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PRIME NUMBERS, TRIANGLES AND PARTICULAR RATIOS 7-11-13 * 11-13-17 * 41-53-67 PYTHAGORAS Pythagoras many centuries after the pyramids will state the relation between the sides of the particular right-angled triangle whose dimensions are respectively equal to 3,4 and 5, that is to say three numbers separated by a constant interval, whose the square hypotenuse is equal to the sum of the squares sides.. There is an infinity, i.e. all the triangles which are proportional to it such as 6,8,10, or 9,12,15, etc... and of which the values of dimensions are separated by a constant interval. The triangle known as of Pythagoras is present on several occasions in the constructions and is not the result of the chance. . . However Ancient Egypt don't know the multiplication nor division, consequently the squares and the square roots. This knowledge for example could be issued from a systematic research or be the consequence of the implementation of particular values. To take an example, the peripheral rectangle which includes the 3 great pyramids is distributed to the 3/5 by the Sphinx, which thus generates mathematically at least a triangle of Pythagoras. (In reality it will be more interesting to consider what occurs starting from the true grid).

What about 7 and 11? The one who is for example a little interested in the pyramid of Cheops knows or could note that its proportions result from the ratio 7/11. (Height 280 cubits either 7x40 and bases 440 cubits, or 11x40). These are two prime numbers which are following and which generate a identical property to the triangle of Pythagoras:7x7=49 plus 11x11=121 equal 170. However the square root of 170 is 13,0384 for the hypotenuse. i.e. with the hundredth precision a triangle 7-11-13 which is unique. Does there exist others? I am not informed of publication or unspecified evocation on this subject, and I met only three mathematical continuations of prime numbers in this case when I had worked out a calculation programme for my computer through the first 15.000 prime numbers and this with continuations consisted of progressive intervals going from 1 to 500. 7-11-13 (hypotenuse: 13,0384)that is three prime numbers which are following, (4/100èmes Error), 11-13-17 (hypotenuse: 17,0294) idem above, (3/100èmes Error),

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41-53-67 (hypotenuse: 67,007) that is also three prime numbers which are following but with an interval of 3, (7/1000èmes Error), and there is no other: the simple fact of taking a multiple (triangle proportional) of these values will lead that we will not use any more prime numbers, or they will not be integers. 2.700 years B.C. it could probably be only a research using the physical measurements which could not have an absolute precision as our current calculators. Only the three evoked triangles generate a continuation of the Pythagoras' type. If one refers to the use of the 5 prime numbers, to a systematic research, and to the fact that the multiplication and division were not known, a fortiori the square roots, this choice of 7/11 for Cheops could not be fortuitous. But especially and it was an enormous surprise for me; the Mathematical continuations 3,4,5 - 7,11,13 - 11,13,17 and 41,53,67 are not only physically present in the pyramids but they are constituent elements from the outset in the design of Giza plateau! Chephren's base lenght is 410, the distance which separates the Eastern bases of Chephren and Mycerinus is 530 the distance which separates the Eastern bases of Chephren and Cheops is 670 The mathematical continuation 41-53-67 is dedicated to the whole Giza Plateau. Continuing my research, it appeared to me that 7-11-13 was dedicated to Cheops each one having seen like an obviousness the ratio 7/11 of the pyramid. (Height 280, Bases 440). Its base can be also written 10 times 7-11-13 plus 13. Concerning Chephren, it is less obvious and I have few elements about it except its dimensions It is probable that the mathematical continuation 11,13,17 is dedicated to Chephren; One can simply note that its base is equal to 10 times the sum of 11-13-17. These particular triangles of Pythagoras' type were well known and present just as I had shown reports/ratios 4/5 of the perimeter of the three great pyramids subdivided into 2/3 by the sphinx... (either generating triangles 3,4,5).

NUMBERS AND RATIOS Proportions

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They result from the use of the grids and a calculation system using the prime numbers. I immediately point out that the values 41-53-67 are present at the basis of the concept, and with the fact that this very particular mathematical continuation to my knowledge had never made the object of any previous publication. The layout which results and which I show on the following diagram are obviously only partially illustrated.

Some of the values which result from the grids that I explained.

The Pyramids' location such as above shows obviously that it is not a question of a chance but of a calculated system. And still for the example in the East-West direction some additional relations:

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The Mycerinus' eastern base is located at 10 times 41 cubits from the first enclosing Temenos' wall; the distances separating the pyramids' eastern bases are respectively of 10 times 53 cubits for Mycerinus and Chephren and of 10 times 67cubits for Chephren and Cheops. Cheops is located at 10 times 53 cubits from the Temple of the valley and as a remind, Chephren has a base equal to 10 times 41 cubits. As for the Chephren Western base which is located at 10 times 53 cubits from North-South line passing by the Temenos. A simple calculation will enable you to note other relations in the distances separating the Western bases: 320 for Mycerinus-Chephren, then 640 (2 times 320) for Chephren-Cheops and finally 960 (3 times 320) for Mycerinus-Cheops, i.e.1 time, 2 times and 3 times the same value. Others quite particular still exist and I show below only some partial examples of them: From base

To base

Distance

Or

NorthCheops

South Chephren

1110

700+410

South Chephren

North Mycerinus

1110

260+410+440

South Cheops

South Chephren

670

260+410

East Cheops

East Chephren

670

230+440

5

East Chephren

East Mycerinus

530

410+120

The above components are among the most frequently met but are not restrictive. Each one can check that these various values which govern a single reality are in agreement with the whole system of 350 cubits grid subdivided in 11 times 30 cubits plus one time 20. (Euclidean division). All dimensions are rigorously in relations, it in the four directions, that confirms once more that Mycerinus was quite present since the design!

Among the used Numbers: Note: A prime number is a number which is divisible only by itself or by 1. Examples: 2, 3, 5,7,11 The values which we know are the pyramids' dimensions (400/280, 410/275 and 200/125) like 350 and 30 which represent the grid that I ha showed If one seeks a relation between the values, a common denominator or something according to our current methods, he will be very quickly stopped by the Chephren's base length which is made up of 10 times 41 cubits, and 41 is a prime number which cannot be common or component of the other pyramids. The system seems abnormal, just like the fact that the pyramids despite their rigour and the great pains taken to their realization are not aligned. This system is deliberated. It was made so that the anomaly causes the question. If the pyramids were aligned, there would be no interpellation and thus not research. I gave the response with one of the grids. It is not they which are important, it is the system which is not only deliberated but designed to transmit knowledge. Idem for this value of 41. If the question seems without answer it exists about it however at least one, but which implies other methods, and the mathematicians will quickly recognize there the analogies with "the algorithm of Euclid " and its positive and negative recursivity. Perhaps its source of inspiration or a collected knowledge? We used a grid of 350 cubits subdivided in 11 times 30 cubits plus one time 20, which can be also written a grid of 2x5x5x7 subdivided in 11 times 2x3x5 and once 2x2x5. If one wants to apply this same method for example to the components of dimensions of the three large pyramids, the measurements could be written: Base

Height

Cheops

(440)

2x5x5x7 plus 3 times 2x3x5

(280)

5x7 x2x2x2

Chephren

(410)

2x5x5x7 plus 2 times 2x3x5

(275)

5x5 x11

6

Mycerinus

(200)

2x5 x5x7 minus 5 times 2x3x5 (125)

5x5 x5

It is still a report. This also has the merit of use only the 5 first of the prime numbers (2, 3, 5, 7, and 11) and to be in coherence with the whole. If one considers the architects' ruler of Ancient Egypt, one will notice that this ruler is mainly divided into 28, in 7 and 2x2 as well as in simple ratios. The constants and relations between the three large Pyramids could be written: Constant 2x3x5 Secondary grid = 2x3x5 (30) It appears as well in the bases as in the main grid. Constant 5x5 Mycerinus' Height = 5 times 5x5 (125) Chephren's Height = 11 times 5x5 (275) Mycerinus' Base = 2x2x2 times 5x5 (200) Constant 5x7 Cheops' Height = 2x2x2 times 5x7 (280) Cheops' Base = 2x5 fois 5x7 plus 3 times2x3x5 (440) Chephren's Base = 2x5 fois 5x7 plus 2 times2x3x5 (410) Main grid = 2x5 times 5x7 (350) One can also note that Chephren is the only one to use two constants (5x7 and 5x5) whereas Cheops uses 5x7 and Mycerinus 5x5. Chephren could be also written by using these two constants, which would give: 11x 5x5 =275 for its height, 11x 5x7 plus 5x5 = 410 for its base, and also 2x2x2x2 times 5x5 (400) plus 2x5 for its base. It would also seem that the first of the great pyramids (Cheops) was designed in terms of dimensions with: 2x2x2x5x 11 for its base (440) and, 2x2x2x5x 7 for its Height (280). 7 and 11, remarkable numbers of which Pythagoras could not be unaware of! The pyramid of Mycerinus would have been designed with:

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5x5x 2x2x2 5x5x 5 One can also note that the height of each of the three Pyramids was designed with: 2x2x2x5x 7 for Cheops (280), it is also 10 times the 28 subdivisions of the Ancient Egypt Architects' ruler. 5x5x 11 for Chephren (275) 5x5x 5 for Mycerinus (125) A little exemple more: Whereas the values of the two other pyramids require the use of the first five prime numbers, Mycerinus requires only two of them: The first and the third. (Does not the ancient name of the pyramid is “Mycerinus is Holy”?) Its Height is written: 5x5x5 (Nowadays we would raised to a cubic power) Its base is written: 2x2x2x5x5 (Nowadays we would say a cubic power by a square power) A lot of relations whose complexity of the result has as a counterpoint only the simplicity of the implementation... There is still there no thesis; it is a simple measurable daily report, and many others one exists. Except the three mathematical continuations, only the first five prime numbers were used to describe the entire whole.

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A RELATION SUCH AS y=ax+b The whole measurements result from a relation such as y=ax+b and we enumerate them in the overall plan and the two tables below (or being the sum between them of various been worth which appear in it).

Value 30 60 90 120 150

a 1 2 3 4 5

x 30 30 30 30 30

+b 0 0 0 0 0

Value 200 230 260 290 320 350 380 410 440 470 500 530

a 6 7 8 9 10 11 12 13 14 15 16 17

x 30 30 30 30 30 30 30 30 30 30 30 30

+b 20 20 20 20 20 20 20 20 20 20 20 20

Example: 670 can be consisted as 440 plus 230 or 410 plus 260, or 200 plus 210 plus 260, and so on. . .

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IMPORTANT REMARK CONCERNING MEASUREMENTS

It is always difficult to obtain precise and reliable physical measurements. As I said, I used only one copy of a Napoleon's chart and the apex angular statements published in the literature. If all the unit agrees, there is an exception which is the following one: I do not have a sure knowledge of the distance which separates the parallels passing by the Mycerinus'Eastern base and the Chephren Western base If we use the angular statements of the Apex, this distance would equal to 120 cubits. I met on Internet distances which can be more or less from 180 to 150 cubits but in these cases there are several degrees of error in the statement of the two apex concerned. Which would be the consequences if we took into account this value of 150 or 180 cubits ? This would remain matching with the grids of 350 distributed according to the Euclidean divison of 11 times 30 and with a complement of 20. On the other hand, in this case we cannot any more speak about the perimetric rectangle including the three pyramids in a ratio of 4/5 nor to say that mycerinus would have been conceived in same time. However, the distribution in 3/5 and 2/5 in North to South direction still remain. It would result from it obviously a difference in the relation of the distances which connects Mycerinus and the bases of the other pyramids, but this does not change anything with the concept. You can consult below these two cases..

Case of the distances (120 cubits) which matches with the angular statements...

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Case of the distances such as 180 cubits then generating a ratio of 485/745, i.e: an angle of 33°3' 52 ("error " of 3°15' 19).

Below, what the Mycerinus’ relations become:

Case of the distances (120 cubits) which matches with the angular statements...

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Case of the distances such as 180 cubits then generating a ratio of 485/745, i.e: an angle of 33°3' 52 ("error " of 3°15' 19). On Line: http://numerus.free.fr/ or http://gotogiza