, on whose thickness PS and SVG seem to differ, as well as verifiable the pre­ci­sion of the numerical values in the angular intersections, of which the al­go­rithm PostScript renders approx. to 3 decimals, obtained from the un­me­di­at­ed exponential calculation.

the Quintessence of the golden spiral

the exact solution of the golden spirals