9. ABOUT STRETCHING THE CHORD

In relation to the question about the practical use of the Phi proportions by the ancient Egyptians, I look into my pile of papers for an old device I invented, lets say reinvented because the ancient Egyptians, or before that time, were already using it. We know from wall paintings and carvings about the Egyptian practice of stretching the chords, therefore, there is nothing new about the chord stretching practice.

The possible method they use to proportionate distances based on Phi, is as follows. My device consists of three nails (they had nails). We need a measuring chord (they also had measuring chords).

In a vertical, or horizontal plane, nailed-up one nail, and at any other desired point, nailed the other. The distance between them no matter, it could be any distance, unless one is specifically required.

At the location of either of the two nails, set a perpendicular distance equal to half the distance between the two nails, and place the third nail (they knew how to establish perpendicular lines). See the Figure.

I have created a possible ancient Egyptian "computer chord Phi generator", as I called it, using the three nails to calculate Phi proportionate distances.

The length of the chord should be equal to the perimeter of the triangle.

This means that the cord's length would be (1+2 + Ö5) = (f³ +1). Note:  f³ is very easy to create and is an important number for the Egyptians, according to my pyramid's research.

To create Phi:

Fix end of the chord at point "a", turn around "b", and reach "c".

You have (Ö 5 + 1) which is equivalent to 2(f):  f = (Ö 5 + 1) / 2

You can see in article 4,What is wrong with Phi? a complete anlysis of the f formulas.

 About the Red Pyramid:

The f³, number, which I think the great Egyptian engineer, Imhotep, work a lot with it, includding Sneferu, and all following pharaohs, has unique properties.

For example: We know that the circumference of a circle is determined by the product of the Diameter and p = C = D (p).

The perimeter (P) of the base = 4b

D (p) = C

D (f³) = P

Think that Sneferu could have wanted for his Red Pyramid that the diameter of the circle multiplied by f³ be equivalent to the Perimeter of the base. 

In other words. C = D (p) = P= D (f³).

The engineer immediately calculated the slope angle for the Pyramid.

We have proved before that C / P for this pyramid, is equal to (p/f³) .

This means that p D / 4b = p / f³

p cancel out in the equation

Therefore, the perimeter P = (4b) = D (f³), and b = f³ / 2

Since the ratio (D / b) = slope of the Pyramid, then,  = 2 / (f³ /2) = 4 / f³.

 So, the engineer determined that Sneferu’s Red Pyramid needs a slope equivalent to 4 / f³ = 0.94427191 for the pyramid. He calculated the angle corresponding to this function and found (as I did), that it is 43.3581975 degrees = 43 ° 21’ 30”.

I have try, by all means, not to get involve with another theories, to keep mine clean and out of contamination. However, although similar theories could exits I guess they will never be equal to mine, it is completely original. I first exposed it in my Spanish book published in 1982.
This is the angle that must people use as the slope.

My theory implies that each pyramid was design base on a determined geometric configuration, which yields the characteristics the pharaoh wanted for his pyramid's design. As an example, and this is very simple, if you considers the configuration of a square circumscribed by a circle. You presume that the square represents the base of the pyramid and the radius of the circle is equal to the pyramid's height. With this data, you can draw the cross sectional view of a pyramid, as seen through the center of its faces.

You set the vertical and horizontal diameter for the circle. The horizontal diameter represents the base line of the pyramid. From the top end of the diameter, set an inclined line, to the vertical line of the square, which intersects the horizontal diameter. Extend these lines to intercept the circumference. These lines represent the slopes of the pyramid.

To finish, trace a horizontal line to join the intercept points of the faces with the circumference.

This is what I called a generic pyramid's design. You can build a pyramid, any size you want with this design. Just set the pyramid's height you want, and calculate every part of it. You need only the pyramid's height.

From the drawing, see that D divided by its side length (b), represents the tangent of the slope angle of the pyramid's faces.

Since the radius is 1, the diameter is 2.
See that D = to the square root of the sum of the two corresponding sides (this represents the now called, Pythagorean theorem). Therefore D = Ö2 multiplied by b.

D = (Ö 2) (b)

b = 2 / (Ö2) = (Ö2)

The tangent of the slope of the faces = D/b = 2/(Ö 2)= 1.41421356. The angle which corresponds to these function is 54.73561032° = 54°44' 8'.

Now you are ready to go. We have just to plug in the pyramid's height, or the base length, and the pyramid will be defined.


Let's assumed that the Pharaoh wants his Pyramid to be 620 ft at its base.

The ratio of the slope angle = D / b = 1.41421356
From the formula D / b = D / 620 = 1.41421356
Therefore the diameter of the circle is 876.8124, and the radius = pyramid's height = 876.8124/2 = 438.4062 feet.

Do you recognize the probable pharaoh that used this configuration?

Yes, Pharaoh Sneferu. This design complies with the lower section of the Bent Pyramid, which was combined (with the most interesting pyramid's configuration you ever imagine), to build his famous Bent pyramid.

The lower angle of the Bent pyramid is given by Petrie, using as reference his drawing identified as "The southern Pyramids and Peribolos" as 54 44'. The base length = 620 ft represents an average of the length’s measurements taken to the base of this pyramid.

Can we assume all of these are coincidences?

All right, let's look for a good one. Point X location in the pyramids, where the descending passage is aligned to, is located at the f = (1.618) ratio of the diameter of the circle. In this case, this point corresponds to a distance 876.8124 / Phi = 541.8999 feet, below this pyramid (not the Bent) apex. If the reversed situation I found in the Great pyramid to determine the location of the Pyramid's entrance, is used here, from this (Phi) set an inclined line with the slope two horizontal and one vertical (slope 26° 33' 54.18") to meet the face sides of the pyramid.

I did this, and the vertical distance from the base to the entrance is 38.05 ft. According to "The Pyramids by A. Fakhry," this vertical distance was measured as 11.8 m = 38.71'.

This situation is very interesting, since shows us the great possibility of the double configuration for the Bent, against the failure theory. This, additionally, could indicate that this pyramid's cross-section was used for the northern side, where the entrance location agrees. Now, what happens if we examine, using the same rules and principles, the other configuration? That is another chapter of this novel. However I guarantee that although it took me a long time because I used the Bent drawings and numbers of a book tht have an incorrect drawing. Finally, I found there was a big mistake in the drawing, the measurements do not fit! I do not know if the corrections for this drawing have been published.

After the correct measurements were used, the designed plan of the magnificent Bent Pyramid emerges from the combined configurations.

Don’t you think there is too much attribution to coincidences, in cases like this? Don’t you think the scholars should take a good look to this theory?

For example, in this Bent Pyramid’s case. It helps to establish the possible location of the entrance at the north, or west side, and the geometric configuration it represents. Is the pharaoh mortuary location at point (Phi) from the apex? Does Khufu copy his father (Sneferu) methods for pyramid designs and construction? That is very possible.

 

The importance of the ratio of the circumference of the circle

and the perimeter of the pyramid's base (C / P).

We know that the original pyramid's height can not be measured. The GP is truncated, and it is not known if it really ever had a capstone, or was finished to the apex. Please refer to the topic "Was the Great Pyramid Completed" where these threads were initiated. We are referring to the GP's height to the imaginary projection of the finished apex, if the GP were completely finished. That is, the theoretical point in elevation intended to be established with the top of the pole (staff), placed on top of the platform in the upper section of the Pyramid, in 1874. In our calculations, with the use of so many decimal places, we are trying to distinguish or separate, clearly, numbers that are very closed together, or very similar in the calculations. In another way, let's say, to help number's identification. It would be a pleasure to clarify any doubt you may have in our threads.

Lets considered the ratio (C / P) in the pyramid's designs. Since C = (D) (p) and the perimeter of its base is P= (4) (b), where b is the base's length, the ratio would be (C / P) = (D)(p) / 4(b). Notice that the factor (p / 4) is constant and the other factor (D / b) represents the function of the tangent of the slope angle of the pyramid's face. Note that using a circle, where its radius is equal to the pyramid's height and b / 2 equal to half the base's length, the slope angle of the faces is (R) / (b/2). Since R = (D / 2), the factor can be changed to (D / b). This is easier to understand, and to work mathematically.

So, the ratio (C / P), for any right pyramid, would be equal to the constant (p /4) times the function of the tangent of the slope face's angle. For the Great Pyramid, the ratio (C / P) would be equal to (p / 4), times the tangent of the face's angle. For the value of Phi, the angle would be equal to 51.82729237° (= 51° 49' 38.25"). The result is C/P = 0.999041897. The obtained result means that the circumference of the circle is very close to 1.000000. I it were equal to one, the value of C would be equal to P, that is C = P, then the circumference of the circle use to design the pyramid, would be equal to the sum of its four sides (perimeter). In order to be equal, if you set C = P, the tangent of the angle would be equal to (4 / p ) = 1.273239545. The angle that corresponds to this function is 51.85397401° = 51° 51' 14.3", this is the gradient angle of the faces using the value of p.

If you consider this method of design for the GP, if the value of p was used, the circumference must be equal to the Perimeter. We can calculate from the measured dimensions, and projections, that they are no equal. It exists a small difference. This difference disappears when we use the value of f. If the angle of the faces is 51.82729237°, the tangent of this angle is = 1.2701965 = (D / b). So, b = D / tangent of 51.82729237° = (480.6636)(2) = 755.7487 feet (the average value measured by Petrie's survey). How come this calculations come to this, just by chance? If you consider that there is no relation between the Pyramid and the circle, is amazing the coincidence.

The apothem would be equal to the Öf times the radius. That is, (Öf) (480.663676) = 611.4136407 feet, which also agrees with the reference's value. As a matter of fact and as a note of interest, this dimension seems to be very important. It is equivalent to 186.36 meters. As a note of interest, when I visited the St. Peter's Basilica in Rome, I noticed a plaque in the floor's entrance, which specifies that the Basilicas's length is 186.36 meters (the same length, I took a photo of it). The tourist's guide informed me that no other basilica in the World can be authorized by the Vatican, larger than this length. As you will figure out, I will never forget that the length of the St. Peter's Basilica is equivalent to the Pyramid's apothem.The construction history of the Basilica is correct. Besides, the apothem dimension of 186.36 meters also seems to be correct. Andre Pochan in his book "The Mysteries of the Great Pyramid", page 12, indicates that the complete GP's apothem is 186.37 meters (611.44 ft). I found what seems to me a relation between the Basilica's dimensions compared to those of the GP. For example: The interior length of the Basilica represents the GP’s apothem, while the length of its central nave represents the GP’s height. The height of the central nave represents the Grand Gallery longitudinal length. The façade symbolizes half the pyramid’s sides, less the hollowing at the base. These information can be shown with numbers.

The plaque over the floor near the bronze door of the Basilica indicates that the dimension of the interior of the Basilica is 186.36 feet (the same as the 186.36 meters of the GP's apothem). This information on the plaque looks to me like a key about the Basilica’s dimensional relations with those of the GP. For this, to happen just coincidental, even shown to the decimal points is intriguing. I am sorry the Basilica's dimensions I found were round off to complete meters. However, they show my point. The length of the central nave of the Basilica is given as 146 meters. This length dimension is very similar to the 146.50 meters corresponding to the GP's height. The height of the central nave is given as 46 meters, which compares to the longitudinal length of 46.63 meters of the Grand Gallery. The facade is 114 meters wide, which is half the side length of the GP's base, less the 0.91 meters corresponding to the hollowing of the faces, that is, (230.35 / 2) - 0.91 = 114.27 meters.

The exact dimensions shown in the Basilica's plans could resolve this argument. However, although I have many reference books on this topic, most of them only present dimensional scaled drawings.

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