The Compositions of « MELENCOLIA § I » and Tarots

Transcription

A new understanding in Art History
__________
__________
Mathematics - History of Art - Esotericism - Visual Arts
The COMPOSITION
of MELENCOLIA
Article I
From the engraving to the cards
----------- Yvo Jacquier ----------------------------------------------------------------------------
COMPARATIVE GEOMETRY
------------------------------------------------------------------------------ December 2012 ------
Yvo Jacquier - Comparative Geometry - Composition of MELENCOLIA (Dürer) - Article I
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INTRODUCTION
This text is not the translation of the french version : I am not able to do
it in a right way. But the images are the same, and they contain 100% of
the truth.
Comparative geometry
The comparative geometry, as a discipline, highlights the geometry of
old masters (up to Renaissance), including its relationship to numbers.
The famous grid of the cathedral builders is the primary tool for
converting forms into digital values. This is the first step to translate into
human language the abstract lines of the composition.
Two notions are at the center of the study.
1) The margins of precision, without what no observation worth
something for science.
2) The notion of structure, which is far away of the poetic conception
that one find in the usual books. A structure is not the sum of simple
geometrical figures; not more the repetitive multiplication of the same
form. This is a consistent set of geometric figures, strongly linked by
remarkable properties. These links hold the symbolic sense of the
artworks. And the properties are the genuine intelligence of images, as
well as the signature of the authors. Music has its harmony, painting
has its sacred geometry.
For the moment these two notions are out of the practice about
painting. At best, measurements are gathered, and the logic remains
accountant until the end. Calculation, without the soul and the spirit of
geometry. For example, the golden ratio is always the result of
considerations about numbers. For Ancients, it was a geometrical
phenomenon. When you take away a square of a golden rectangle, the
remaining one is in gold too.
One method gives its name to comparative geometry : one compare the
works from different epochs, different authors up to reveal common
structures. This practice is not changing its tools like fashion or like war.
Since its early paleolithic beginning, the grid remains the support, and
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the figures don't change their rules with the weather. The contemporary
opinion of “subjectivity” in art is not fitting with that. In music too, people
have their opinion. But very few musicians pretend that a triumphal
march could be played in Re minor ! They know the structures of music
when painters and historians ignore that painting has the same.
This culture is deeply homogeneous because of the nature of geometry
- especially with eyes, on a grid. And second because of its translation.
The key of symbolic is the bridge between forms and numbers which
allows to understand the most abstract lines...
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PRESENTATION OF THE WORK
The « Dürer's Project »
Thus Rublev and Botticelli are the ancestors of Dürer. He uses the
same systems, and makes many references to their lessons.
Melencolia belongs to a set of four prints called Meisterstiche. This
historical concept is now extended to Adam and Eve (1504). The other
two prints are The Knight, Death and the Devil (1513) and St. Gerome
in his cell (1514). These four works have a similar format, that has been
noticed as unusual.
The tarot of Marseille, specifically those that the cartier Nicolas Conver
continues to reproduce faithfully in 1760, have been designed with the
Meisterstiche. The whole has the mission to present the will of an entire
civilization, a culture of image that has lived since Pythagoras (for the
Western stream), until the Renaissance, period of its apotheosis. This is
a didactic device, designed to open gradually. Each step delivers
information that helps gradually to learn the art of composition of the
Ancients.
A language of image, a real one, reveals itself. These tarot cards are an
ecumenical encyclopedia of symbols. They are the synthesis of all
spiritual movements, religious and esoteric, accumulated over a long
history. From Vedic India to the icy norths of Europe, since the ancient
Celts to traditional Christians, no influence escapes to this universal
crossroads of the symbolic.
Dürer is precise, up to anticipate the errors of the researcher !
The last, and most recent scale correction has been displayed on the
heaviest file of engraving Melencolia (1514), with the value of 262.510
pixels per unit, instead of 262.222 - established a few years earlier. This
little revolution puts a difference of 0.288 pixel per tile, 1.1 ‰. A new
figure, recently revealed, gave the secret of two sides of the work, with
this ultimate rectification.
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The most famous work of Renaissance
MELENCOLIA § I
Albrecht Dürer (1471-1528)
Gravure au burin - 1514
238 mm par 187 mm (± 0,75 mm)
British Museum de Londres, Royaume Uni — Musée Jenisch, cabinet cantonal des
Estampes, Vevey, Suisse — Cabinet des Estampes et des Dessins, Strasbourg, France.
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THE GREAT SQUARE - GREAK CROSS
From the circle to the square,
then the rectangle, then the circle
Two marks of composition:
- The center of the sphere
- The arrow of the signature
on the right side (in “1514”)
In inverted sense one can read « † IRI »
Other marks :
- The shadow of the square angle
on the polyhedron
- The 3-4th line in the magic square
All of these initial developments are
re-adjusted later through a figure even
larger than the large square: it will
fetch both edges of the engraving.
The accuracy will become one ten
thousandth (relative / height).
Now, we will share the square with
the golden ratio :
φ = (1+√5)÷2 ≈ 1,618 034
φ2 = φ + 1
Look at ω !
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The diameter of the circle is equal to the width of the square, divided by
the golden ratio - φ. It fits perfectly with the rainbow. Look at the
diagonal, the knee of St Michael, his compass etc. !
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FROM THE SQUARE TO THE CIRCLE
Basic vocabulary. How to put a rectangle in a
square ? After the golden rectangle, Dürer
choses that the diagonal has the width of the
square. The values ‘3’, ‘4’ and ‘5’ are
provisional.
The rectangle takes several position at the
center. and two triangles share the same
inscribed circle.
All the positions together draw a rosace and
the typical 7 of Jupiter is in one almond. The
surface of its triangle is 7 (the one of Sun is 6).
π/7 belongs to Mercury.
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Right : the yellow circles are build with φ.
The green lines are the bisectors of the
triangles coming from the square without φ.
But for of them fit with the circles.
Geometry says what calculation ignores.
Left :
all the rectangles (4) with one
diagonal vertical or horizontal. 8
triangles.
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THE SCALE OF MELENCOLIA
Geometry with eyes
The main properties of the triangle 3-4-5 are :
1=
Radius of the inscribed circle
(called intimus)
2=
Diameter of the inscribed circle
3&4=
Sides with right angle
5=
Hypothenuse, diameter of the
circumscribed circle
6=
Surface of the triangle
7=
Slope of the hypothenuse when the triangle turns of 45°
like in tarots.
2.φ = Distance from the vertex to the inscribed circle on the bisector 2.
See the picture
- The bisectors are the natural diagonals of a single, a double and a
triple square.
- The bisectors of order 1 and 3 are the natural diagonals of a golden
rectangle.
Composition of tarots
The composition of tarots comprises a system seven
triangles linked by a homothety (they are similar).
Everyone is a star, from the moon up to Saturn.
The set of figures is built on a grid that rotates finally
of 45 °. The double square that serves as a
framework for cards is shown here in a spot.
Two triangles are useful for understanding
Melencolia :
The Triangle of the Sun - its inscribed circle d = 2
The triangle of Jupiter - its large circle
D = 35/6
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Perfect meeting, perfect accord
The two circles of tarots that we have seen
have exactly the same ratio than the two
explicit circles of Melencolia. The sphere and
the rainbow are like the Sun and Jupiter...
So wee can try to accord the two composition
and through them, to accord the works... Ô
surprise ! The golden bisector of the Sun
shows the upright of the ladder. Miss Chance
is not so much talented !
This meeting of the two composition fits with
the reality : the real works, engraving and
cards, have compatible formats. This is the
reason of why the Meisterstiche have an
unordinary size.
The magic of angles
The half of the great angle of the diagonal in a double square
is equal to the small angle of the diagonal of a golden rectangle.
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THE GREAT TRIANGLE OF ZEUS
We can measure all, especially the great triangle that we study. It is the
triangle of Jupiter multiply by φ. So it is devoted to Zeus !
Measurements of the great Triangle of Zeus
d
= (7/3).φ
≈ 3,775 413
‘Sin’
= (7/2).φ
≈ 5,663 413
‘Cos’
= (14/3).φ
≈ 7,550 413
D
= (35/6).φ
≈ 9,438 413
S
= 7.φ
RD
= (7/3).φ2
≈ 6,108 413
Inscribed circle
Small side
Medium side
Circumscribed circle
Surface
L. golden rectangle
Fixed on the cross formed by two lines of the great square, here is the
golden rectangle of Zeus (which takes birth on the golden bisector with
the circle inscribed). It is presented in a horizontal position - before
taking off with the triangle, and it shows curious affinities with the
polyhedron!
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The strange position of the triangle of Zeus
The inclination of the triangle of Zeus
1) The golden rectangle, in
its favorite position with
the vertex and the circle is
particular : one diagonal is
horizontal and the other
th e n a t u r a l o n e o f a
double-square.
2) The angle β of the small
side of the triangle with
horizontal has a tangent
equal to φ.
Translation of “triangle de Zeus”
The left vertex is on the middle of the great square and the center of the
inscribed circle is on the diagonal of the same great square.
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Bis repetita placent...
The frame of tarots
Dürer efforts to promote a double
square : the one of tarots ! We know its
measurements and we need to
understand its exact place. Each card
has a favorite place on Melencolia. This
place is for Le Monde, number XXI,
which is on the top of the hierarchic
pyramid...
There will be an other place, available
for all the cards. But let us see the
World at first.
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The apparition of the World (XXI)
Faced with this show, it is
understood that the engraving and
the card game called "Conver"
were designed as a coherent
whole : the « Didactic Project of
Dürer ».
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A good point
& a question
Golden angle: We
know now why the
character of the world
lifts his foot!On the
other hand, there is an
angle of ~0,7° between
the centers of the
circles, drawing the
mandorla.
The two "levels" cross on the back of the hand, specifically around the
circle that the character exhibits. Does he speak about tides ? It fits with
the horizon. Dürer is so precise that the slightest anomaly takes the
form of a guess.
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THE GREAT RECTANGLE OF ZEUS
The citation of π
There is a rectangle between the
left side of the great square, its
medium horizontal, and the top
and right side of the engraving.
- The proportion is 4x3.
- The length of its diagonal
equals the circumference of the
circle of Zeus !
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THE CENTRAL PLACE OF TAROTS
The grid
Contact :
Vertex of the polyhedron = Vertex of the great triangle of Zeus.
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The triangles of the Sun and Jupiter are showing once more the ladder
by their bisectors. The tangent point is on a golden line of the great
square. Below : The meeting points of the grid with the triangle give a
figures with the surface of the frame of tarots.
The Ankou
Figure of reference
The character of the Nameless
Arcane is the only to confront openly
the gaze of the Archangel...
In this picture, all the elements of the
composition combine like by magic. A
golden rectangle is born on the
diagonal that is the short side of the
triangle of Zeus, and adds a square
on the right. Its green diagonal bisects
the large golden triangle.
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The center of the circle of Zeus (inscribed) is on the line which encircles
the head of the Ankou. On the opposite side of the framework of the
tarot, the diagonal of the first golden rectangle crosses the bisector of
the right angle (triangle), and the edge of the rectangle of Zeus.
Appointment if he is on the circle of the scythe !
Christophe de Cène notes that a line joins the center of the sphere in
the middle of the foot of the hourglass with an angle of 36 ° from the
vertical. It passes the center of the sphere, through the hole in the
millstone, and extends to the corner of the magic square to the top of
the bell. You can still add the belt of Cupid (maternal inheritance) to the
inventory! A conjunction of planets would be in this symbolic assembly,
rare enough to bring a date. Could it be the exact moment ends this
article?
This visual is the final bouquet of an article which provides little space
for interpretation. However you can build few frames of reflection and
meditation. About religion, different layers overlap. The paleolithic
symbol of Venus (black triangle) cuts the foot of Death. Jupiter is pipped
by Zeus, Ankou calls all the Druidic culture, and through St. Michael,
Dürer describes the basics of Catholicism.
Science has the place that it can in 1514: that of a foreshadow. Through
the quality of mathematics that the master develops in his compositions,
but also the themes underlying his arguments : an hourglass counting
the time, a scale for weighing, a compass to measure distances, a bell
to give La, and even candle to evaluate light (the folds of the garment of
the Archangel looks like this material). These measuring instruments
are tools for comparing more than the materials that science will need
to progress.
What does the Archangel think about all this ?
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The Compositions of « MELENCOLIA § I » and Tarots

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