Imprimis, only one is golden section, not two: and it is Φ, Phi 0.61803…
the other, φ, phi [1.618] although the most commonly adopted, is its first de­riv­a­tive, mathematically an ad­den­dum, or its specular projection, which can be referred to as golden ratio, but not the ‘golden section’ whose proper definition by Euclid himself, (“Elements”, 300 BC), was division of a segment “in extreme and mean ratio”, so that the proportion of the whole length to the section equals the ratio of the section to the shorter length – that is its own ratio; which the title Φ + Φ × Φ = 1 maximally synthesizes.
I know very well that this statement may irritate some; but can they explain to me how they can consider 1,618 a section? A section of what?
It is derived from 0.618, either as its division by 1 / Φ, or as its sum 1 + Φ.
The comparison of these terms: 1/Φ = 1+Φ gives the essential true equation.

Always defined as ‘section’ since ancient Greece, thus described up to Pa­cio­li, and by Kepler as ‘mean and extreme ratio’ (of a segment), eve­ry­thing sug­gests its internal ratio to any quantity whatsoever, although in the many titles that confirm it, to name a few _read – as well as in the pro­lif­er­a­tion of web ar­ti­cles, more or less attracted by the magic of the number – the mean­ing of section is not taken into account at all.
1.618 is preferred in­stead, perhaps because it is more flashy, but it is only the reflection of the section applied to the 1.

Author	Title
Fletcher, Rachel (2006)	"The golden section". Nexus Network Journal. 8 (1): 67-89.
Frings, Marcus (2002)	"The Golden Section in Architectural Theory". Nexus Network Journal. 4 (1): 9-32.
Jones, Ronald (1971)	"The golden section: A most remarkable measure". The Structurist.
Howat, Roy (1983)	"1. Proportional structure and the Golden Section". Debussy in Proportion: A Musical Analysis. Cambridge University Press. pp. 1-10.
Le Salon de la Section d'Or	October 1912, Mediation Centre Pompidou
La Section d'Or	Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, no. 1, 9 octobre 1912, pp. 1-7 Archived 2020-10-30 at the Wayback Machine,
Camfield, William A. (March 1965)	"Juan Gris and the golden section". The Art Bulletin. 47 (1): 128-134.
Scimone, Aldo (1997)	La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo:
Walser, Hans	[Der Goldene Schnitt 1993]"The Golden Section". Peter Hilton trans. Washington, DC: The Mathematical As­so­ci­a­tion of America.
Knott, Ron	"The Golden section ratio: Phi". Information and activities by a mathematics professor.
Emma C. Ack­er­mann	§. The Golden Section - Michigan State Normal School
It is worth noting that Leonardo da Vinci himself, illustrating Luca Pacioli's book, with the generic title «Divina pro­por­tione» (1509) referred to this ratio as “sectio aurea”, that is, golden section; later “sectio divina” for Kepler.

THE DIVINE PROPORTION      

In secundis, locution that comes ad hoc! from a her­me­neu­tic point of view, the Source of Cre­a­tion, the U­nique Prin­ci­ple or how­ev­er one can con­ceive it, to give rise to the po­lar­i­ty of pos­i­tive­/neg­a­tive, yin/yang, Shi­va­/­Shak­ti, Fa­ther­/Mo­ther; what es­o­ter­i­cism has al­ways taught, as well as the CHRIST in the 5th of his 9 LET­TERS (1998), i.e. re­cent Les­sons to hu­man know­ledge, not sur­pris­ing­ly the fifth! could only split with im­mense will, not ex­pand.
For the ONE noth­ing ­is in being out­side of the Self: it has been riv­en, di­vid­ed, not mul­ti­plied, thus only de­ter­min­ing the pos­i­tive-neg­a­tive an­tith­e­sis: ex­pan­sion­/con­trac­tion consistent with the Φ; at­trac­tion/ ­/repulsion in grav­i­ta­tion­al e­qui­lib­ri­um reg­u­lat­ed by the π which is its square root (for our use the perimeter 3.1446 of the square with that side, i.e. with area Φ), terms which are in essence deeply and ab­so­lute­ly correlated and inseparable.

1 = x + x × x

An expressly symbolic notation, with semantic figures, for those who un­der­stand it, and try to scan it out loud. Φ enunciates in cipher the only Pow­er [x] that can amplify the One, integrating with It in expansion to φ, which takes place exactly in dividing it; and maintains its identity with unchanged decimal tail even in the union, as well as in its square where the 1 comes 2.

Interpreting it from a dialectical perspective, a sort of 'partnership' of the Golden Number with the Unit is confirmed by having divided it, which allows a two-way comparison, both by addition with Φ, and by subtraction with φ.
By factoring the union from ¹/_Φ [1.618] for which Φ is added to 1, the out­come of the operation is indifferent in the double perspective: whether Φ adds 1 to itself [Φ +1], or whether Φ is added to 1 [1 + Φ]. It changes instead for φ, if it subtracts the 1 from itself [φ -1], or if it is subtracted from 1 [1 - φ], resulting negative in one of the cases ± Φ, which however lead back to Φ as the basis. The resulting statement is the most linear of all:
Golden Section is the unique number that dividing the 1 adds itself to it.

To better assimilate its fundamental and mysterical Criterion:
the golden number represents – not only ab­stract­ly – the only partition of the ONE that in dividing It, results at a time added and unified to It, in the perfect un­fath­om­a­ble equilibrium and mutual restraint.

We also find it in the very famous “Emerald Tablet”, one of the most important and discussed myths of ancient Egypt, having established over the centuries the commandments of alchemy and the most committed esotericism.

I translated it from a now classic Latin version, but a last one among who knows how many others in every language, and all even only partially mis­un­der­stood and then subject to interpretative alterations; which seems obvious after all, otherwise at least one would have prevailed, with global effect.
It was certainly not intended for the general public.
However, its first three paragraphs, which have become a universal emblem, sound clearly and unequivocally – understandable to everyone – what max­i­mal­ly completes our assumption: ONE is the Whole, from which everything descends.

The same cannot be said of φ, for which multiplying the One does not carry much additional meaning, while dividing it only leads to Φ. The alternate for­mu­la, i.e. 1 = x² -x as we will see, would be devoid of interpretative sense.

So That's why the Golden Section is one, unique and part of the ONE in such Pro­por­tion as to be considered Divine.
It is found universally expressed in the great golden triangle, that One Divine Triangle which perfectly represents it in geometric reality, as well as in the construct of the Great Pyramid of Giza.
Once this is understood, it is clear that the formula x² - x = 1 is inverse and in con­tra­dic­tion; but let's watch its development.

The rational description generally adopted is nothing but its scholastic re­duc­tion, a widespread version, which emerged from discovering in an easy but indirect way how the Golden Proportion manifests itself in theory and prac­tice; but it is referred to at least two distinct numerical or geometric quan­tities, and to the relationship with their sum or subtraction.
The resulting and varied methods of calculation are sometimes complex and superfluous, displays tainted by the innate need to complicate simple things, precisely because we do not yet know exactly how to frame them.
Indeed, it seems that the most widespread way to make it intelligible is to embroider on the elementary school series of Fibonacci integers, which makes me think of it as trying to build the Great Pyramid with LEGO^® tiles;

If nothing else, though to dispel the myth of the golden ratio, the University of Edinburgh Science Media states that: In fact, as is known, said series had a completely different formation and func­tion; but then the problem of making sense of the golden section re­mained open…
The fact that the numbers of the series converge to the Golden Section as they grow is only the demonstration that this exact proportion is the virtual mother of every event, numerical and geometric, as also demonstrated by my fascinating discovery of the ‘spiraloid’; but it will never be able to replace it, in any theoretical or practical context.

Given an interval of any value AZ, real or abstract but global i.e. unitary, there is within it only one point P that cuts it, establishing between the parts the proportion of the type AZ ÷ AP = AP ÷ (AZ-AP).

As if on cue, it is precisely the alphabet A^{bcdefghijklmno}P_qrstuvwxyZ that re­sponds admirably to representing it, with ‘P’ in the position 16 out of 26, where 26 × Φ = 16.07! Furthermore, in the positional system with base 16, i.e. hex­a­dec­i­mal, P has code 0050, whose 5, as I have already explained, is an ex­clu­sive recurring module of the golden section.
Although only one is the unknown, that is the Φ portion, we can retrace it in the three cases of rotation of the 1 value applied to each of the three quan­tities.

1. The main and most natural and eloquent case, therefore, sees the Unity and its Golden Section. Searching it [inside] of the quantity AZ, AZ will be the whole, of which the section-trait AP represents the unknown Φ.
   We will have with the greatest simplicity and coherence that reflects the formula:
   

   1 ÷ AP = AP ÷ PZ 1 / AP = AP / (1 - AP) 1 = AP2 / (1 - AP) 1 - AP = AP2 1 = AP2 + AP

   1.0 Abcdefghijklmno PqrstuvwxyZ 0.618 ¡0.382

   whose quadratic solution (easily reproducible online) is:
   ( √5 -1 ) ÷ 2 =  0.618,
   and, obviously less significant:  (-√5 -1 ) ÷ 2 = -1.618.
   -φ is antithesis; its positive is the emblematic expansion deriving
   from the subdivision of the One by its golden section: 1 / Φ.
   Nonetheless both satisfy the final assumption:
   sia 0.618 + 0.382 = 1 che -1.618 + 2.618 = 1.
   To satisfy the inverse equation (1 / AP) + 1 = (1 / AP)², while
   AP = 0.618 gives the result φ, the value AP = -1.618 gives Φ² that is 0.382.
   
   Beyond the usual scholastic formulation of this relationship, this ineffable Divine Proportion is what is wonderfully exteriorized and can be contemplated in the autonomous and univocal geometric reality of the Great Golden Triangle.
   It is reproduced here with base b and e­qual sides a, c. The three sides, from the base, re­al­ize the pro­por­tions b ÷ (a + c) = (a + c) ÷ (a + b + c) = Φ, i.e. (π_­/4)², with the right π 3.14460; for which un­par­al­leled com­plete­ness I in­tend­ed to call this tri­an­gle at the pin­na­cle of per­fec­tion: Divine.
   What is more, its vertex involves two circles: the cir­cum­scribed one, which touch­es it, whose quar­ter is e­quiv­a­lent in length to the sides a and c, and the one hav­ing its cen­ter at the ver­tex of the tri­an­gle and its height as its ra­di­us, whose quar­ter of cir­cum­fer­ence is e­quiv­a­lent to the base, to which it is tan­gent.
   Double squaring of the circle, the second of which recalls the legacy embodied for millennia in the Great Pyramid of Giza. The two circles are in proportion 1 : (1 + Φ³), like the diameters, which is then the same as the ratio between the side and the base of the triangle.
   I played kaleidoscope-style with this hidden pattern at the bottom of my page on measurements of the Great Pyramid of Giza.
2. In the 2nd case the monad contemplates its union with its golden section.
   It is an inverse process that of AP = 1, which searches for the golden ra­tio outside the unit (no longer a subdivision, but a sum), for which it will be the unknown φ to complete the AZ segment.

   AZ ÷ 1 = 1 ÷ PZ AZ / 1 = 1 / (AZ - 1) AZ = 12 / (AZ - 1) in short: AZ2 - AZ = 1

   1.618 Abcdefghijklmno PqrstuvwxyZ 1.0 ¡0.618

   In this mode, the equation leads to the negative value -0.618 and to the so-called golden ratio 1.618, which adds up to a value greater than unit, where the unit becomes AP, golden ratio of the whole AZ, a con­tra­dic­tion in terms of teaching and statement; at the same time the negative golden ratio cannot have a square root, thus lacking the implicit and le­git­i­mate ge­ne­al­o­gy of the π and the constant , while the √ φ = 1.272 is physically less rel­e­vant, it only reflects calculations that can be obtained in another simpler way.
   It can indeed reproduce the interatomic gravitational constant ( by my at­tri­bu­tion) = 0.78615 via a further step, which is 1/√ φ, but it just re­flects the direct result of √ Φ, calculating it in a convoluted way.
3. The same occurs in the 3rd triangular case, the extreme presence of the golden ratio, in which PZ will be the Unit, relating the unknown to the seg­ment AP (1/ Φ) [φ], while the entire AZ will correspond to φ².

   (AP + PZ) ÷ AP = AP ÷ PZ (AP + 1) / AP = AP / 1 AP + 1 = AP2 AP2 - AP = 1

   2.618 Abcdefghijklmno PqrstuvwxyZ 1.618 ¡1.0

   This reflects the exaltations of squaring for the two extremes of the Pro­por­tion: in PZ the monad is in the position of Φ² _[0.382]; of the 1st case;
   in AZ we have that of φ² _[2.618], the union with its section of the 2nd case.

   A triumph of the Divine Proportion, which manifests its unrepeatable and irreplaceable potential in the world of numbers and shapes.

In addition to the rightly praised concepts of harmony of forms, albeit of of surface elaborations, the Golden Section plays a fundamental role in every pro­cess of becoming, perfectly expressing the generative aspect of the mo­ther figure 5 the same convex and concave pentagonal figures, or of in­no­va­tive­-sub­sti­tu­tive construction and destruction.
The criterion of reciprocal ratio expressed by the relationship between ex­ter­nal and internal sides, can be found in the formula in fact dependent on the number 5:  √5 × 0.5 ± 0.5.

The two roots interact in a unique functional symmetry, which seems to os­cil­late between the extremes of past and future, synonyms of contraction and expansion; if their mutual multiplication leads to 1, each one, dividing the other, enhances its property by multiplying it by itself:

φ / Φ = φ2 [2.618] Φ / φ = Φ2 [0.382] 1

In essence, as I have been expounding for years on the astrological ground, and attested by historical events documented on this site, the c­re­a­tive-de­struc­tive component of the pentagon is confirmed and concentrated in a u­nique way in the golden ratio between the two sides, where the minor section (convex side) is always the creative one, the golden section of the major one which is substitutive-regenerative (concave star-shaped side), from whatever perspective one considers them.
It means acknowledging the expansive-active value of Φ and the pas­sive-enveloping value of φ which is its mirror and functional connection.
In this regard, I suggest comparing it with my own development of the vital principle of the golden spiral.

Antonio Alessi © The Watch Publisher, 2003-25