__________A new understanding in Art History__________

Mathematics - History of Art - Esotericism - Visual Arts

The Sex ofNumbers

March, Venus, CupidA family story

------------ Yvo Jacquier ------------------------------------------------------------------------------

COMPARATIVE GEOMETRY ---------------------------------------------------------------------------------------- March 2015 -----

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 1 on 25

PART I

TRADITIONS

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 2 on 25

MYTHOLOGIES

For what the Symbolists call « the tradition », the number 3 is celestial,

feminine and sacred beside the number 4, terrestrial and masculine. I

have personally sought (and found) a lot of arguments in the artworks

which go in this sense, until I realized, about 3, that the evidence of

Celestial are direct, while those of its female property remain indirect.

Eg, the Vesica Piscis, the divine and female vulva is marked by √3, and

not directly by 3... Another embarrassing example : the magic square of

order 3, from Byzantine origin, is attributed to Saturn. This planet is the

frontier of the visible space, without telescope, and the Celestial option

is legitimate. However the virility of the "old taciturn" doesn't fit with

femininity, as « the tradition » attributes to 3. Saturn should be relieved

of this wrong role, and THE Celestial must take a "reasonable distance"

in front of what that becomes THE Terrestrial.

1 - Traditions – The tradition

The so-called tradition often contradicts itself in the name of the

mystery. 3 and 4 are neutral in essence, and the genders will be

defined differently besides. The doubts about the femininity of 3 have a

broad historical roots (the same about the virility of 4). Thus the myths

of ancient Greece "reverse" the roles of Heaven (Uranus, male) and

Earth (Gaia, female). The mythology even cuts the phallus of Ouranos.

That of Osiris does not know a better fate, Isis owes its fertility to a

subterfuge. The Isis Egyptian triangle attributes the side 4 to the

goddess, and the side 3 to her husband Osiris (the hypotenuse 5 is for

their son Horus). The terrestrial Isis, symbol of the hidden laws of

nature, faces a celestial Osiris funerary god and judge of the souls.

These definitions will evolve over time in Egypt, including for political

issues. For their part, the Pythagoreans would associate the 2 with woman, and

the 3 with man. Finally, the Western tribal Africa attributes the 3 to the

man, and 4 to the woman. In many cases, the male and female

qualifiers are inverted with respect to the claims of “tradition".

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 3 on 25

2 - The lesson of tarots

The tarot cards are a true encyclopedia of the symbolic. The reference

game is that of Nicolas Conver, falsely dated from 1760. And the

original model of these cards is much older. The project is of Byzantine

origin and it crossed the north of Italy to reach the expert hands of

Albrecht Dürer. The Conver is actually the replica of the (woodcut)

original master of Nürnberg.

http://www.melencoliai.org/Tarots/Costumes.html Christophe de Cène have restored the pyramidal structure of the cards

by their dice combinations. The complete article is at :

http://www.melencoliai.org/compo-tarots/Christophe_de_Cene-

Numerical_pyramid_tarot.pdf

The "visual" pyramid is at the address :

http://www.melencoliai.org/farandole/numerique/Pyramide_numerique-2.jpg

2.1 – The wings of Celestial

The Empress is the card number III, and her associated combination is

(3.3). Five other cards have this 3 in common : XVI-6.3 La Maison Dieu

- XX-5.3 The Judgment - VI-4.3 The Lover - XIV-3.2 The Temperance -

XII-3.1 The Hangman. Methodically, these six blades show the various

ways to invest the celestial space. It must be either : • 3.1) Hang oneself physically, by the feet, as this Hangman, and enjoy

this world upside down, meditate. • 3.2) Be an angel standing upright like the Temperance (one of the two

representations of St Mikael), wielding the magical fluids. • 4.3) Be a winged God, for example Cupid on the card of the Lover.

And practice archery. • 5.3) Be an angel on a cloud, like that of Judgment. The practice of

trumpet seems optional. • 6.3) Jump from the top of a tower like the two human characters of

the Maison Dieu.

A detail is hidden in this card in the plume of smoke/lightning : the dove

of the Holy Spirit. The myth of Icarus is not far. • 3.3) Have a dream, like the Empress, and let grow the filigree wings

of spirituality.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 4 on 25

N.B. 1 : here the wings are the prerogative of the angels and the birds.

The wings of the Empress are blue shadows, and the bird of the Star

flaps its wings, but it does not fly. The accuracy of these images invite

us to exercise the utmost rigor, concerning numbers. N.B. 2 : The only purely female blade of this set is the Empress.

Celestial femininity is far to be an evidence in tarot cards.

2.2 – Celestial and terrestrial couples

In the pyramid reconstructed by Christophe de

Cène, nine couples of cards are placed

symmetrically to a vertical axis. Three cards are

on this axis: The World (XXI), the Chariot (VII)

and the Lover (VI). These couples are gendered. The three cards of

the central axis, including the Lover, are not

concerned. We can point out on this occasion,

that the sex of the character into the card of the

World has probably no real importance.

Each card feeds from two archetypes at the base

of the pyramid, as represented here by arrows -

white for the Empress and black for the Emperor. The verdict is clear : three cards on four claiming

the Ce lest ia l Empress are mascu l ine.

Conversely about the Emperor.

The idea of a feminine Celestial and a masculine Terrestrial collapses.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 5 on 25

2.3 – When the tradition looses its way

The card of Lover (VI) is at the intersection of the

Empress (III) and the Emperor (IIII). It is the union of

Heaven and Earth. At left, the Conver sets the scene of

a marriage contract under the auspices of Cupid. His

arrow can not be perfectly vertical, no more than the

scepter of God in the Holy Trinity by Rublev : human

freedom depends on it. On the right, the woman wears

a hairstyle typical of young brides, and her mother

knights the young lover.

In a very personal version,

Oswald Wirth draws a kind

of fop between two young

women with a tantalizing

allure.

The purpose of this trio leaves room for

interpretations beyond the boundaries of

marriage. From the origin, the classic card

is too often interpreted as that of a false

choice : the young man would hesitate

between an old and wealthy woman and

another young and poor. That choice does

not care about love, it brings into

opposition the greed and the hunger for

flesh. This freedom despises the human to

the benefit of the ego. And Cupid has no

place in such a charade. In addition, his

arrows don't fit with the notion of doubt!

In the Conver above, the heavenly love finds an earthly dimension and

the history ends by a marriage contract (see the hairstyle of the young

woman). The tradition presents the alliance of heavenly love with the

terrestrial social like a vertical transaction. Love and Heaven are not

feminine, but they are shared objects - and the social terrestrial is not

masculine. Wirth distorts this verticality in favor of an arrow aiming the

head of the man instead of landing between the legs of the characters.

The map shows a woman with an "aristocratic look and a "country girl".

These allegories install an horizontal competition that erases the

purpose of freedom : the commitment.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 6 on 25

THE LESSON OF EGYPTIANS

Aristotle proclaims in his « Metaphysics » :

In this way, the Egypt has been the birthplace of the mathematical arts

Thales (-625, -547) is the first of the Greeks to study geometry in Egypt.

Pythagoras (-580, -497) even learn the Pharaonic language and "the

secret doctrines concerning the gods". Democritus (-460, -370) made

his classes there (in geometry). The astronomer Eudoxus (-406, -355)

spent fifteen months with Egyptian Priests. Finally, following his meeting

with the Pythagoreans, Plato (-428, -347) stayed in Egypt with the

priests of the high clergy.

The Egyptian geometry has preceded that of Greeks, and

methodologically it also demonstrates a consistent corpus. One thinks it

with the eyes avoiding the calculation, because it is primarily used to

build art and architecture. Few axioms are enough, the similar triangles,

to which Thales will give his name.

The pyramids of Giza

In 2012, the study of the floorplan of the Giza pyramids was an

opportunity to trace the source of sacred geometry. This plateau offers

a festival of the highest level. The reconstruction line by line of the

corpus of "Geometry with the eyes" Egyptian had just ended, under the

benevolence of Jean-Paul Guichard (IREM-Poitiers). The very new

mathematical arguments were eager to get involved into the concrete of

the composition. As my teaching guide had suspected, the "arithmetic

version" of the plan, carefully prepared by the physicist John AR Legon,

had not revealed all its secrets. The accuracy of this synthesis of

surveys was however an asset : a guarantee of certainty for any reliable

geometrical proposition. An introductory article is available at :

http://www.contemporary-painting.com/Yvo_Jacquier-3_squares_Giza.pdf

The Egyptians have the spirit of simplicity. They make a place to

complexity when it makes sense : about the sacred. The logic of

geometrical figures and their formidable sequence were the evidence. I

then experienced the feeling of a lack...

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 7 on 25

The fourth pyramid of Giza

The three men of the family, Khufu, Khafre and Menkaure were

involved in a large geometric scenario, each line had a referent, with

logic, but one line was desperately orphan. That was impossible to

move it, even less to remove it, and it was staying desperately

anonymous. For common sense, the three great pyramids of the father,

son and grandson were not enough. So I tried to find the exact place of

the pyramid dedicated to the Queen Henutsen, the wife of Khufu and

the mother of Khafre. And all has lit up.

Two lines, led by two principles

related to the golden ratio, φ,

and to the √3, respectively from

the pyramids of Khufu (the

father) and Henutsen (the

mother) cross precisely at the

foot of the pyramid of Khafre

(the son). The two numbers φ

and √3 build a speech about

procreation !

The composition of Giza, its footprint, is an extraordinary source for our

understanding of both the symbolic and the Egyptian culture. In

addition, sacred geometry demonstrates its wonderful unity in time. And

for good reason : it relies on mathematics that it considers as the

language of God(s) !

The resolution of the masculine and feminine principles of the

symbolism began by the study of pyramids. These tell us primarily a

family story. A little common sense was enough to foresee it, in front of

the names of the son, the father and the small son. It was enough to

find the woman to conclude...

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 8 on 25

PART II

PROPORTIONS

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 9 on 25

THE GEOMETRY WITH EYES

Un article de référence avec l'IREM

The study of a particular sign, the letter "G" in the tympanum of

Conques - jewel of Roman art, has triggered a very productive work.

Here is the result of the reconstituted dialogue between the spirit of the

Egyptians and that of the Greeks :

http://www.contemporary-painting.com/Y_Jacquier-IREM-Figure_Tympan_Conques-en.pdf The original purpose of this study was pedagogical. However, the

amount of accumulated properties has significantly strengthened the

status of the root of three, as a proportion, to the point of bringing it up

to the number of gold. This one was the only one, so far, which had

attracted the attention of the scientists.

The corpus of the Egyptian geometry

The reconstitution of the Egyptian geometry and its corpus has required

many years. It was necessary to establish a catalog of very unusual

methods into the demonstrations, and to engage them into the study of

classical figures of geometry. The 3-4-5 triangle reveals a series of

completely inedited properties, absent from textbooks. The golden

appears in four different ways, in the said triangle.

http://www.contemporary-painting.com/Yvo_Jacquier-Geometrie_egyptienne-2014.pdf

The fact to avoid the calculation does not produce less results, but

another way to understand the geometry.

Construction of √3 and φ, the golden ratio

The geometry with eyes of the Egyptians understands the proportions

of rectangles by the angle of their diagonal. Both numbers √3 and φ are born of the same double square by a game

of diagonals. This common origin is full of symbolic meaning. We find

the idea of soul mate, by which an original entity divides to form two

complementary expressions.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 10 on 25

Left : The root of three is obtained by the copy of the two units of the

double-square which become the diagonal of a rectangle, called type ∆.

We construct indeed a right triangle with the measures 1 and √3, with 2

as hypotenuse. And that's half of an equilateral triangle (hence Δ). Right : The golden ratio is obtained by the half of the angle of the

diagonal in a double square. The bisector intersects the horizontal of

the first square to the distance φ. This is also the average of 1 and ∂ -

diagonal of the double-square (=√5).

Reproduction

In nature, the reproduction starts with a division. But in this process, the

two rectangles that we presented do not have the same behavior,

especially by the withdrawal of a square. We are in a geometry with a

grid, and we started with a double square. This square belongs to a

logical development.

The withdrawal of a square from a

golden rectangle produces another

golden rectangle. In biological terms,

this division remembers the cloning.

But this identical reproduction (except

the scale) is not what we expect better

in reproduction.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 11 on 25

The rectangle of proportion √3 is called "type Δ" because its diagonal is

the side of an equilateral triangle. The withdrawal of a square from the

rectangle Δ is a rectangle of type Δ ', of proportion (1+√3)/2. This value is a relative of the golden ratio φ = (1+√5)/2

and it comes out of the belly of the √3. Several simple equations emphasize this relationship :

If H = (1+√3)/2

H 2 = H + 1/2

H.√3 = H + 1

Which remind :

φ² = φ + 1

This value H comes out of its mother's belly and looks like its father.

The Little angel, Cupid, in the famous

engraving « MELENCOLIA § I » by Albrecht

Dürer, is designed according to this

proportion. Its vertical measurement is 5/√3 -

in a setting that is 14 units high. In the composition of the Giza Plateau, the

pentagram that unites Khafre to his father

Khufu puts its center in the middle of a same

type of rectangle (which extends the square of

the son Khafre).

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 12 on 25

The genetic code of numbers

The reduced fractions

Any number can be decomposed as a sum of integers and residues,

with a unique process (in this example, the root of 2) :

http://upload.wikimedia.org/math/d/1/2/d12c854854016d41841cedae13951cd2.png

The sequence of integers of this “step fraction” is :

√2 = [1, 2, 2, 2 etc]

The « reduced » build a set of fractions that develop closer and closer

to their number. They neglect the residual distance with the referent

number. This, to be rational, ie the type of a/b, where a and b are

integers. Each fraction is identified by its place in the series, called rank

n, from n = 0 (rank of the integer part of the number).

The reduced of √3

The primes are in red

0 1 2 3 4 5 6 7 8 9

1/1 2/1 5/3 7/4 19/11 26/15 71/41 97/56 265/153 362/209 10 11 12 13 14

989/571 1351/780 3691/2131 5042/2911 13775/7953

• Synoptic and didactic visual of the reduced of √3 :

http://www.art-renaissance.net/Schools/Inferno_Canto_XXXI/Fractions_reduites-racine_3.jpg

• 9th position in the listing (n=8, precision 10-5) :

265 = 5 x 53

153 = 9 x 17

153 = 1 + 2 +...+ 17

• 13th position in the listing (n=12, precision 10-7) :

3691 is prime

2131 is prime

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 13 on 25

The ratio of the numerators

A second train of proportions is revealed within this sequence of

numbers. Christophe de Cène was interested in the reports of the

numerators (denominators have the same behavior). These ratios

approach even faster their limit, namely :

(1+√3)/2 when n+1 is odd, et 1+√3 when n+1 is even 0 1 2 3 4 5 6 7 8 9

1 2 5 7 19 26 71 97 265 362

2 2,5 1,4 2,714 1,368 2,732 1,366 2,732 1,366

2% 2‰ 10-3 10-4 10-5 10-5

10 11 12 13 14

989 1351 3691 5042 13775

2,732 1,366 2,732 1,366 2,732

10-6 10-7 10-7 10-8 10-8

The formats

The proportions which have interesting behaviors by division or removal

of a square are very few. Let us consider the most common rectangles

of sacred geometry. The rational values, like 4/3 or 3/2, do not produce

anything interesting, we have to turn to the "quadratic" values, the

square roots.

N.B. :

A rectangle of proportion √n,

with n integer, contains n

rectangles of the same type.

In final, three proportions emerge :

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 14 on 25

• The √2 produces, by division or by duplication, the same proportion.

This is the case of common formats, of Type A, that we use daily.

A4 = 21,0 x 29,7 cm A3 = 2x A4 = 42,0 x 29,7 cm

A0 = 84,1 x 1 18,9 cm —> The surface of A0 is de 1m²

√2 = [1, 2, 2, 2 etc] Diagonal of a simple square, side 1

• φ produces by removal of a square, the same format.

(1+√5)/2 = [1, 1, 1 etc] Golden Ratio

• √3 produces by removal of a square, the format (1+√3)/2

√3 = [1, 1, 2, 1, 2, etc] Height of the almond of a Vesica

(1+√3)/2 = [1, 2, 1, 2, 1, 2 etc] Ratio of [Triangle equi.+ Circle]

Note the similarity of the codes of these two proportions.

Thus the child (1+√3)/2 resembles to its father φ = (1+√5)/2 by its

algebraic expression, and to its mother √3 by its genetic code.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 15 on 25

PART III

SYMBOLIC

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 16 on 25

The Vesica Piscis of Venus

Construction

Definition

Two twin circles put their center

on the line of the other.

If the circles are radius 1, the

height of the almond is √3. and

the blue rectangle is 1 x √3.

The rectangle contains two

equilateral triangles, that can be

placed as a lozenge or, as here,

as a sandglass.

The « déïque vulva » of the almond makes from this figure both a

feminine and fertility symbol (see below).

Constructions on the grid

The Vesica Piscis belongs to the chapter of the

equilateral triangles. The basic figure is a circle

diameter 4 on a square 4x4, that provides the

trisection of the right angle - there where the circle

intersects the grid. The hexagram is the

accompl ished geometr ical f igure of th is

development. It consists of two equilateral

triangles that share their centers of gravity, of the

inscribed and circumscribed circles, and their

symmetry axis.

With the Vesica Pisces of Venus, we enter another phase of the study.

In a first approach, we considered the numbers as proportions. Their

geometrical translation was rectangle - the primary version. Other forms

appear.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 17 on 25

The exclusive symbol of Venus

The Vesica Pisces is the symbol of Venus. It has a very ancient origin:

the cult of fertility Palaeolithic (Gravetian Venus). In the composition of

tarot cards, the two circles are diameter 7. This geometric figure is the

prerogative of Venus.

Contrary to the received idea, there is

no representation of Jesus-Christ

which borrows this scheme to build its

mandorla. The centers are still out of

the circles. Sacred geometry respects

somehow the anteriority of Venus.

Left, the tympanum of Conques -

XI/XIIth century - roman art. The

mandorla of Christ is 3 times the

distance between the centers of

their circles.

If this distance is 1, the radius R1 of

the circle is (√10)/2 = √(5/2).

The archaeologist Genevieve von Petzinger has sought the traces of

the vesica piscis in the Paleolithic, but in vain. In fact, the figure of the

vesica piscis is typical of the geometry on the grid, which is a

component of the « Neolithic beam », the set of factors that

characterize the Neolithic revolution (or “Néolithisation”). Men

fundamentally pass from magic to religious/sacred. The figure of Venus

is the example of the emergence of a new thought, as much as of a

change in the expression of the spiritual.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 18 on 25

The Pentagram of March

Summary

The √3 shows its femininity through the study of numbers and shapes.

Beyond the rectangle Δ and its proportion, its referential figure is the

Vesica Pisces of Venus. The proportion Δ’ comes out of Δ by the simple

removal of a square. Δ [= (1+√3)/2] has an algebraic affinity with the

golden ratio, φ = (1+√5)/2. A family resemblance. Δ’ has as much

affinity with the √3. Their genetic codes (reduced fractions) are closely

linked. For all these reasons, the proportion Δ’ can be considered as the

child of √3 and φ.

It seemed logical to link the Hexagram to the √3 and its femininity. Let's

be careful. The star needs √3 to grow, certainly. This does not mean

that it is the expression of femininity. The 6, that characterizes it, is an

indice. This time, we will have to listen the discourse of the "tradition,"

which sees in this figure the symbol of universal love.

Now we will concern ourselves with the golden ratio, we will seek its

geometrical expressions beyond its rectangle proportion. We will use

the comparison with the hexagram, which is a kind of vis-à-vis. On one

hand the fixity and balance of the hexagram, the triangles are pooling

their centers and their symmetry axis. On the other, the mobility of the

pentagram, its orientation and its ability of combination (seen in

chemistry, especially with quasicrystals now Nobel prize-winning). The

orientation of the golden ratio is evident in the simplest expression: the

rectangle. There are two ways to put the dividing lines, and they result

in two spirals. Dextrorotatory or levorotatory - in the manner of 6 and 9.

All the progress of the geometry with eyes have been initiated by a

discovery in the works of art and architecture. The "G" of the tympanum

of Conques is one of the more didactic cases. This time, we'll call at the

helm of the ship one of the three great artists in art of composition, with

Andrei Rublev and Albrecht Dürer, for whom he was the master :

Sandro Botticelli (1445-1510).

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 19 on 25

Venus, or the lesson by Sandro Botticelli

This work has the status it deserves: one of the biggest hits of all time,

mainly due to the emotion caused by this painting. Botticelli not talking

about beauty, it shows us what it is. Even summarized, compiled,

shortened by a series of simplifications, the mastery of this work does

not lose its strength.

The equation of Venus

The three major issues of painting are : What is God ? Rublev literally

evades to his Prince Vasili with "The Holy Trinity". What is figuration?

Dürer responds through a "didactic project" that includes four copper

engravings, called Meisterstiche, and the woodcuts of an entire set of

tarot cards. Before him, Botticelli answers the question: "What is

beauty ? ". The "Birth of Venus" pushes this equation to its perihelion:

more than an answer, there is a revelation.

Presentation of the work

The grid has its origin at the top right.

The frame is 2φ3 in width, and 2φ2 in height - golden rectangle.

The navel of Venus (no one symbol could be more relevant) is at :

- 2 units of the top of the frame and 2φ from the bottom.

The navel cuts the height with the golden section.

- 4 units from the right edge. 2√5 from the left edge.

2 double squares on the right, 2 diagonals of the same figure on the left.

Venus does not stand upright,

the horizon invites to surf and the

shell stands up vertically. In

general, all the background

seems a stage set stretched

behind the characters.

The original title of the work warns us. "Venus Anadyomene" id est

"coming out of the water." The scene takes place underwater. The hairs

and draped testify it; they float in the space with a sort of characteristic

slowness. N.B. The wind, like the light, must choose a direction ...

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 20 on 25

The story/history of Venus

When Botticelli begins his painting,

his model (Simonetta Vespucci) is

dead for ten years, swath in her full

youth. An inverted pentagram, which

covers the height of the frame, pats

her on the shoulder and crashes into

ground. The hope will be born from

an upright pentagram, which will offer

a wide golden triangle...

Simonetta has the milky skin of Greek

marbles. Sandro studies them, in the

collection of the Medici. She will be

reborn in the guise of a Goddess.

Venus is the daughter of Uranus. A

golden spiral is lost in his heaven.

The divine seed will cross the scum

that it will fertilize, like an alevin's milt.

Chloris sprints on this path.

Venus is marine. A Vesica Piscis

takes the mermaid's body, and it offers

all the harmony of the world. The

circles are diameter 5, in which two

rectangles 3x4 are bowing. Their

crossed edges join the angle of the

twin. How can the triangle 3-4-5

manage such marriage with the

Vesica ? This is a lesson of the

pentagram...

This composition is a set of superimposed figures. They bind

themselves with compelling logic, and irrigate the entire canvas by a

real will. Several key figures round off the system. We present only the

most accessible to the speech.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 21 on 25

Two foci wear a set of lines at 9° from

each other. These whalebones keep

the mermaid. The movement of the

characters binds to this frame, and the

closer they get close to the foci, the

more they accord to the fan.

Especially the feet of Zephyr, Chloris

and Hour.

The spiral of the pentagram

concludes the composition like a

signature. The curve enters into the

déïque vulva and goes out at the top.

Ouranos fecundates the foam, not

without having tasted the joys of

femininity.

The metaphysical lessons of this work are now under the responsibility

of a real author. I bring the scales of this wonderful fish, and I wish them

to shine in the light of genuine talent, and by other eyes than mine.

The sexuality of numbers

We suffice to push the investigation on the sexuality of numbers ;

purpose of this article. As they showr in this example, the male and

female principles are not expressing in the same way. The golden ratio

takes two forms, pentagram and spiral, and each time it is moving. It

designates Simonetta, it crashes into the ground, it stalls two rectangles

and directs the toes of the characters. Finally, it shows the path from

the depths of the sky up to the feet of the Belle. The idea of a "passive"

female and an "active" male is not new. In this work, this becomes

literally didactic.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 22 on 25

The orientation and the movement

We addressed this characteristic of the golden ratio - and therefore the

logic of all of its developments. φ is by nature oriented. This trait can not

be separated from its ability to act. There is no movement without

orientation. Basically, any vector has a point, an origin and a direction.

As we have studied and observed, the √3 does not wear this principle.

The Vesica is symmetrical in both directions. Dan Shechtman, Nobel

Prize in Chemistry, in 2011, has faced this problem with the

quasicrystals. Certain metals have a non periodic structure. They renew

their arrangements indefinitely without being broken. These principles

are known by ceramists since ancient times; they use parts of

pentagram, golden or silver, triangles and lozenges, to renew their

paving. So we need to distinguish the roles, and the logic of proportion to which

we summarize often the art of composition does not fit. The geometry

carries an idea of movement, and the multiplicity of the figures

contradicts the "squared conception". The proportions are the

symptoms of a much more elaborate system, as well as the grid is the

support of many figures that are not only the rectangle. The reflection on geometric expressions of the golden ratio, as

masculine, is far from to be complete. But it has began.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 23 on 25

La figure de passage

The pentagram and the Vesica Piscis

The Vesica Piscis produces literally a pentagram.

To reveal a real relationship, a coincidence of figure, he

must grow from the center of the circles of the Vesica.

φ = (1+ √5)/2 = 1/2 + √5/2

The translation is a square (of 1/2) plus the diagonal of a

double square. We defer the total to the vertical height

through the almond.

Now simply extend this bar of golden ratio until

the circle to complete a branch of the pentagram.

The addition of 1 unit by the circle establishes its

center as the point of intersection of the

branches (this is the point of separation

according to the golden ratio). Then, from the

bottom point, we pass through the symmetrical

center, on the left of the Vesica.

These demonstrations need only the axioms of similar

triangles. With this modest equipment, our ancestors were

able to get all the secrets of the pentagram, here in

harmony with the Vesica Piscis.

N.B. : 8 of 10 points of the pentagram are on the circles.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 24 on 25

LE SYSTÈME MASCULIN-FÉMININ

Now the principles of femininity and masculinity should be part of a

whole, in a vast symbolic representation. The Heavenly and the Earthly

must be confronted to the masculine and feminine. Their direct

association is not convincing, we can not say that the Heavenly and

Earthly are marked by one of the two natures. The symbolic offers too

much "against-examples". Except not to consider the consistency of this

noble discipline, we need to find a generic explanation that involves all

cases. There is an example of this kind of problem which is completely

solved : the four elements, and how they have been explained by the

tradition... Two double-principles combine themselves to generate the four

elements. Hot and cold, dry and wet. It should be noted that all these

notions have some sort of affinity, just as the Heavenly and Earthly with

the feminine and the masculine. Thus :

- THE AIR is the combination of hot and wet

- THE EARTH is the combination of cold and dry

- THE FIRE is the combination of hot and dry

- THE WATER is the combination of cold and wet Similarly one can compare the concepts of Celestial/Terrestrial those of

feminine/masculine. The Tarot cards, the encyclopedia of the symbolic,

inform us :

- THE EMPEROR, IV, combination 4.4

is a male representation of Terrestrial

- THE EMPERESS, III, combination 3.3

is a female representation of Celestial

- THE JUSTICE, VIII, combination 5.4

is a female representation of Terrestrial

- THE TEMPERANCE, XIV, combination 3.2

is a male representation of Celestial

These examples are not the only ones, but this first approach opens the

way to reflection, addressing the classic themes of the relationship

between the authority and the law, the one of Justice and jurisprudence

when more generally an ideal of celestial order is confronted to an

earthly reality... Curiously, the shades appear with sexuality.

Yvo Jacquier - Comparative Geometry - The sexuality of numbers 25 on 25