‘'Sacred geometry' means living geometry,
living geometry is sacred geometry. The most admired Golden SpiralHere we are finally with the bestknown one, whose 360° vaulting is worth φ^{4} 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 lowquality 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 nonoverlapping 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 nonrandom, 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 …
maintained uniformly by the Ellipses 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: geometricgraphicWe will then attend the appearance of a second rectangular perimeter, with proportions much closer to the square, to be exact in scale1:^{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 ellipsesIt 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 setup 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 correspondenceIf 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:
φ^{12} [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.
Meditate to understand, on the fact that the spiral is not a closed and complete entity, defined as any regular polygon is.
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.
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,
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 [ Bd, orange] and rectangle [green], comforting the initial expectation.
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).
Said and done, I planned it in the graphic above, at the very least to lighten the SVG code on thousands of instructions.
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,

%!PSAdobe3.0 EPSF3.0 %%Title: a 'new' Golden Spiral exact drawing formula, with ellipses %%Creator: Antonio Alessi %%Copyright: The WATCH Publisher © 2022  https://goldenspiral.eyeofrevelation.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 % 
% (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 