SUMMARY OF CONCLUSION ABOUT THE GREAT PYRAMID OF GIZAH

I. My interpretation of the dimensions of the Pyramid is based on three new points, not previously made in other interpretations.

  1. The value of the cubit of the Pyramid is 524.1483 mm. This datum has been obtained from the study of Egyptian units of measurement and is confirmed by Egyptian geographical records: From tests of the King's Chamber, Petrie arrived at a value of 524.052 0.10 mm.
  2. The study of similar monuments of Egypt, Mesopotamia and Greece proves that the Pyramid constitutes a geographical projection of the Northern Hemisphere. It is a projection in which each quadrant of 90 degrees of longitude is projected on a face of the Pyramid. The Apex of the Pyramid corresponds to the Pole and the Perimeter corresponds to the Equator. It is a projection in which the Equator is at the right scale.
  3. All ancient authors who provide data about the Pyramid, except Herodotos, prove to be derived from the description of Egypt written by Aristagoras of Miletos, about half a century after Herodotos. The collation of these texts reveals that Aristagoras provided the following information:
    1. The side corresponds to the length of 1/8 of a minute of degree, and the perimeter corresponds to of a minute of degree.
    2. The apothem is 356 cubits.
    3. The length of the apothem corresponds to 1/10 of a minute of degree.
    4. The Pyramid ends with a pyramidion which may be included in the calculations, or may be excluded. The apothem is 356 cubits with the pyramidion and 351.6 cubits without the pyramidion. The apothem of the pyramidion is 4.4 cubits.

II. Two main interpretations of the dimensions of the Pyramid have been advanced to date.

  1. The interpretation championed by Petrie, according to which the Pyramid incorporates the relation pi.
  2. The interpretation based on the text of Herodotos, according to which the Pyramid incorporates the relation ø.

In my opinion both theories are correct. The relation pi and ø were incorporated into the Pyramid, but the results were adjusted in order to indicate the correct values of the degree.

Petrie assumes that the height was set at 280 cubits and the side was set at 440 cubits, in order to indicate a relation 7/22 between the diameter and the circumference of a circle. He is perfectly correct because the height of the Pyramid represents the polar radius and the perimeter represents the equatorial circle.

According to Petrie the height was 280 cubits and the side was 440 cubits in order to indicate the value equal to 22/7. This is true in the sense that the planning began with a height of 280 cubits and a rough figure of 440 cubits for the side. According to these figures the slope would have been 5150'34". Petrie found that the slope seems to be 51 50' 40" 15" on the North face, which is the best preserved. A slope 51 51' 14" would give a perfect relation.

According to Petrie's figures the apothem is 356.0898 cubits. But the apothem was set at 356 cubits not only to achieve a round figure, but also to indicate a relation 89/55 between the apothem and the semi-side. The number is the limit of the Fibonacci series 1/2, 3/5, 8/13, 21/38, 55/89...The Fibonacci series, in which each number is the sum of the two preceding ones, was made known in our world by Leonardo Fibonacci of Pisa in 1204 A.D., on the basis of what he had learned in his travels in the Levant.

If the Pyramid had been calculated exactly according to the value of phi, it would have been somewhat squatter, with a slope of 51 49' 38".

The Pyramid can be calculated both by pi and by ø, given the similarity of 2/ø-1 (2 divided by the square root of ø) with pi/2

 

2/ø-1

equal to 1.5723

 

pi/2

equal to 1.57079

 

11/7

equal to 1.5714

 

2/(89/55)-1

equal to 1.57223

III. The Pyramid, however, was not built with a side of 440 cubits, but with a side of 439½ cubits equal to 230,362 mm. The Cole report of 1925 provides the following data for the length of the sides.

Maximum possible error in establishing the position of the corner blocks

 

West

230,357 mm.

( 30mm. at either end)

 

North

230,253 mm.

( 6 mm. at either end)

 

East

230,391 mm.

( 6 mm. at either end)

 

South

230,454 mm.

( 10 mm. West end

     

30 mm. at East end)

 

Average

230,363.75 mm

 

The West side was intended to have the exact length of 230.363 mm. The East side in principle should have had the same length, but for reasons that I shall explain later, there was added to it an amount of 34 mm., so that it came to be 230.397 mm. Possibly this amount was reckoned as 1/14 of cubit or 37.5 mm.

The manner in which the length of the East and West sides was arrived at explains also why the Golden Section was incorporated in the Pyramid. The Egyptians assigned cosmological meaning to the number, because it gives the trigonometric functions of several important angles:

sin 18

= cos

72

= 1/2ø

sin 54

= cos

36

= ø/2

sec 36

= cosec

54

= 2/ø

sec 72

= cosec

18

=

In geography and in practical land surveying, the Egyptians made extensive use of a right triangle with an angle of 36 in which one side is 100 and the the hypotenuse was 100 (2/ø). Often the hypotenuse was reckoned as 123, but more precisely it was reckoned by the mnemonic number 123.456. The correct figure would have been 123.6068.

The length of East and West sides was obtained multiplying the set value of the apothem, 356 cubits, by 2/ø or 5-1-1 taken as 1.23456... This results in a side of 439.5058 cubits, which apparently was rounded to 439½ cubits.

If the apothem of 356 cubits had been multiplied by the exact value of 2/ø the sides would have been 440.040 cubits.

The figure for the length of the East and West side was obtained by multiplying the apothem of 356 cubits by 1.23456 ..., but, if the height is 280 cubits and the side is 439½ cubits, the apothem must be 355.319 cubits:

282x 218.75 = 355.319

In order to obtain an apothem of exactly 356 cubits, the South side was lengthened, given that:

356 - 280 = 219/.8545²

Apparently in the measuring rods, the royal cubit of 28 fingers (16 fingers to the foot) was divided as usual into 7 hands, but these were not divided into 4 fingers, being divided instead into 6 parts which would be half-inches. In other words, the foot was divided into 21 inches,.so that the royal cubit was divided into 21 inches The inch was divided into ha;ves of 12.480 mm. As a result the length of the South side which should have been 439.709 cubits, was measured as 439 29/42 = 439.6905 cubits = 230,463 mm.

The amount of 100 mm.= 8/42 Cubits, added to the South side, was deducted from the length of the North side, making it 439 13/42 cubits= 230,263 mm.

According to this reckoning the South and North sides averaged 230,363 mm., which is the length of the West side. Cole reports the following findings:

   

North side

230,253 mm.

   

South side

230,454 mm.

 

Average

 

230,353 mm.

       

Later, we shall deal with the reason why the South side was made longer and the North side was made shorter. For the same reason the East side was made 3 half-inches or 37.4 mm. longer than the West side, so that it came to be 230,400 mm. The East side was 439 24/42 = 439 4/7 cubits.

In conclusion, the length of the sides, as I have calculated it, corresponds most closely with Cole's findings:

Intended Lengths Cole's Findings

 

North Side

439 13/42

cubits=230,263 mm.

230,253 mm. 8

 

West side

439

cubits=230,363 mm.

230,357 mm. 0

 

East side

439 24/42

cubits=230,400 mm.

230,391 mm. 3

 

South side

440 29/42

cubits=236463 mm.

230,454 mm. 8

Cole's findings are consistently lower by a few millimeters (6,10, 9 and 9mm.) than the figures I have obtained by calculation. This suggests that the discrepancy corresponds to the error of the Egyptian builders in making straight lines on the ground. The most common error in surveying is that of coming short, because of the difficulty in following an absolutely straight line.

From what I have said it follows that the intended figures were:

 

West side

2' 30"

W of true N

 

North side

2' 30"

N of true E

 

East side

5' 30"

W of true N

 

South side

2' 00"

N of true E

These figures appear even more amazing when we compare them with those reported by Petrie for the Second Pyramid of Gizah:

 

West side

4' 21"

W of true N

 

North side

5' 31"

N of true E

 

East side

6' 13"

W of true N

 

South side

5' 40"

N of true E

First of all we notice the same contraction of the North side as in the Great Pyramid. These is also an improvement in the pattern, which however, makes the construction more difficult. Apparently the North sides were intended to be parallel and to be at an angle of 5' 30" with the meridian: this is the same angle that occurs in the East of the Great Pyramid.

Possibly the West and East sides were intended to be an angle of 1' 50" , 1/3 of 5' 30" , with each other. If the figures of Petrie are correct, 1' 10" of this angle was assigned to the West side and 0' 40" was assigned to the East side. The figures could be also 1' 00" and 0,450"

In conclusion, the orientation of the sides of the Second Pyramid may have been:

 

West side

4' 30"

W of true N

 

North side

5' 30"

N of true E

 

East side

6' 15"

W of true N

 

South side

5' 30"

N of true E

In order to interpret these figures it would be necessary to know the exact orientation of the galleries of the Great Pyramid. At present my tentative interpretation is that the sides were at an angle because they were related to sighting devices.

An angle of 0' 15" corresponds to a second of time in the apparent motion of the stars.

An obtuseness of 0 00' 30" in the SU angle causes a lengthening of the East side in relation to the West side which is reckoned by tan 000" 30"=0.0001455 as 33.5 mm. If we suppose that lengthening was rounded up to 3 half-inches or 37.4 mm., the obtuseness of the SW angle becomes 00" 33", the tangent of which is 0.0001600, implying a lengthening of the East side by 36.85 mm.

In conclusion, it appears that the sides of the Pyramid were calculated with phenomenal precision.

IV. According to Cole, the orientation of the sides of the Pyramid was the following:

 

West side

2' 30"

W of true N

 

North side

2' 28"

N of true E

 

East side

2' 28"

W of true N

 

South side

1' 57"

N of true E

From what I have said it follows that the intended figures were:

 

West side

2' 30"

W of true N

 

North side

2' 30"

N of true E

 

North side

5' 30"

W of true N

 

North side

2' 00"

N of true E

Those figures appears even more amazing when we compare them with those reported by Petrie for the Second Pyramid of Gizah:

 

West side

4' 21"

W of true N

 

North side

5; 31"

Nof true E

 

East side

6' 13"

W of true N

 

South side

5' 40"

N of true E

First of all we notice the same contraction of the North side as in the Great pyramid. There is also an improvement in the pattern, which, however, makes the construction more difficult. Apparently the North and South sides were intended to be parallel and to be at an angle of 5' 30" with the meridian" this is the same angle that occurs in the East side of the Great Pyramid.

Possibly the West and East sides were intended to be an angle of 1' 50" , or 1/3 of 5' 30" with each other. If the figures of Petrie are correct, 1' 10" of the angle was assigned to the West side and 0' 40" was assigned to the East side. The figure could be also 1' 00" and 0' 5.

In conclusion, the orientation of the sides of the Second Pyramid may have been:

 

West side

4' 30"

W of true N

 

North side

5' 30"

N of true E

 

East side

6' 15"

W of true N

 

South side

5' 30"

N of true E

       

In order to interpret these figures it would be necessary to know the exact orientation of the galleries of he Great Pyramid. At present my tentative interpretation is that the sides were at an angle because they were related to sighting devices.

An angle of 0' 15" corresponds to a second of time in the apparent motion of the stars.

V. the four sides of the Great Pyramid have a basic length of 439 1/2 cubits. This length was intended to indicate the length of the degree at the Equator.

they indicate a minute of degree of 1,842.905 m. and a degree of 110;574 m.

According to modern calculations a minute of latitude at the Equator is:

 

by Clarke Spheroid of 1866

1,842.787 m.

 

by International Spheroid

1,842.925 m.

The addition of the pyramidion was intended to indicate the assumed elongation of the Earth at the poles

The apothem indicates 1/10 of minute of degree. The apothem without the pyramidion gives the same values as the basis. The apothem with the pyramidion (356 cubits) gives the length of the degree at the Pole"

 

356 cubits =

186,596.8 mm.

 

minute of degree

186,5968 m.

 

degree

111958 m

Present value for minute at Pole

 

Clarke Spheroid of 1866

1,861.656 m.

 

International Spheroid

1,861.666 m.

However, the apothem on the West side conformed, the side which conformed to the basic length at the basis, was 355.319 cubits, would indicate a minute of 1,862.398 m.