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Art and Mathematics

THE GEOMETRY OF MASTERS

The civilizations of image



Capital letter A, of the word Art. rt is the asylum of emotions, a favoured space for feelings. Music is considered as a universal language. It is also extensively didactic. An instrument cannot touch one's soul if the player doesn't vibrate with it. BUT if this instrument is not tuned or else if the notes don't respect any harmony, the musician can shake as much as he wants, he will succeed at best to irritate the audience. The music needs a minimum of intelligence to be understood, and without it, our ears become closed very soon. Is it a fact or an opinion ? Even contemporary, the public doesn't listen a so-called music which ignores harmony.
Up to Renaissance, the practice of Painting was like the one of Music - from the neolithic dawn. The language of image was structured, as it was the case in poetry. The development of printing broke definitely this family complicity. The text took soon the leadership, and the image dived into the statute of illustration. Now, when we speak about harmony of a painting, it concerns more the subject than the work of art. This constitutes a definition of materialism and thus this question could arise : Can we confuse, for example, a beautiful painting of a woman and a painting of a beautiful woman ? (And in which is the real opinion?)
Symbol of geometry in art. The writing emerged in Mesopotamia during the fourth millennium BCE. In spite of this, men will continue to think with their eyes as well as with their tongue up to Renaissance. The printed book, with other factors, will alert the guilds of artists in northern Italy. Their way of thinking, inherited from antiquity (especially Egypt) through Byzantium, was condemned to disappear. They decided to put the essential in a testament.

Master in sacred geometry. This was already the project of Byzantines, but it needed a technic such as engraving to multiply the copies and secure their survival. Alas, Byzantines didn't know any process of reproduction. Albrecht Dürer has been the great achiever of this mission, with four engravings called Meisterstiche and one set of cards now called « Tarots de Marseille ». This « Didactic Project of Dürer » has been like a floating bottle on the sea for five centuries. But this flacon is a champaign bottle !

Proofs and opinions



Capital letter O, of the word One. ne enters the zone of proofs. It means that the beginning of this travel through the epochs of art will avoid opinions to the profit of established facts.

One analysis is clear : up to now, there is not any reliable publication about sacred geometry. For proof : the basic figure of sacred triangle, the modest triangle 3-4-5 wears four times the golden ratio, φ.

The golden rectangle of the sacred triangle in the composition of Vladimirskaya. The golden proportion appears in four ways into the lines of the triangle, but there is no trace of these properties in the « however so talkative writings » about it. These properties are facts, not contemporary opinions, and when you ignore these properties, you cannot pretend to know anything more.
Link to the articles about mathematics in art. The first step has been the restitution of the mathematical bases which bear sacred geometry (link on the left). This old practice on a grid avoids the calculation. This, for very good reasons. First, the mathematics in question were born before calculation, which begins with writing (try to calculate without any notes...). Second : these are special skills for artists (contemporary painting) and architects. Visually minded.

We cannot justify the magic of the golden ratio for Egyptians with equations. However a lot of « Big Hats » develop high concepts with this fundamental error. In fact the Egyptians didn't understand the golden ratio through numbers. They built its geometry in a simple way (other finding). The first definition of golden ratio is purely geometric.

The first lesson



Capital letter T, of the word The. he first lesson doesn't concern painting, and it is Egyptian. This doesn't mean that the Mesopotamian artists were awkward. But the Sumerians and later the Babylonians were obsessed by the calculation, and they built another discipline, maybe ancestor of Kabbalah. The fact is that their high skills in numbers remain the lessons of geometry, but at the moment we are not concerned by the mysteries of Plimpton's tablet N° 322 - a contemporary name, isn't it ?
Article about the composition of Giza, ground plan.
Plateau of Giza - Pyramids
Composition of the ground plan.
2500 years BC, the precision of geometers is already fantastic. In the early Antiquity, the « rope stretchers » are in a particular category - harpedonaptai in Egypt, Sulbakas in India. This precision brings a double proof. The accuracy of the realization meets the exactness of the plan.

This article belongs to a new « generation », in which the didactical aspect takes the control of the text. For years, this has not been possible. The preoccupation of the facts and the proofs was leading the progress. The conception of this website is also the occasion of a great progress.

Three Masters - Three works



Capital letter D, of the word During. uring twelve years of research, three painters have emerged from the main stream. The practice of sacred geometry shows a great unity in the time as well as in the space. The figures combine their properties with a mathematical logic, so it would be difficult to escape to their dogmas. The triangle 3-4-5 is dimensionally stable and incorruptible. It doesn't invite the student to inspirational drifts !
From the early neolithic up to the Renaissance, not a single major work of art (painting and architecture) escapes to the process of creation involving the composition with this « geometry with eyes ». As we saw before, there is no serious publication about it : the main properties are missing in the contemporary writings. Therefore, it was impossible to identify into the works what we ignored totally.

It becomes ridiculous to argue that there are other approaches than mathematics when we discover that these skills are everywhere in the works. A so great effort can not be considered as the frivolous hobby of the authors. The History of art imposes a dogma that we could resume as : « Art is subjective because it is subjective; for proof it is subjective », but a lot of people feel nervous in front of that. If the artists imply their subjectivity in their creation, the historians should not usurp their place.

The Musicology requires very strong competencies in the domain of musical structures, and these skills are comparable to the ones of geometry. This type of investigation is not a choice : it is a necessity.
This framework being laid, we can address the greatest masters of Sacred Geometry. The level of their practice is worthy of the aerospace. They are distinguished in particular by their composition keys.

The composition of Vladimirskaya, by Andreï Rublev.
Andreï Rublev (1360-1430)
« Vladimirskaya II » (circa 1400)

This Russian Icon is an adaptation of the Byzantine model (dating from 1130).

Article about the composition of Vladimirskaya by Andreï Rublev. More complete and recent.
Pdf about « Vladimirskaya II »
More complete and recent article. The « first stone » of a studio which aims to restitute the original models.

Article about the sign of Rublev in his icon The Holy Trinity.
The sign of Rublev
« The Holy Trinity » (1420/28)

In french at the moment. Less magistral than the G of Conques, this rectangle is however a precise invitation to study the composition of the « Holy Trinity ».

The study of a drawing of Botticelli, « The Giants chained to the edge of a well »,
                        illustration for the « Divine Comedy » by Dante.
« The Giants chained... » (1480/95)
Illustration of Dante by Botticelli

« The Giants chained to the edge of a well » by Sandro Botticelli belongs to a series of drawings illustrating the « Divine Comedy » by Dante Alighieri (1265-1321). The artist took fifteen years to accomplish this tribute, which foreshadows the Didactic Project of Albrecht Dürer.

Article about the sexuality of numbers, with elements of the Birth of Venus by Botticelli.
Sandro Botticelli (1444-1510)
« The Birth of Venus » (1485)

This article presents the latest topic under study : the feminity of √3 and the masculinity of φ. You will find, inside this wide analysis, the main construction elements of the Birth of Venus.

Article about the compositions of Melencolia by Dürer and the Tarots by Conver.
Albrecht Dürer (1471-1528)
« MELENCOLIA § I » (1514)

Like harmony in music, the practice of Sacred Geometry has its top school. The composition of MELENCOLIA § I by Dürer meets the composition of Tarot by Conver on the same grid of measurement. The famous grid of the builders of Cathedrals.

Methodology

Presentation of the Comparative Geometry, a new understanding of history of art.
Many « a priori » have polluted the History of Art during centuries. The depiction of Giorgio Vasari by Stendhal shows a very bad beginning ! The founder of the discipline was more at his place in the salons than in a painting studio. At the beginning of this study in 2002, the goal was to understand painting more than to understand the historians of art. Ten years later, the question of their methodology arises.

Can we speak really about a methodology, when the only objective elements are biographical. As soon as we enter the works of art, we undergo the diktat of subjectivity. And this subjectivity is not the one of the legitimate authors or the artists, no : it is the one of the historians ! The joke doesn't stop there : at the end, we wonder if the artists know, if they understand what they do. The intelligence of their work comes with the bright light of interpretation !

Best of all, if the imagination of the « interpreter » is not enough, one appeals to the spectator ! Like an echo, a large part of the contemporary art is based on the same principle. The artists propose a mirror to the public, on which anybody can guess what he wants. No risk, especially the one of creation !

Capital letter A, of the word As. s luck would have it, these considerations were far from the study in 2002. The subject was painting, the question was composition, the preoccupation was evidence. And this is still true. We could resume all the questions in one : « How to be sure of a proposal ? ».

Some proof can suffice to convince the specialist - they spread a kind of smell. However, in the context of a virgin territory, it is necessary to consolidate every element, even impressive ones, by a complete set of checks. This series of concordant facts brings didactical elements which will be anyway necessary. For many reasons the methodology which has been forged, through twelve years of research, is terrifically exigent.

The first requirement relates to the duality of the field of research. « Composition » means geometry as a particular chapter of mathematics, but also as a particular practice of drawing. We must logically find proofs in the both sides, with a common meaning. Concretely : geometric properties and their translation in the artwork. Even more : we might observe the same kind of forms in other works, like patterns or standards of composition. In the field of pure geometry, we will pay attention to the logical proofs, to the mathematical reasoning. In the field of observation, as an applied science, we will consider the accuracy, the margins of precision.

The methodology of Comparative Geometry, a new understanding of history of art.
With a large investigation, at different epochs and in different civilisations, we will even build a repertoire of classics ! The comparison of different works has brought a lot of precious elements during twelve years. When the progress about one work was in standby, or paralyzed, the new discoveries about other works have always untightened the situation. Three works have been in this game for years : Trinity (Rublev), the cards of Tarots (Conver) and Melencolia (Durer). For this reason, the discipline which studies composition is now called « Comparative Geometry ».

Don't be surprised. The figures and their associated logic can not evolve like wild shapes. The concept of subjectivity will be soon obsolete. Two notions join at the end of this investigation : the « single » logic of geometry, and the universality of sacred values which depend on the previous. Human beings have all the same kind of chromosomes. We have some similar triangles in our mind.