Star watching was usual and a calendar year had been set up with 12 months, 30 days each, and an additional 5 days (no leap year). Thus the notion of angle and the concept of geometry were known.

4000 Years B.C., Babylonians had already a very advanced system of classification comprising numbers'60 bases and the question of the Triples known as Pythagoreans certainly arose before the construction of the pyramides,(-2550). It appears in form written in the shelf babylonean wedge-shaped known as Plimpton 322,(-1800), then will be treated towards -500 by Pythagoras with its famous triangle, followed in that by Euclide, Diophante d'Aléxandrie and even more close to us by Pierre de Fermat, Euler, Gauss, etc...
and will show that to be checked, the relation x²+y²=z² must comprise at least one of with dimensions of the right angle with an even value what contradicts the triples made up only with prime numbers that I expose.
I point out however that at the time measurements and experiments could probably be only graphic.

I thought it was interesting to face geometry and its bases the same way Man faced his initial environment.

Today, shapes and figures look obvious for everybody and bring no question. It was probably not so by that time.

Basic geometrical figures:

Two straight lines will define a point at their intersection.

With three straight lines, we obtain a triangle (polygon), with four a quadrilateral with special cases like trapezium, rhomb, rectangle or square... and so forth.

And this is only plane geometry.

If we map, in three dimensional space, the triangle over itself, we get a triangular based pyramid. If we map the 3 side figure (triangle) over the 4 side figure (square), we get a quadrangular pyramid (like the Giza pyramids), and if we map the square over itself, we get a cube.

A kid can do that.

Furthermore, in the same way a square can become a rhomb, a pyramid can be regular and symmetrical or irregular. It can also have different heights over a given base.

A lot of new questions arise about these simple figures and the corresponding ratios and proportions.

Ratios and proportions:

A simple triangle in plane geometry raises a set of questions like ratios between angles, ratios between angles and sides (today known as trigonometry), ratios between sides.

Triangles with identical angles but sides of different length will look like, but not be equal: they are called similar, and if their points correspond 2 by 2 along strait lines passing through a fixed point, they will be called homothetic.

Each property leads to a new one.

Several rectangles, laid side to side, generate new shapes, perfect or not, etc.

Even without explanation and demonstration, these simple facts could not be missed by the designer(s) of the Pyramids and raise all kinds of new questions.

Anyone can observe that the Pyramid stones are perfectly cut and assembled, using the right angle which was not only known by itself but also with its properties.

The so called Pythagorean triangle can be observed many times in the buildings and this is not just hazard. Though, since they did not know multiplication, and thus neither squares nor square roots, this knowledge could only be the result of iterative research and tests:

- a triangle with 3 equal sides one Cubit long,

- a triangle with 2 equal sides one Cubit long,

- a triangle with one side one cubit long,

- a triangle with one side one Cubit long, one side 2 Cubits long...

- idem including a right angle etc,

- or using the diagonals of a rectangle or square...like we probably all did on our school exercise book.

This is a system based on ratios and these are probably the questions that were studied by that time.

No sophisticated techniques, but essentially permanent observations and a lot of thinking.

A lot of geometrical knowledge is here, even without theorems and proofs.

The culture of that aera included this knowledge.

The Rhind Papyrus:

The British Museum hosts the rhind Papyrus, dated from approximately 1750 BC, that is to say about 700 years after Kheops building: it states the following question: "A pyramid is 93 Cubit and 1/3 high. What is the face slope angle if the face height is 140 Cubits ?"

The resulting angle value is indeed given by a ratio of measures using a unit: the Cubit.

The ancient Greeks later used to a great extent the culture and mathematical knowledge of ancient Egypt.

Around 500 BC, Thales, back from a trip to Egypt, stated the basics of geometry, his theorem on proportions and ratios of angles and triangles. This took place ten centuries after the rhind papyrus !

Euclid appeared later, on the third century BC, and synthesized the previous contributions within the so called "Euclidean geometry" still used today.

However, one can note, and I'll prove it, that the Giza lay out, designed much earlier than Euclid, obviously uses "the Euclidean division" and the notion of "Euclid algorithm"

Many knowledges were lost and it was necessary to remake all the advance of it.

The momification and the products which were used are an example.

It is also the case of the necessarily knownledge of anatomy by the embalmers , but it took thereafter many centuries for the occident to practise the dissection and (to re)discover the anatomy.

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