__________A new understanding in Art History__________

Mathematics - History of Art - Esotericism - Visual Arts

Plateauof GIZA

THREE PRINTS ON THEGROUND BY PYRAMIDS

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

COMPARATIVE GEOMETRY ---------------------------------------------------------------------------------- MARCH 2015 ------

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PART I

INTRODUCTION

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THE EGYPTIAN SPIRIT

Geometry with eyes

1 • The origin of geometry

At the Renaissance, artists and architects used skills of composition

which still kept the “Egyptian style”. Why ? These mathematics rely to

visual proofs, and they avoid calculation — by fear to frighten the

numbers. This knowledge is perfectly adapted to visual arts. Many other

reasons enter into the total explanation, especially the religious

meaning, but at first in a practical way, the grid of cathedrals' builders is

the continuation of the Egyptian grid. A lot of testimonies refer to this

origin, by writing but also by image. Albrecht Dürer - Pyramids

1514 – MELENCOLIA § I (left)

1511 – The Adoration of the Trinity

With the mathematicians of french IREM (Institutes of Research in

Mathematics Education), we reconstruct the entire corpus of this very

particular chapter of mathematics. We prepare a series of publications

aiming the pedagogues teaching in Middle School (Junior High School).

First article :

http://www.contemporary-painting.com/Y_Jacquier-IREM-Figure_Tympan_Conques-en.pdf

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2 • The high level of the Egyptian knowledge

These skills are not empirical (axiomatic), and the corpus is fully

coherent. Thus the ancient Greeks didn't invent the principle of proofs

and demonstration. They developed another system of thinking, very

didactic, with additional definitions. Euclid embodies this progress, but

all the Greeks mathematicians went to learn at the Egyptian school ! The fabulous properties of the « Geometry with eyes » are all unedited –

they are absent of the textbooks and publications. Eg : the four

manifestations of the golden ratio in the triangle 3-4-5. A complete

article is accessible at :

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

Pyramids : the mistake of accounting !

About Egyptian architecture, the usual studies focus on the measures,

and they try to explain some “secret” through pure calculation and

without religious meaning. Of course, Egyptians practiced accounting.

But at first, unlike the Sumerians who were obsessed by numbers, the

Egyptians were deeply artists and they were thinking with their eyes.

Even the writing shows this difference. The cuneiform scripture is in the

abstract spirit of numbers, while the hieroglyphs are images. During

Neolithic and Antiquity, the knowledge was traveling and we guess that

all the skills were accessible for the both — Mesopotamian and

Egyptian stream. But a Culture involves also choices, expressions of

personality, all the differences that make the result recognizable. Other aspect : in the study of these epochs, our contemporary

preoccupations and reflexes have to fade. We dream to walk on March,

but Egyptians have dreamed about eternal life. The pyramids have

been built to cross an other space than the physical one. The technical

performance, as remarkable as it is, is not necessarily the main purpose

of a any sacred building. Our approach of Egyptian Civilization through

numbers, without any religious meaning, is a double error. For strategical reasons, I separated the pure geometrical aspect from

its interpretation. This attitude has been relevant. It is possible to study

Egyptian geometry with a scientific approach. This attitude brings a lot

of proofs but not only. Through this learning, we begin to understand

the Egyptian mentality. Especially their common sense.

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THE EXPRESSION OF SACRED

How works sacred composition

The process of representation is like a reciprocal chain between the

final drawing/plan and the original meaning (in human words)

Representation <•> Geometry <•> Numbers <•> Interpretation

Object Meaning

Composition

In a practical way : 1) The geometrical figures guide and define the final

lines of the artwork. 2) The meaning of this frame/composition is given

by the symbolic of numbers. These numbers allow to translate the

values of the grid into human words. For the Ancients, symbolism and

esotericism are synonymous. We separate these notions today

because we lost the skills of Egyptian geometry. In a paradoxical way,

accounting doesn't bring the symbolical meaning of numbers.

The chain of symbolism/esotericism/composition

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The previous schema shows the chain of sacred geometry. On the left,

all the figures are built on a same grid. Note that the simple triangle 3-4-

5 has four manifestations of the golden ratio - φ. These figure will lead the drawing of figurative symbols, according to

the lexicon of numbers, here very resumed. These words express the intentions of the authors, then numbers enter

in a dialogue with the grid to produce geometry. Very important point :

measurement is not the only one translation of a number. Eg the

diagonal of a double square can be the expression of number 2.

After Renaissance

The Renaissance period is the apotheosis of this culture, and also the

beginning of its oblivion. For a while, the artists will practice sacred

geometry and the perspective system. This other has never been a real

system of composition. It is a way to make real the elements in a

drawing. But there is no symbolical values, neither aesthetic. After

Renaissance, symbols have lost their geometrical structure and they

became allegories, id est undefined expressions (great confusion). Some skills escaped in northern Italy, where painters like Poussin and

Ingres went to study painting.

« La grande Odalisque » by J.-D. Ingres - 1814

In this composition, the great circle is diameter 3, measure of Celestial.

The grid is, as usually, revealed by the triangle 3-4-5.

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BEFORE EGYPTIANS

Gobekli Tepe, Anatolian region

So we guess the shadow of a very

wide culture in early neolithic, with

megalithic constructions. They reflect

the complexity of a society. This

“megalithic stream” is alive ten

thousand years B.C.

Atlantic Megalithic

Other example of

this “land art” with

geometry : the

megaliths of Mont-

Saint-Michel in

Brittany. How did

they put points in this

precise way ?

Triangulation ? The

frame takes the size

of a country.

Eanna temple, Uruk IV - IVth Mil. B.C.

This practice invests Mesopotamian

architecture, during the IVth millennium B.C.

The geometry is already perfectly mature,

and writing is on the way to birth...

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PART II

FRESCOES

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Ay and Osiris

Tomb of Tutankhamun

Valley of the Kings at Thebes - tomb 22

Ay (1704 - 1690 BCE)

The Grid

The triangle 3-4-5

The Golden Rectangle 2x2φ

This rectangle is the natural

one of the triangle 3-4-5

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The fresco of Benia

Benia or Pahekamen (ou Paheqamen)

Tomb TT 343 of Benia

Fresco/wall engraving - East Wall, southern part

18th dynasty - circa 1500 BCE

The Golden Ratio

Rectangle 2x2φ

Relations scripture — drawing

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The fresco of Nofretari

« Nefertari playing Senet »

Painting in the tomb of the Egyptian queen Nefertari Meritmut.

Technique = Wandbild / Fresco - Dimensions = 61 x 70 cm

The Grid

Triangles 3-4-5

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Golden Ratio

In yellow : 2φ

Circles diameter 1

Pentagrams into circle 5

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PART III

PYRAMIDSFLOORPLAN

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INTRODUCTION

A golden gift for study

The plan of reference is the work of the physician John A.R. Legon. He

gathered all the data of well-known egyptologists into a synthesis.

Especially M. Lehner, 1985 - J.H Cole,1925 - Flinders Petrie, 1880. This

plan has been approved by he Archaeology Society of Staten Island.

http://www.john-legon.co.uk/gizeplan.htm This is a real gift for the study. The comparison of the geometry to the

reality of a work is the most delicate stage of the study. Above all, this

step is the most “questionable”. In the particular case of Giza, we will

confront the geometry to the values of the plan ! No contestation

possible. We respect the margins of precision or not. That's all.

Thank you so much Mister Legon !

Overview

The frame of this study is firstly based on the analysis of John A.R.

Legon. On that basis, we will build a geometrical structure according to

the corpus of geometry with eyes. The results will be all compared with

the values of the plan, into their margins. Then, we will compare the

meaning of the geometry with the Egyptian thinking, more precisely

their religion. So, in this process, we have two occasions to verify that

we are right. First, with the measurements and second, with the

mythology. A third argument has to be underlined. The structure is not a

sum of figures. They depend of each other by a chain of construction. If

you change one figure, all the other are disturbed and they stop to fit

with the plan — without any other to replace. This particular process is

called “crash test”. We will begin by two squares for Khafre and Khufu. Then we will place

them on a great golden rectangle by a very sophisticated process. Then

we will built the third square, Menkaure, and we will find its/his place.

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KHUFU, KHAFRE, GREAT RECTANGLE

The square of Khafre (C2)

A unit of 250 royal cubits builds a first grid to the field.

The side of Khafre is constructed with the ruler and the

compass, by plotting the difference between the

diagonals of a double and a single square and

multiplying by two this measure :

C2 = 2 x [(√5-√2) x 250] ≈ 2 x 205,4636

≈ 410,9272 cubits

The square of Khufu (C1)

The relationship between the son Khafre and the father

Khufu is marked by the golden ratio. Giza teach us that

√3 is a feminine value and φ is masculine, especially in

the context of reproduction. So we build at first a circle

from the previous unit of (√5-√2) and we get a radius R :

R = φ x (√5-√2) x 250 c.

The circle builds a particular almond with a rule :

We can inscribed a double square with the centers.

So, the centers are separated by R/√2

The square of Khufu will chose the intersections of the

almond. We need four points so the figure is multiplied

in a cross (classical development in sacred geometry).

But these points build a lozenge ? It is solved by a

simple rotation of 45°.

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The final result of the side is :

√7.φ x (√5-√2)/2 x 250 cubits

≈ 439,78617 cubits

The great golden rectangle

The main frame of the composition of Giza is a large golden rectangle,

height 4 (x 250 cubits) and width 4.φ (x 250 cubits). So 1000 x 1618

cubits. Note that the choice to consider φ as 1.6180 may, at the end of

the study, to be questioned. The accuracy of results demonstrate the

need to grow φ up to 1.618034 ! Who pretends that Egyptians were not

good in calculation ? Not me.

This frame has a physical reality. Its West edge is aligned with that of

Menkaure, its North edge passes through the center of Khufu, and its

geometrical center is on the line joining the peaks of Khufu and

Menkaure. This last property is a “tip top multi-argument”.

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Φ & √3 - THE GREAT MEETING

Egyptians think primarily with angles. Proof is the miraculous alignment

of the pyramids according to the cardinal points. If for us, the proportion

of √3 refers to a rectangle, it is expressed here by a straight line (DE) at

30° of “horizontal” (Est-West).

An article with the french IREM develops this chapter :

http://www.contemporary-painting.com/Y_Jacquier-IREM-Figure_Tympan_Conques-en.pdf The Golden Ratio has several kind of expressions. In geometry, it can

take the form of a line which angle α has a particular tangent : 2/φ2. We

get it with two crossing golden rectangles (see on the figure).

The red line is coming from B, corner of the great square (1000 x 1000

cubits - in golden rectangle), and it has the angle α (golden figure).

The green line is coming from E, on the east side of the rectangle. This

point is distant from the north by the half of Khufu square. The angle of

the green line is 30°(/horizontal W-E of the plan), and it refers to √3.

These two lines represent the masculine (φ, in red) and the feminine

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(√3, in green) principles. And they cross precisely at the border of

Khafre. We understand the logic of the red line, when we know that

Khafre is the son of Khufu. But what about the green line ?

The Queen Henutsen

Henutsen is the wife of Khufu and the mother of

Khafre. Her monument, called G1C, bears an

inscription epigraph :

« The living Horus Medjou Hor, King of Upper and

Lower Egypt, Khufu, received life. It is next to the

tabernacle-the-Sphinx, northwest of the home-

d'Osiris, Lord of Rôsétaoui, he established the

House-d'Isis. It is next to the temple of the goddess

that he built his pyramid. It is next to the temple he

built a pyramid for the king's daughter, Hénoutsen » Curiously, the azimuth of the solstice at Giza is 62,25°/ N ... If we plot

this angle from the couple's melting point (C), this new line passes still

much more clearly on the pyramid.

BC is a line of a golden cross, whose lowest line coincides with the

south line of Khafre. This square is now totally defined. About Khufu, we

know only that the upper side of the great square passes in its center...

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EXACT PLACE OF KHUFU

The rectangles type ∆ and ∆’

The rectangle type ∆’

The ratio (1+√3)/2 is a parent of φ = (1+√5)/2

and it is very linked to √3, which is feminine. Khafre is fixed. We will use this square to place

Khufu. At first, we add a equilateral triangle

from the center of the square, towards the

north. And we get a rectangle type ∆’, with the

proportion H/L = (1+√3)/2

The center Ω of this rectangle ∆’ becomes the

center of a wide pentagram, which is built with a

special criteria. Between the points μ and ν,

there is exactly the side of Khafre,

2 x (√5-√2) x 250 ≈ 410,927 c.

According to this, the circumcircle is radius :

R = 2φ/√(3-φ) x 250 cubits

≈ 2,752 7639 x (√5-√2) x 250 c.

≈ 565,592 79 cubits

FINAL : The corner of Khufu, called α point, is on the circle.

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The sexuality of Symbols

In March 2012, when I cracked this system of composition, I didn't know

the feminine nature of √3. I was still listening what we call “Tradition”. In

fact this so-called tradition is so much corrupted that it doesn't deserve

even its name. Thus, the 3 would be Celestial and feminine, and the 4

would be Terrestrial and masculine... This is not serious.

Too much myths, symbols and traditions invert the components.

Especially the Triangle of Isis with her side 4. We can not rely on a

system which forget one of three of the reality.

One proposal solves the problem, if we desire to face it. There is an

example in symbolic : the four elements, and how they have been

explained by the tradition...

- 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 associate the concepts of Celestial/Terrestrial with

those of feminine/masculine. This system produces four possibilities,

that the cards of tarots, encyclopedia of symbols, could translate :

- THE EMPEROR, IV (noted IIII)

is a male representation of Terrestrial

- THE EMPERESS, III

is a female representation of Celestial

- THE JUSTICE, VIII

is a female representation of Terrestrial

- THE TEMPERANCE, XIV (Noted XIIII)

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.

The golden ratio can now be considered as a masculine value

and √3 his feminine partner. Two equations are comparable :

If we call H = (1+√3)/2 and φ = (1+√5)/2

H√3 = H + 1

φ² = φ + 1

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Symbolical meaning of Khafre' figure

A male Pentagram is covering (around its center) a rectangle type ∆’,

female, in which sleeps Khafre, the son. The circumcircle of this

pentagram is touching the square of Khufu, the father.

The presence of the pyramid G1C (very poetical name) on the diagonal

line type ∆ (√3) achieves the picture. These buildings tell et first the

conception of Khafre by Henutsen and Khufu.

Last image : the star of the pentagram will join, according to the

Egyptian religion, the sky of Nut.

Special Thanks to John A.R. Legon

John A.R. Legon proposes an algebraic solution for the relationship

between the middle side of Khufu and the corner of Khafre. This

proposal rejoins the final result of symbolical geometry. If we call a,b,c

the little side, the diagonal, and the great side of the rectangle (see the

picture), the theoretical values of the study are close to 1, 2, √3 (at 10-4)

a # 249,98205 # 250 at 10-4

----------------------> b/a # 1,9998 # 2 at 10-4

b # 499,92330

----------------------> c/a # 1,73191 # √3 at 10-4

c # 432,94626

We find one more the trace of the rectangle ∆, with its feminine √3, as a

link between the father Khufu and the son Khafre.

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MENKAURE

The figure right is explicit. The diagonal of

Khafre, in green, intersects the one of Khufu

to show the entire side of Menkaure, in red.

Then, bellow : the diagonals of the great

golden rectangle intersect at the point R. It is

exactly on the line between the peaks P of

Khufu and Q of Menkaure.

• The precision of this meeting is impressive (~ 7,5 cm)

See at the end of the article the consistency of this Key figure.

• The total height of the field, with the three pyramids, is 1732,334 c.

# 1000.√3 cubits, with a precision of 1,6.10-4 (~ 0,2 cubit or ~ 15 cm)

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CONFRONTATION WITH THE PLAN

Confrontation with the survey of Cole and Petrie (basics)

KHUFU (C1)

Measure of Cole ≈ 439,8 Cubits ± 0,1 Cubit

Comparative Geometry ≈ 439,78617 Cubits

—> 0,014 Cubit from Cole (7 mm)

KHAFRE (C2)

Measure of Pétrie ≈ 411.00 ± 0,073

Comparative Geometry ≈ 410,9272 Cubits (schema of Legon)

—> Result at the limit of the range

MENKAURE (C3)

Measure of Pétrie ≈ 201,44 Cubits ± 3,0 inches

maximum limit ≈ 201,58549 Cubits

Comparative Geometry ≈ 201,5465 Cubits

—> Result within the range

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CONCLUSIONS

General information

This didactical presentation (of this royal family story) is specially

addressed to the Egyptologists. This article resumes the study, by Comparative Geometry, of a major

example of Egyptian Culture, through the links between art and

mathematics. It has been necessary to make choices to promote the

main points of the file. A lot of findings stay behind this summary, and

also a lot of interesting questions. About the accuracy of numbers : all the folder (the extensive version)

has been reviewed by Jean-Paul Guichard (IREM).

The relevance of the Key figure

All the great works, where the practice of sacred geometry is involved,

include at least one figure which covers the others. This geometrical

key is a signature of the work. The key to the Giza plateau is typical of

Egyptian culture, of a high purity and of an essential symbolic force.

http://www.art-renaissance.net/Gizeh/10.jpg I - This figure is not arbitrary. Three factors determine the large golden

rectangle : the north side settles itself on Khufu. The west side is

aligned on Menkaure, and its measure is 1000 royal cubits. These three

elements are anchored in the plateau's reality (including measurement,

typical of this Egypt). The other two sides (South and East) are the

logical consequences of the above — similarly the middle of the

rectangle. Finally, the line joining the tops of Khufu and Menkaure is

clearly a line of the plan - as well as the drawing of the squares. II - The figure is not trivial. The central point represents here the golden

rectangle. One can easily identify the rectangle from this simple point.

In seeking the most westerly line, we get the exact measurement of 500

x φ royal cubits. The idea of the large golden rectangle becomes

obvious at the simple consideration of that point.

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The figure thus confronts a clear line, almost appearing on the plan, to

the base of its geometric structure, summarized in one point. Finally, we

must add to the geometrical relevance of the figure (not trivial), the

relevance of its meaning, inside a coherent beam of symbolic

affirmations. The Giza Plateau tells us a royal family story, especially

their kinship. The golden ratio shows here its masculine nature on an

expression of the lineage. Thus the center of the golden rectangle

punctuates the line that unites the grandfather to his grand-son. Khafre

demonstrates his delicateness when he moves aside the direct path

that connects his father and his son. III - The accuracy of this coincidence, on the order of a few centimeters

(on a field of over 67 ha *) is obviously the convincing criterion of this

beautiful demonstration by the Egyptian architects. This figure gains an

other status. The “proposal by the analyst” becomes a “perfect evidence

by the designers of the site”. And we must remember this lesson : the

architects wanted that we find this geometrical key. This key attests

their art of composition, as much as the meaning that they put in it. (*) 907,162 m x 742,353 m = 67,343 ha

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