Conference in Charles University April 2nd 2013 Faculty of Mathematics and Physics Dpt of Mathematics Education. Geometry & Art

Transcription

1 Conference in Charles University April 2nd 2013 Faculty of Mathematics and Physics Dpt of Mathematics Education Geometry & Art Geometry with eyes Part 1/3 - Introduction & The obsession of evidence Yvo Jacquier COMPARATIVE GEOMETRY February A tribute to the great pedagog Jan Amos Komenský (Comenius) Geometry and Art Part 1/3 Introduction Page 1 on 28

2 Conference at Charles University «Geometry and Art» April the 2nd, h 20 for one hour plus questions Yvo Jacquier, invited by Zdeněk Halas and Alena Šarounová Department of Mathematics Education Faculty of Mathematics and Physics Sokolovská 49/83 Prague Three articles are a support for the conference. Several reasons, including the weight of the files, invite to separate the main subject in this way. The order of these items may differ according to the readers. Here's a suggestion : For scientists : For literary-minded persons : For artists : Articles 2, 3, and 1. Articles 1, 3, and 2 with a mathematics teacher. Articles 1, 2, and 3. 1/3 The first article is an introduction to the practice of composition in art. We have first to distinguish two practices : sacred geometry and perspective system. 2/3 The old practice of sacred geometry needs special competences, especially if we want to understand the works of Renaissance. These mathematical elements build a particular corpus of geometry, called «geometry with eyes». These pre-euclidian mathematics are the subject of a second article. 3/3 With this tool of understanding, we can approach the masterpieces of Renaissance, including «The Holy Trinity» of Andreï Rublev (the final fireworks of at least five thousand years of images and buildings). The «Didactic Project of Albrecht Dürer» is the testament and also the encyclopedia of this forgotten culture. Geometry and Art Part 1/3 Introduction Page 2 on 28

3 INTRODUCTION What geometry and what art? There are so many ways to take elements of geometry, and to put them in art. We will not approach all these aspects in the deep. Just a quick panorama, beginning by some applied mathematics, that everybody knows : the perspective system. VISUAL N 1 - The vanish point We are used to think that composition in art concerns only this precise practice. But this opinion has been built during the XXth century. Philosophers and historians thought that the main progress in history of art is in the mastering of reality with drawing. For sure, we can rebuild what we see in a precise way with the lines of perspective - meeting on vanish points, and dealing the measures of the work with the plans of architecture. VISUAL N 2 Perspective building Geometry and Art Part 1/3 Introduction Page 3 on 28

4 The theoretical skills, i.e. the final principles of this practice have been established during the Renaissance. Albrecht Dürer came in Italy, which was the melting-pot for art at this time, and he entered in this fabulous adventure. However this pioneer in the perspective system, who even invented a famous machine to catch the truth in drawing, was also a great master of an other practice : sacred geometry. VISUAL N 3 - Dürer's machine Albrecht Dürer explains the presence of the both systems in his famous engraving «MELENCOLIA I». The work respects mainly the rules of a simple perspective (with one vanish point, located on the horizon at the vertical of the sphere). Left - VISUAL N 4 The perspective of Melencolia Geometry and Art Part 1/3 Introduction Page 4 on 28

5 But some subjects are escaping to the general perspective. The ladder is the more impressive case. The bars are parallel, and that is impossible with this disposition of the uprights. The bars are even not equidistant! Left - VISUAL N 5 The ladder of Melencolia As a conclusion for this provocation : if we apply the perspective's rules to this ladder, they show that it is over-sized. This could not be an error from Dürer, and aesthetic arguments are very precarious to explain his insistence. Below - VISUAL N 6 The ladder and St Michael In fact, there is a second composition over the one which builds the concrete reality in the picture. The preoccupation is not the same : perspective system cares about the concrete appearance, sacred geometry builds the symbolic meaning of the work. Geometry and Art Part 1/3 Introduction Page 5 on 28

6 Dürer puts precise marks in Melencolia to guide the student. The digit 4 and the center of the sphere are the firsts. We can build a square with them, and then divide the square by the golden ratio. The lines build a greek cross and the circle, diameter side/φ, is fitting with the rainbow in a perfect way. φ = Golden ratio = (1+ 5) 2 # 1, This is the first elementary step of what I call «The Didactic Project of Dürer», which is an encyclopedia of sacred geometry and symbolics. A huge set of images, including four engravings (called Meisterstiche) and a game of Tarot (the model of Nicolas Conver), teaches, step by step, how works this «geometry with eyes». All the instructions are in the images, and there is none writing outside of these images. VISUALS N 7 & 7 bis - The true composition of «MELENCOLIA I» Geometry and Art Part 1/3 Introduction Page 6 on 28

7 First reflections about methodology in research An appendix is put at the end of this article : «The obsession of evidence» We are at the crossing area of several disciplines : - Art with Dürer - History of art with Renaissance - Mathematics with composition (we will focus on this today) - Esotericism and history of science with the meaning of Melencolia It means that we have to build a new methodology. The criterions of each discipline forbid a necessary progress for the others. The best example is the floor-plan of Giza. The study by comparative geometry gives one complete proposition which respects the standards of at least three disciplines : Mathematics, by the figures of «geometry with eyes». Physics, by the margins of precision which fit with the survey results of Egyptologists. History, by the religious interpretation of the geometrical structure. In the three cases, the results are conclusive, but no one discipline is able to conclude on the other fields, because they are outside of their competences. In fact, the final certitude requires several components. We can not rely to a proposal which doesn't respect one of the aspects of the file. Farther, the propositions which care only about one aspect of the file are suspicious. By the way, climatology shows a very cold period between 1450 and called the minimum de Spörer. Italy was at this time the best refuge in Europe for the artists who created Renaissance. Dürer speaks about these circumstances in a letter to his best friend - Pirkheimer : «I will feel so much cold, thinking to the sun!» Geometry and Art Part 1/3 Introduction Page 7 on 28

8 WHAT IS COMPOSITION? The two systems of composition In architecture, perspective is the way to translate the plans of a building (the different views) into a two dimensions picture. The architect applies rules in which there is none idea of composition, none decision. The options of the author about the harmony, the measures, the proportions etc are outside the perspective system, they don't depend on its lines. This observation is basic but fundamental. The indications in composition escape all, by definition, to the perspective system. The only one goal of this system is to represent the reality as it is to our eyes. During Renaissance, painters practice two systems to compose their paintings. And the perspective system is a way to make true the results of the ancient system that we call sacred geometry. When a painter shows a beautiful background, like Raphael in his «School of Athens», its harmony comes from an other construction, which uses more than the neutral lines of perspective. These other lines may coincide with the one of perspective, as we see in this fresco by Raphael. E.g. the two white lines going down make an angle of 36 with the vertical. This refers to the pentagram. The feeling of harmony doesn't depend on the one of reality. This angle belongs to an other construction. Geometry and Art Part 1/3 Introduction Page 8 on 28

9 Previous page : VISUAL N 8 - The composition of The school of Athens - by Raphael cm - Chambre de la Signature (les Stanze) des musées du Vatican VISUAL N 8bis - A lesson, by Plato himself! The size of this fresco is too big : it is not the ideal to be analyzed at this first step. Just look at the measure of the golden ratio, given by Plato, and confirmed by the hand of Aristotle. Right - VISUAL N 9 St Michael slaying Satan cm Musée du Louvre, Paris So let we chose a simpler one in the works by Raphael : «Saint Michel and Satan». Here the sacred geometry is simple, and as ever with this painter : terrifically efficient. The arrow, drawn by the geometry, is more effective than the spear itself! Geometry and Art Part 1/3 Introduction Page 9 on 28

10 At this occasion, we understand that composition (of a work) could be compared to the wood of a formwork in a construction. The characteristic of this wood is to disappear at the end of construction. A set of geometric line guides the artist or the architect, up to be digested by the work. Sacred Geometry is the oldest practice of composition. VISUAL N 10 St Theresa Church Port Said, Egypt This geometry with eyes is built on a grid, which allows to measure the forms. The numbers open to a symbolic translation in a human language. In this work, by J.-D. Ingres, the great circle is diameter 3, measure of Celestial. The grid is, as usually, revealed by the triangle The side of the green square is 2.φ. VISUAL N 11 - The composition of «La grande Odalisque» by J.-D. Ingres cm - Musée du Louvre (Aile Denon, 1er étage), Paris Historical filiations Verroccio (Florence, Venice, 1488) has been the master of Leonardo da Vinci and Sandro Botticelli. Vinci has been the master of Raffaello Sanzio/Raphael ( ). Sandro Botticelli ( ) has been the master of Albrecht Dürer ( ) Raphael has been the historical reference of Nicolas Poussin ( ). Nicolas Poussin has been the historical reference of Jacques-Louis David ( ). Jacques-Louis David has been the main master of Jean-Dominique Ingres ( ). Poussin, David and Ingres have been in Italy to learn painting. This is a stream. We have the same kind of phenomenon with greek mathematicians visiting Egypt. Geometry and Art Part 1/3 Introduction Page 10 on 28

11 HISTORICAL PANORAMA - I The field of «comparative geometry» Situation of geometry in research We have to place this particular geometry in the process of art research, and also on the scale of history. The restoration of the theoretical skills has been very soon a necessity in the study of art works. The best strategy is to compare the compositions of different works to identify their common points. Even the one of a neolithic temples and the one of a painting from Renaissance. All these projects have obvious common points, which constitute a mutual fund. So it was primordial to reveal the theoretical aspects, if we assume that people, throughout these millennia, knew what they were doing! It is the only one hypothesis of what is called «comparative geometry». Situation of geometry over time Now what is the schedule of this geometry all along millenniums? This fabulous puzzle is under construction. We guess the shadow of a very wide stream in early neolithic, embodied by megalithic constructions. The geometry has the size of a country, and the constructions reflect the development of the society in its complexity. With the anatolian site of Gobekli Tepe, we discover that this megalithic stream is alive at least ten thousand years B.C.. VISUAL N 12 etc - Gobekli Tepe Actual Turkey Y. B.C. For me, these stones are the support of a wide wood-frame. The holes in the stones, even on their top, might find from this simple hypothesis a simple explanation. Geometry and Art Part 1/3 Introduction Page 11 on 28

12 VISUALS N 13, 14 & 15 - Megaliths of the Mont-Saint-Michel Other example of this land art in geometry : the megaliths of Mont-Saint-Michel in Brittany. How did they put points in this precise way? Triangulation? Below : «La maison des Feins» - Tressé, Brittany, France Geometry and Art Part 1/3 Introduction Page 12 on 28

13 This practice invests mesopotamian architecture, during the fourth millennium : Left - VISUAL N 16 - Eanna temple Uruk IV - IVth millennium B.C. And very soon, Egypt shows the same skills : Right - VISUAL N 17 Plateau of Giza 2500 years B.C. Below - VISUAL N 18 The tablet Plimpton th century B.C. The famous tablet Plimpton 322, dating from 1800 y. B.C., is now explained as a list of pythagorean triplets, reduced to couples. These couples produce a list of primes, and the missing lines of the tablet respect the rule! The principe seems extendable, two values of the triplets becoming the couple of a next triplet. Geometry and Art Part 1/3 Introduction Page 13 on 28

14 The Pythagoreans gather the both influences, mesopotamian and egyptian. During centuries Greece is the crossroads of knowledge. [ VISUAL N 19 - Parthénon ] The plan of Parthenon - Athens, Vth Century B.C. We need a very beautiful picture - maybe the university could provide Sacred Geometry tries to escape to the roman realism that Vitruve will express in his architectural treatise - Ist Century B.C., and it spreads in the celtic world. The archaeologist Jean-Loup Flouest and mathematician Marc Bacault have shown the influence of Pythagoreans on the celtic world. This very recent research concerns the latenian phaleras in Champagne, France. A Phalera is a decorative piece, generally with a circular shape. 190 circles and arcs are needed to draw the elements of this object. Based on the numbers 8 and 27, elements fetishes of the numerology taught by Pythagoras, this object also shows the movement of intellectual exchanges between the latin scientific elite, and celtic druids, who were considered at the time as Pythagoreans. The level of these decorations requires four years of university studies in mathematics - to be understood. VISUAL N 20 - Celtic phalera XIIth Century Geometry and Art Part 1/3 Introduction Page 14 on 28

15 In early middle ages, this culture finds a harbor in Byzantium (Icons and ceramics). We see here an accomplishment with the gift of the Patriarch of Constantinople to the great prince of Kiev in (The orthodox church separated from the catholic one in 1054.) VISUAL N 21 - The Madonna of Vladimir XIIth Century This quite orthodox world is shaken by iconoclasm : between 730 and 787, then between 813 and 843. Below - VISUAL N 22 The geometrical key of Germigny France - IXth Century At this period, a lot of artists and builders escape towards the west part of Europa, and they save their life but also their culture. In the oratoire of Germigny, they put the basics about the forms in a didactic scale. The extreme solution will be once more the celtic world. The irish monks welcome this culture and rewrite the Holy Bible. By the way, the famous «Book of Kells» is known to take a great liberty with the text, as if the message had to be clear : geometry is the main purpose of this work. Left - VISUAL N 23 - The Book of Kells Ireland - late VIIIth Century Geometry and Art Part 1/3 Introduction Page 15 on 28

16 VISUAL N 24 Root of 3 and Phi According to the grid, there is a proposition about root of three and golden ratio. The France of cathedrals is maybe the result of this explosion - due to iconoclasm. A new idea is emerging in my researches : I was so obsessed by foreign influences that I didn't see that France was able to participate and to give something to this art, as well as northern Italy or Novgorod. VISUAL N 25 - The cathedral of Dol-de-Bretagne France - mainly XIIIth and XIVth Century Geometry and Art Part 1/3 Introduction Page 16 on 28

17 The french influence will be seen in the frontage of the «House with the bell» in Prague. VISUAL N 26 - The House with the bell in Prague Early XIVth Century The Spirit of French is maybe seen in the work of Jean Fouquet in the fifteenth century. VISUAL N 27 Le portrait de Charles VII Jean Fouquet (1415/ /81) Portrait of Charles the VIIth (1450/1455) Now we enter in a period of apotheosis for sacred geometry : the Renaissance. Three pieces embody this accomplishment : «The Holy Trinity» by Andreï Rublev, in 1420/28, «The birth of Venus» by Sandro Botticelli, in 1485, and «MELENCOLIA I» by Albrecht Dürer, in The third belongs to a wider project which is the conclusion, the testament of this civilization of image. Geometry and Art Part 1/3 Introduction Page 17 on 28

18 After Renaissance, the perspective system that emerges during the Renaissance, will remain the only one in composition - after a period of cohabitation with sacred geometry in the same works. Later, the «art of the diagonals», with classical painters, will try simple rules, but without finding the lost secrets of sacred geometry. And slowly, allegories will take the place of symbols... We could speak about XXth Century, but I am afraid that after the magnificent fireworks of Renaissance, your eyes will be disappointed, and your mind, as mathematicians, a little bit insulted. I would prefer to expose the elements of geometry with eyes. The must! «The Holy Trinity» by Andreï Rublev /28 Andreï Rublev signs his Holy Trinity by a simple rectangle, instead of the traditional monogram of Christ - or a sentence from the Bible. This sign is not involved in the narrative of the work, it is not more a signature or an ornament. By this simple rectangle, Andrei Rublev claims explicitly for sacred geometry. VISUAL N 28 - «The Holy Trinity» This rectangle refers to the golden ratio. Historical detail :In the early twentieth century, the british critic and fencer Theodore Andrea Cook ( ) agrees with his friend, the American mathematician Mark Barr, to introduce the Greek letter φ as a mathematical symbol of the Golden Ratio - in reference to the geek sculptor Phidias (Vth century B.C.). The rectangle of Rublev shows at first the golden ratio up to the floor under the Altar. VISUAL N 29 The rectangle of Andreï Rublev The golden ratio of the rectangle Geometry and Art Part 1/3 Introduction Page 18 on 28

19 The front panel is a lesson of arithmetic values. Beginning by the equation, here in red by the geometry : φ2 = φ + 1 VISUAL N 30 Front of the Altar - 1 And ending by the convergence with the measure of the grid, which will be found out with a totally other method (!) VISUAL N 31 Front of the Altar - 2 We can identify this grid by two points, called Alpha and Omega. Geometry and Art Part 1/3 Introduction Page 19 on 28

20 VISUALS N 32 & 33 - Omega Omega is the crossing point of the light. The dais bring a first confirmation, and the Holy Spirit gives his benediction by a line of 45. VISUAL N 34 Confirmation of Omega In spite of the state of the work, a lot of elements confirm the point. The three angels represent from the left to the right : the Father, the Son and the Holy Spirit. Geometry and Art Part 1/3 Introduction Page 20 on 28

21 VISUALS N 35 & 36 Alpha and its confirmation A second point is called Alpha, and it is the crossing point of several lines, passing between the pupils of the eyes (Father and Son), and the lengthening the scepter of the Holy Spirit. Other lines will confirm this option. Left - VISUAL N 37 - The grid The two points Alpha and Omega are the key of the grid. Remark 1 : the definition of the grid is never an ordinary attempt. As we see through this example, its definition needs a complete set of evidence. This case is very important, because the grid has ever been the base of sacred geometry. All the figures and the links need its support to build. Remark 2 : it would be impossible to translate the symbolic meaning without any measure. Because of that, the grid is irreplaceable. Remark 3 : Before to go further in the exhibition of this art, we need elements of mathematics. Right - VISUAL N 37bis - Confirmation of the grid The dais produce two lines in the center, with an angle of 8 and 9. The first is π/45, and second remembers of the pentagram. Their two crossing points, on the vertical of Omega, are separated by the radius of a circle, which surface is one tile on the grid. Squaring the circle. In the low part of the Icon, Rublev exposes arithmetics. In the upper part, he practices geometrical principles. Spiritual/Concrete. Geometry and Art Part 1/3 Introduction Page 21 on 28

22 APPENDIX The obsession of evidence The reliability of results is the prime objective of any scientific study. It is for this reason that I have chosen to approach the composition in the arts with scientific methodology. I wanted evidence, not opinions. There are many specialized ways to take on this subject, but none uses for example the concept of margin of accuracy/precision. Then, the comparison of structures has never been extended to the whole history. But this is the only way to bring out the common fund, of this true language - the symbolic geometry. This historical panorama helped me at end, to reconstruct the corpus of pre-euclidean geometry. I should explain what determines the validity of my results, beyond the personal opinion that the experience and intuition allow me to claim. When I present a composition as the right one, I gathered the means to be sure of that, converging evidence and fully consistent. 1 - Systems in composition First, we deal with systems of composition, and not only figures, even sums of figures. These structures bind the figures by their apparent elements (sides, circles), but also by their internal lines (including bisectors). 2 - The Grid The identification of the grid is the first evidence, to validate a system of composition. All figures must be accorded to the same diapason, the same unit of measurement. A process called "crash test" is to change the unit of 5 and see the impact of this variation. In almost all cases, the composition systems, all attached to the same grid, cease to be valid. 3 - The margins of precision This is the precision with which we observe the geometric shapes on the works or the plans, or else the results of surveyors (very learned in the case of the Giza Plateau). By Dürer the accuracy becomes a subject of astonishment : the conventional softwares for image are not sufficient to account its accuracy. Eg we must translate the angles, by Pythagoras in pixel values. In the study of tarot, the results becomes statistics : the 22 major plates meet all the criteria of composition. Another aspect is developing : some coincidences of figures are so precise, but not exact, that the difference is swallowed by the finest line of the sketch. Geometry and Art Part 1/3 Introduction Page 22 on 28

23 4 - The comparison > Comparative geometry We find the same types of systems, of organization, in works very far in the history. Eg the house with the bell in Prague (early fourteenth century) takes a pattern entering the temple of Eanna - Uruk civilization, Mesopotamia, the fourth millennium BCE. In the early sixteenth century, Dürer fully incorporates an entire system of composition belonging to the «Holy Trinity» by Rublev, to draw his model of tarot : seven astrological triangles, type 3-4-5, held together by an homothety. The separate development of the same system in works that have nothing to do a priori, is particularly convincing. A complex structure can hardly stick with two totally different subjects, by an arbitrary operation. The great puzzle of sacred geometry gradually reveals a very homogeneous face, this culture shows a great unit The geometrical key of the composition This is the most salutary aspect of the study. After a painstaking reconstruction, a figure often stands out. It connects all the others and demonstrates unique properties, surprising on the mathematical plan. This is a key of composition. And unlike systems that can move from one work to another as standards, this key has a unique character that is like a signature to the artist. For a long time, artists did not sign their works, probably for this reason. The geometry is in the mind of the Ancients, the language of God. NB: Dürer will impose its cartouche, to prevent the looting of his engravings. 6 - The marks of composition This is the simplest aspect, and the most convincing, in the study. A number of signs are placed on the works without any status : they are not in the narrative register, do not take more a role in aesthetic, they are not readable by the codes of writing. In summary they are absolutely nothing, while their accuracy confirms a significant effort on the part of the artist. These signs are actually benchmarks that allow us to reconstruct the composition. The best example is given by Rublev, when he puts a rectangle in front of the altar of the Holy Trinity, instead of the usual writings of the Bible or the monogram of Christ. In fact, the top line of the rectangle begins a golden vertical rectangle : it is enough to extend the lines up to the ground to reconstitute it. And Rublev tells us that his work is entirely geometrical. Here are some examples of these marks. Geometry and Art Part 1/3 Introduction Page 23 on 28

24 VISUAL N M1 - The marks of tarots These signs need too much effort to be useless. Their discretion enhances this idea. Composition will give them a real statute. VISUAL N M2 - The Book of Kells This book pays attention to the punctuation of drawing very much more than to the one of the text. VISUAL N M3 - Trinity Rublev puts a rectangle in front of the altar of the Holy Trinity, instead of the usual writings of the Bible, or else the monogram of Christ. The top line of the rectangle begins a golden vertical rectangle : it is enough to extend the lines up to the ground to reconstitute it. Rublev tells us that his work is entirely geometrical. Geometry and Art Part 1/3 Introduction Page 24 on 28

25 VISUAL N Rozenkranzfest M4 - Look at the paper in the hand of Dürer. VISUAL N M5 - Alpha et Omega Melencolia : two lines inclined of 45 are above the head of the angels, meeting Alpha of the magic square and Omega of the scales. VISUAL N M6 - Signature The 4 of 1514 is the most important mark of Melencolia. The great square depends on it. Geometry and Art Part 1/3 Introduction Page 25 on 28

26 7 - The interpretation The interpretation must evidently confirm the validity of a geometrical proposition. In some cases such as the «Virgin of Vladimir» or the Giza plateau, this translation takes the form of a didactic explanation of sacred values. In many others, the comparative geometry limits its development to a minimum. This knowledge is being rebuilt, and this interpretation presents a double-risk : it may influence the study and cause a possible opposition from specialists, who have rightfully the mission to extend the mathematical study. This phase must be addressed by a full team, and the methodology remains to be established. It is not sure that the theological grids, or even what we call tradition would be sufficient. The geometry of composition is a language in its own right, and it is likely that its codes and operations differ from others, in the same way that music can not be the integral translation of a written or spoken language. A - The double-proof This aspect is not directly in the process that evidence claims. It concerns the internal cooking in the research of composition. Any system produces harmonics, as well as any figure carries with it internal lines like diagonals and bisectors. These structural lines appear implicitly in the study, without that we know a priori the intention of the authors. How could we know if the authors have integrated them in their development? In following the tracks of their development. If the observation of these lines led to new findings, basically : if they create a surprise, then we are sure that these lines are incorporated into the composition. In addition, the harvest of these new findings offer a double proof to the original proposals. This process may seem complicated, but it actually simplifies the study: systems produce a lot of harmonics, due to their complexity. The double-proof is useful to isolate the main framing girders. VISUAL N D1 - Measures on Melencolia The measures of Melencolia, according to the grid of tarots. The sphere is the diapason. Geometry and Art Part 1/3 Introduction Page 26 on 28

27 VISUAL N D2 - The angle of π/7 First confirmation An angle of π/7 appears from the golden rectangle of height 7, and its name is on the sleeve of the angel's robe. Second confirmation The golden rectangle shows the vertical of the home behind. Left - VISUAL N D3 - Meeting the Square Third confirmation The line of the angle π/7 is meeting the angle of the great square. Right - VISUAL N D4 The mark π/7 on the sleeve. Geometry and Art Part 1/3 Introduction Page 27 on 28

28 B - The corpus of geometry with eyes (pre-euclidean) Through the study of art and architecture, it has been possible to reconstruct the corpus of pre-euclidean geometry, now called geometry with eyes. This theoretical result gives great credit to the entire study. It was believed that this proto-geometry was limited and approximate. It reveals a coherent and well developed face. Its properties didn't wait the Greeks to demonstrate, and its lean corpus is now extended by other amazing properties, including: - The four expressions of the golden ratio in the triangle (unpublished). - The structural connections between the symbolic figures on the ancient grid. - The Pythagorean theorem/diagonal becomes a theorem of writing. - Several fields of study, such as the presence of a list of prime numbers, between the lines of the Babylonian tablet Plimpton 322 (unpublished). - And also the geometry of Giza : its structure responds to the surveying of Egyptologists, with an incredible precision (unpublished). This theoretical corpus is the hard foundation on which any study can build. By its coherence and its intelligence, this geometry before writing can build concretely the idea of a language of images. Synthesis of evidence The aspects described above 1 to 7, must come together to validate a geometrical proposition (the last two letters marked A and B, are naturally apart). Thus, it is not an opinion, a path, a reference which decides the status of a proposal, but a comprehensive and consistent ensemble. Geometry and Art Part 1/3 Introduction Page 28 on 28