9. ABOUT STRETCHING
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
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.
Set nails at corners a, b and c.
I have created a possible ancient
Egyptian "computer chord Phi generator", as I called it, using the three nails to calculate Phi
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
5 + 1) / 2
see in article 4,What is wrong with Phi? a complete anlysis of the
f³, number, which I think the
great Egyptian engineer, Imhotep, work a lot with it, includding Sneferu, and all following pharaohs, has unique
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
The ratio (D / b) represents the function of the tangent
of the slope of the pyramid. This ratio is equivalent to 4 / f³ = 0.94427191. The
angle corresponding to this tangent’s function is 43° 21' 30", exactly the same
angle attributed to the upper section of the Bent Pyramid.
If the formula (D / b) = 4 /
is rearranged as (D) (f³)
= 4 (b), it can be observed that (4b) represents the perimeter (P) of the square
or base of the pyramid, therefrore D f³
Note the interesting
fact of this configuration. The diameter of the circle multiplied by
p represents the circumference of the circle,
while the same diameter multiplied by f3,
represents the perimeter (P) of the base of the pyramid.
Think that Sneferu could have wanted for
his Red Pyramid that
the diameter of the circle multiplied by
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 =
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 /
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
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
(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
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.
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
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.
TO ADVANCED CONTENT SECTION PAGE