Conference in Charles University | April 2nd 2013Faculty of Mathematics and Physics | Dpt of Mathematics Education

Geometry & Art

Geometry with eyesPart 3/3 - The Renaissance

& Letter to a Vitruvian

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

COMPARATIVE GEOMETRY ------------------------------------------------------------------------- February 2013 -----

A tribute to the great pedagog Jan Amos Komenský (Comenius)

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HISTORICAL PANORAMA - II--------------------------------------------------------------------------------------------------------------------------------------------------------------------------

The apotheosis of Renaissance

« The Holy Trinity » by A. Rublev - « The birth of Venus » by S. Botticelli

Now, we can really study Trinity

and other masterpieces. In Trinity,

two angels bow the one who is

necessarily the Father. The Christ

is at the center in a classical

iconographical position - called

“enthroned”.

VISUAL N° 38 - Sacred Triangle

of « The Holy Trinity » by Rublev

The translation of this picture

could be : From the center of the

incircle, the bisector of order one

shows the Father, the second

(golden) shows the Son, and the

third point out the Holy Spirit.

In the other sense from the points to

the sides, the first brings unity (1) to

human (5), the second brings

inspiration/faith (2) to the Earth (4),

and the third brings Celestial (3) to the

sky (3) in a mirror effect.

The golden point of the inscribed

circle is on the hand of the Holy Spirit.

And the basic rectangle which takes

the measures of the intimus circle (2)

and the golden measure (2φ) is

reaching the top of the frame.

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VISUAL N° 39 - Golden rectangle

A great Masterpiece This icon gives the occasion to precise a fundamental aspect of symbolics : symbols don't

live among geometrical figures which communicate through some “analogical thoughts”.

These images are a real language, with real structures. So, what is a structure ? We could

compare with music. Harmony puts several accords one after one. Sacred geometry puts

several layers one over one. The links between the different figures is like the ones

between different accords. The “Music of the spheres” dear to Plato is not a phantasm.

The Grid is the first step of this particular culture. Then, we can approach each figure,

each internal line with the measure of this grid.

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CERTAINTIES OF THE STUDY--------------------------------------------------------------------------------------------------------------------------------------------------------------------------

The results of a scientific method[ See the Appendix - II « The obsession of evidence » ]

Margins of precision

How to be sure of a

composition ? How to be secure

in the process of study ? Science

brings answers for a part. The

first of them is “margin of

precision”. The more you are

precise, the closer you are to the

truth. We will choose an other

example to illustrate this aspect, a

work which is free of any damage

of restoration. A Self-portrait by

Dürer - dating 1500. He explains

the combinations of golden ratio

with the circle 1/φ2.

VISUAL N° 40

Self-portrait

Albrecht Dürer - 1500

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VISUALS N° 41 & 42 - The golden logic

Comparison of different works This basic attitude has to be mixed with other ones. In particular the principle of

comparison. The main problem with geometry, especially with organized systems, is in

what I call the “Harmonics” (it is a musical vocabulary). A complex of geometrical forms

produces naturally a lot of other forms. But these extra-forms are not in the initial will of

the authors. They come like the heat with the movement.

How to sort the fundamentals from the

harmonics ? The comparison of several works

which use the same kind of structure enables

to isolate the fundamental lines. The hard part

of the system. Hence the name of

« comparative geometry ». The harmonics are

not the same in all the cases, as well as the

works are different. The minimal common

point of certain developments avoid to be

definitely lost in their labyrinth ! In this long

process, I have been surprised that Rublev and

Botticelli were the masters in composition of

Dürer. The tarots are one of the proofs.

VISUAL N° 43 - Homothety of Tarots

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Left - VISUAL N° 44 - The main elements

The homothety produces seven astrologic

triangles, bounded by geometrical links.

One interesting point is in their values -

including the golden ratio. The “cardinal”

planets (Sun, Jupiter, Saturn) have rational

values and the others have irrational values.

The key-figure of a composition The complexity of this art is endless... In

addition of these common points, a very special

figure is often like the mother of all the other in

a work. This figure, with unexpected properties,

is quite the signature of the work. The key.

Painters didn't sign their work for a long time

because of that. For the Ancients, geometry was

the language of God(s).

Right - VISUAL N° 45

Geometrical key of the self-portrait 1500

« The birth of Venus »,

by Botticelli is also a

huge work. We have at

first to show the grid.

Left - VISUAL N° 46

The Grid of Venus

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The triangle 3-4-5 shows the

syntax of its elements, in

particular the golden bisector

coming to the navel of Venus.

Left - VISUAL N° 47

The triangle 3-4-5, queen figure

Below - VISUAL N° 48

The fans of Venus

Then a fabulous construction

combines two beautiful fans,

two sets of lines separated by

the angle of 9°. Second

element : a vesica piscis of two

circles diameter 5. Third

element : two rectangles 3x4

inclined of 27°. Fetish number

for Pythagoreans. It means 4

triangles 3-4-5. And the key is

that one line of the rectangle

finds the point of a symmetric

triangle. Very special property,

because it is due to the “fan of a pentagram”.

Left - VISUAL N° 49

The key of Vesica

The head of Venus is outside the

almond. Vesica piscis non caput.

Piscis primum a capite foetet. VISUAL N° 50 - link :

Venice : The pescheria of Rialto Ryba smrdí od hlavy.

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Right - VISUAL N° 51

Venus is a mermaid The original title goes in this

sense (Vénus anadyomène)

In the case of « The birth of

Venus » by Botticelli, one key of

the work is hanged to a kind of

shell called Cyprée in France (//

Cyprus). Like Dürer in his self-

portrait, Botticelli speaks about

the golden logic, but this time by

angles. This is a second key in

the work.

Left - VISUAL N° 52

Golden circles

Right - VISUAL N° 52bis

Golden development

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At last, one concrete aspect helps

to secure the proposals. The

marks of composition. We have

seen a first example with the

rectangle of Rublev. These

particular objects have no own-

meaning, no interest for the

decorative, nor for the

understanding in a narrative way.

In spite of the real effort that they

need to be produced, these objects

are definitely useless. So their

importance is necessarily apart,

above the work, in the category of

titles, commentaries or plans.

Here at first the didactic square of

Dürer's self-portrait.

Left - VISUAL N° 53

The didactic square of Dürer

Big dealsAll is not solved. A wide field is still

under study. Here is a typical problem

in sacred geometry. We come back to

Trinity. In this case, the diameter of a

circle has a value close to √1+ √2+ √3 ≈

4,1463 and π +1 ≈ 4,1416. The

difference is 1,1‰; and it is swallowed

by the thinest line in a drawing. So,

who knows ?

VISUAL N° 54

Circle of π+1

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The circle finds a second place,

by a process too much

complicated for being exposed

here. The triangle takes the slope

of the scepter of the Holy Spirit.

VISUAL N° 55

Inclination of the scepter

The subject of measures, in the

concrete reality and on the works,

is a file in the file. Nowadays, we

confuse two conceptions in one.

But for the Ancients, the values of

sacred geometry (spirituality) are

not the same like in the life

(science). There is an appendix

about it : « Letter to a Vitruvian »

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HISTORICAL PANORAMA - III--------------------------------------------------------------------------------------------------------------------------------------------------------------------------

The Didactic Project of A. Dürer

Artworks as a true manual

In his self-portrait, Albrecht Dürer insists

on a precise engagement : he will never

speak about his practice, his knowledge

and his conceptions of sacred geometry.

Left - VISUAL N° 56

Dürer's engagement

Right - VISUAL N° 57

Circles of golden ratio

Look at the light on his eyes, laid on

the circles of radius φ...

This work has been painted in 1500, in

his new studio, between his two trips to

Venice (1495-1505). The top level of

the geometry makes the encounter of

Dürer with his master Botticelli,

plausible from his first voyage in Italy.

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However, the didactic project will begin few years after, in 1504, with a first of four

engravings, which are called Meisterstiche : « Adam and Eva ». The three other

engravings are : « The Knight, Death and the Devil » dating from 1513,

« MELENCOLIA § I » and « St. Jerome in his cell », both dating from 1514.

VISUALS N° 58 & 59 - The Meisterstiche and the tarots of Nicolas Conver

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This surprising set of images complete the base : the model of tarots of Marseille, that

Nicolas Conver will continue to copy in 1760 !

The amount of accumulated evidence about this project is purely colossal. And this body

of evidence exceeds the surprise caused by this revelation. The real interesting fact is not

in the paternity of Dürer about the tarots of Conver. The historical and mathematical

particularity is in this unique plan : a didactic encyclopedia of symbolism. Due to the

project of Dürer, we can reconstitute step by step the rules of sacred geometry.

Just images - Meisterstiche (with Melencolia) The images build a real language, that could concern linguists. The four engravings of

Meisterstiche are put one over the other, in one same rectangle. So we look through and

the subjects don't move : they just mix themselves layer on layer.

St Michael from Melencolia becomes a

Griffon in the cell of St Jerome. Left - VISUAL N° 60 - The Griffon

By the way MELENCOLIA § I

is the anagram of

LEO IN MIC§AEL

Below - VISUAL N° 61 - Anagram

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And the date 1514 refers to the cards of Tarots :

Devil and St Michael. Left - VISUAL N° 62 - The date 1514

The horse of the Knight mix with Adam

on Melencolia.

Right - VISUAL N° 63 - Centaurus

Adam takes the pose of contrapposto in

the cell of St Jerome. Left - VISUAL N° 64 - Contrapposto

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St Jerome, who is the patron saint of the

translators, takes the compass of St

Michael to write with sacred geometry. Left - VISUAL N° 65

The compass of sacred geometry

Love-love with Cupid

Right - VISUAL N° 66

Love-love

So much feeling. This refer to the « Ship

of Fools » that Dürer illustrated by the

style. But behind the joke, there is the

perfect explanation of human

reproduction ! Left - VISUAL N° 67

A certain feeling

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Cupid prepares the war ! VISUAL N° 68

Cupid with sword and snake

The precision of these meetings is

impressive. Here, the sword and

the snake show their exact

appearing part on their respective

engraving (Knight, and Adam).

Just images - Tarots Below - VISUAL N° 69 - Emperor and Judgement

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The cards are also built to discuss between them. A very interesting image mixes the

elements to give this message : Angels tell to humans that they have to pray for dead

people, so their soul goes up to God the father. Look how the hands are touching the legs

of the eagle, as well as the trumpet in its neck.

All the cards discuss with the others in the

same way. The first dialogue is into the

couples. N.B. : Temperance and Justice are

the two representations of St Michael. Left - VISUAL N° 70

Pope and female Pope

Couple of spiritual power

The strong points of this meeting are the

contact of the King with his symbolic son,

and the hand of the Queen caring of the

sexual parts of her husband (same circle).

Right - VISUAL N° 71

Emperor and Empress

Couple of temporal power

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These characters are connected at a point, a

very little circle. it is on the wall of the Force.

This point comes inside the object (maybe a

circle) standing in the hand of the Baleleur. Left - VISUAL N° 72

Bateleur and Force

Couple of knowledge power

These are the two representations of St

Michael. They need Melencolia to reveal

their relative position, more precisely the

Omega of the scales.

Right - VISUAL N° 73

Temperance and Justice

Couple of St Michael

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They need too the same element : the omega of

the scales to find their relative place. Left - VISUAL N° 74

Devil and Maison Dieu

Couple of religion

Right - VISUAL N° 75

Moon and Sun

Couple of rhythm

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As a conclusion, for this little show.

This one is not a couple : they are just partners... Left - VISUAL N° 76

Devil and Temperance

Tantric result

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Just images - Tarots on Melencolia

VISUAL N° 77 - First cards on Melencolia This is one very easy proof of Dürer's Project. Melencolia is a backdrop for the cards of tarot,

with their real size. The middle of the double

square is on the horizon. The cards make a

show on the engraving. At first : the number 7,

8 and 9 find their place on the picture. This presentation could be an esoteric

commercial show ! But not : it is for sure a step

that Dürer puts underneath our feet. This confrontation needs other elements - more

precise. Especially the scale of tarots, available

for the both compositions. Tarots and

Melencolia are linked by the golden ratio !

Right - VISUAL N° 78

The farandole of cards

VII - Chariot

VIII - Justice

IX - Ermite

Look at the triumph of the prince

(Chariot) under the rainbow, the

Justice with her weight scales, and

the Hermit holding the sandglass.

The symbolic of three successive

cards is fitting perfectly with the

one of Melencolia : rainbow for the

Chariot, scales for the Justice,

sandglass and magic square of

Jupiter for the Ermite.

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Left - VISUAL N° 79 VI - Lover on Melencolia The Lover takes his place

lower, and the two images

of Cupid join themselves.

The frame is aligned to the

lowest line of the

sandglass.

The Compositions

VISUAL N° 80 - Great square of Melencolia All the composite images that we have seen

are the result of a great composition. Each

work has its own frame, and more : the

compositions of tarots and Melencolia are

crossing in a precise way. The composition of Melencolia begins by a

great square respecting two marks on the

picture : the center of the sphere and the '4' of

the date in the signature. An article explains this real phenomenon :

http://www.jacquier.org/IREM/

Yvo_Jacquier-Composition_Melencolia-I-

en.pdf

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Left - VISUAL N° 80bis - The Greek Cross

This great square is divided by the golden

ratio, and the first result of this division

produces a circle, fitting with the rainbow.

Below - VISUAL N° 81 - Triangle of Zeus

A triangle type 3-4-5 take the width of the great

square, as the measure of its hypothenuse.

Left - VISUAL N° 81bis

Placement of the triangle of Zeus This triangle, called “triangle of Zeus”, turns to

get a special propoerty : The side ‘3’ becomes

the diagonal of a golden rectangle.

Then the triangle puts the lowest angle on the

horizontal (middle of the great square). Second

condition : the center of the incircle is on the

diagonal of this great square.

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Under these conditions, the golden rectangle of the

triangle (over the golden bisector, here in red and

yellow) has one of its diagonal strictly horizontal. Left - VISUAL N° 82

The “classical” golden rectangle of Zeus

And the other diagonal becomes the one of

an upright double-square.

This double-square is the one of tarots.

Right - VISUAL N° 83

The double square of tarots

Left - VISUAL N° 84

The personage of the World

Card XXI of the tarots

The mandorla fits perfectly with the rainbow.

And of course with the circle of Melencolia.

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The great links of the compositions

Left - VISUAL N° 85

The astrological composition of tarots A very strange coincidence appears :

The composition of the tarots and the one of

Melencolia meet exactly. Two important circles are

the same for the both.

• Concerning the tarots, on the scale of the seven

astrological triangles, the intimus circle of the Sun,

and the circumscribed circle of Jupiter.

N.B. : this composition turns of 45° to be placed on

the double square - as the picture shows • Concerning Melencolia, the circle of the sphere

and the circle of the rainbow. This second one is

coming from the great square, in dividing the side

of the great square by the golden ratio.

VISUAL N° 86

Proof of the sphere

We can compare the both

compositions, placing the

one of tarots at the scale

of the sphere.

The sphere becomes then

the intimus circle. and the

circumcircle of Jupiter

becomes the one of the

rainbow.

In clear terms, the two

compositions have two

circles in common.

And the golden bisector

shows the direction of

one ladder's upright !

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VISUAL N° 87

Central disposition of tarots And that is not all. We can place

the composition of tarots on one

apex of the polyhedron, the one

which is the left point of the great

triangle of Zeus. Then the golden bisectors of the

Sun and Jupiter are taking the

precise space of the ladder ! The hypothenuse of the sun shows

the middle of the comet. At last, the high point of the

triangle of the sun is the crossing

point between the ladder and the

polyhedron. A real structure.

This new place is available for all the

cards, especially for the Death ! Look at

the right foot, that the symbol of fertility

has cut.

VISUAL N° 88

The Arcana without name

Why the names of Zeus and Jupiter ?

The measures of the great triangle of

Zeus (Melencolia) are φ times the ones of

the triangle of Jupiter (tarots) !

So, the sphere is a diapason with its

diameter 2. The circumcircle of Jupiter is

diameter 35/6 and the circumcircle of

Zeus is (35/6).φ = 35φ/6.

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APPENDIX------------------------------------------------------------------------------------------------------------

Letter to a Vitruvian Alberti Since 1420, the italian architect Leone Battista Alberti (1404-1472) rediscovers the

writings of Vitruve, a 1st Century colleague of him. However, the first edition of « De

Architectura » will be printed in Rome in 1486 : the year when Venus was reborn in the

hands of Sandro Botticelli (1445-1510). As very great builder, Alberti gains his celebrity

at Florence in his late years, and he meets the master of the place : Lorezo the

Magnificent (Lorenzo il Magnifico).

Vitruve The profile of this typical Roman is between architect and engineer. He is very practical

and secure in his taught, and full of common sense. Don't wait esoteric or spiritual

considerations from him : he is a legionnaire of building. The myth of “golden spirit

revival” about him is definitely an apocryphal story. We are at the heart of Renaissance :

every revolution needs its heroes and people prefer the dead ones because they can speak

more easily on their behalf.

« De Architectura » There is no mention about golden ratio in the Book of Vitruve, which is considered by

mistake as the Bible of a new age for composition. On the contrary, Vitruve rationalizes

the problem of beauty in a scale of proportions very practical, and without any

preoccupation about symbolic. By the way, the perspective system emerges during

Renaissance, and these skills will bring materialistic effects. So we can assume that the

“Renaissance of Vitruve” has a literary inspiration about freedom in a very realistic

background in the reality. This is a real imposture, perfectly dressed with these beautiful

lies that people call dreams.

Two kind of measures The measure has two different applied fields : first the concrete reality of the nature, and

second the objects of representation in the artworks. Nowadays, we make a confusion

between them, we think to solve with only one perspective system. But for the Ancients,

sacred values of geometry (belonging to spirituality) were not the same like the measures

in the life (leading to science).

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The lesson of Sandro Botticelli

Left - VISUAL N° 89

Venus measurements Remember the visual N°

49 in this article, page 6.

The size of the drawing

on the canvas. Venus is

close to 5√3/2, and

Chloris gives the last

correction to this value.

Below- VISUAL N° 90

The height and the head

The grid on the right is the one

of the total painting, the

referential.

• The height of physical Venus

is, according to this unity,

exactly H = 1+ √2 + √3 !

• Then the grid of Venus, on her

measure, is 7 tiles height on 2

width

• Under that conditions, the

head of Venus is r= H/8, as the

circle shows. On that point,

Botticelli agrees with Vitruve.

VISUAL N° 91

Venus vs Vitruve The engraving is from the

book « The proportions of

humain body, Architecture or

art of good building by Marc

Vitruve Pollion », by Jean

Martin, 1547. The little difference is just a

matter of beauty...

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VISUAL N° 92

The face of Venus

Vitruve says that the face is 1/10

of the total height H. But Botticelli

uses the ratio 1/9.

...

So, about the same size of the head

(1/8), Sandro, the master of

beauty, affirms that the face of

Vitruve is too short !

Moreover, the opposition is quite

religious. Venus doesn't respect

even anatomy :

VISUAL N° 93

Venus is she drunk ?

She doesn't

stay upright.

Her neck is

longer than

possible. Her

left shoulder

is going

down and her

arm becomes

the one of

Popeye. In

spite of this,

or more :

because of

this, the

woman is the

embodiment

of the grace.

The geometry of Venus is one of the most sophisticate of all the times :

http://www.art-renaissance.net/Botticelli/en-Presentation.html

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