8. EGYPTIAN CAPSULES
I developed the following formulas, based on Phi = f, for some of the most important pyramids of Egypt and for the benefit of the readers.
Pyramids Slope Side Length Height Angle
D/b (b) (R) ° /' /"
Khufu Öf (D / Öf) = 2R/ Öf (b Öf) / 2 51.82729337°
51° 49' 38.3"
Chephren (4Ö2) / f³ (f³R Ö2)/ 4 2Ö2 b / ( f³) 53.17273225°
53° 10' 21.8"
Mycerinus 2 / f (R)(f) (b) / f 51.02655266°
51° 01' 35.5"
Red Pyramid 4 / f³ (R) f³ / 2 (2 b) / f³ 43.35819755°
43° 21' 29.51"
Lower angle Ö2 (R) Ö2 (b) / Ö2 54.73561032°
54° 44' 8.2"
Upper angle 4 / f³ (R) f³ / 2 ( 2 b) / f³ 43.35819755°
43° 21' 29.51"
R= radius, D = diameter, C= circumference, f = 1.6180339...
Slope = D/ b, Slope angle = Tan inv. (D/b)
Examples: Calculations of the pyramid's heights
Khufu (b Öf) / 2 = 755.7488 (1.27201965) / 2 = 480.66366 ft
b = 755.7488 ft
Chephren 2 Ö2 b / ( f³) = 2 Ö2 (706.24) / f³ = 471.5572 ft
b = 706.24 ft
Mycerinus (b) / f = 335 / f = 207.04 ft
b = 335.00 ft
Red Pyramid (2 b) / f³ = (2) (722.25) /f³ = 341.00 ft
b = 722.25 ft
Lower angle (b)/Ö2 = 620 (Ö2) = 876.8124ft (Extended-height)
b= 620 ft
Upper angle 2(b) / f³ = 2(722.25) / f³ = 341.00 ft
b = 722.25 ft
I do not consider myself as an expert in math. I consider an expert in maths a man like Imhotep, that had the ability to design three different calendars for Egypt, established the procedures for the architectural design of temples, pyramids, walls, columns, ceilings, and all kind of stone work necessary to build the Egyptian monuments. Besides being proficient in math’s, geometry, physics, almost all sciences, he was a high priest, with the rank of a Pharaoh. How come some Egyptologists credit the old Egyptians capabilities with maths, that evidenced in old papyrus, which seems only elementary work? From another point, other Egyptologists believe in the Egyptian knowledge and abilities to work with the cosmos, such as the star alignments, the constellations, and others sciences that require high math’s calculations and equipment to determine their exact location in the space.
It is amazing the struggle to put Pi and Phi out of their calculations. For me there is not reason to leave the value of phi. It's formula is seen in nature, in the galaxies, even in the human proportions. It comes out easily from the triangles in the ratio 1: 2 in their sides. Comes from the square, from the circle, helping in establishing beauty an art in the build structures, and make easier the calculations and field surveying. One Egyptologist told me once, that f and Pi are completely out of any pyramid's study. I believe that the exact location of f in any structure, was considered sacred and in line with their religious dogmas. Why the center of the niche in the eastern wall, in the Queen Chamber, was set at a distance f from the northern wall, instead of the center of the wall? Why the descending passage of the Great Pyramid is exactly aligned to the f point in the vertical axis of the pyramid (using a circle for its design with radius equivalent to its height)? Are these coincidences? The triangles with the ratio 1: 2 in its sides can be found any place in the pyramid's design. The f function is inherent to this triangle.
Besides the ventilation shafts in the
Great Pyramid, there are other mysteries about the pyramids. Where are the
working ramps the builders built for their construction? Well, as we assumed,
the owners of the pyramids want the labor force to clean up all the premises
after the job is finished. They want to see their finished property as better
looking as possible. This is so in this time, and I believe, as well, in ancient
times. I am sure the pharaohs required from the builders to remove all debris
and trash from the finished works, and deliver them somewhere. The better place
to place this material, in those times, were the original place where they were
excavated. The excavation of these ramp materials, soil, mud, sand, etc.,
creates deep trenches, where the material could be re-deposited and the surface
finished again, to erased all traces of its use. Maybe this is the reason, that
even the material used for the ramps is not found.
Nevertheless, scholars and Egyptologists have the ideas that they were constructed for ventilation, others to align the shafts with the starts, and many to allow the pharaoh's spirit to go out to the sky. You still can select your point of view since "none" of the "defined purposes" have been proved to be the correct one.
IN 1/3 AND 2/3 SEGMENTS
HOW TO DIVIDE, GEOMETRICALLY, ANY LINE IN 5 SEGMENTS
Now, if you are interested in dividing any line in (1/5 units), you can use this
similar method, which I also developed. The line can be divided in 2, 3, 5, and
many other whole numbers, an their multiples.
If I am required to divide the width of a wall in thirds.
case I will use the string and pin method to solve the problem.
Formulas based on f, which represents whole integers.
f2 - f = 1
f + (1 / f2) = 2
f2 + (1 / f2) = 3
f3 - (1 / f3) = 4
f4 + (1 / f4) = 7
f5 - (1 / f5) = 11
f6 + (1 / f6) = 18
f7 - (1 / f7) = 29
f8 + (1 / f8) = 47
f9 - (1 / f9) = 76
f10 + (1 / f10) = 123
f11 - (1 / f11) = 199
f12 + (1 / f12) = 322
f13 - (1 / f13) = 521
f14 + (1 / f14) = 843
f15 - (1 / f15) = 1,364
f16 + (1 / f16) = 2,207
f17 - (1 / f17) = 3,571
f18 + (1 / f18) = 5,778
f19 - (1 / f19) = 9,349
f20 + (1 / f20) = 15,127
f n = f n-1 + f n-2
Junction between the Descending Corridor and the horizontal passage.
Note that the cross-section of the descending corridor is larger.
My idea that the reason why each pyramid
has a different slope angle is that the Egyptian designers used different
geometrical configurations for their designs. Just as an example: the
configuration of a pyramid that is created by the circle which circumscribe a
square. Let’s say the square is the base of the pyramid, the radius of the
circle symbolizes the pyramid’s height. You can design a pyramid using this
configuration. The resulting slope angle, as can be easily calculate is 54° 44’
8’. Using this configuration, since the radius is one, any measurement, with our
favorite unit of measurement, can be used for its height. All the dimensions of
the pyramid will be automatically set, since they are proportional to its radius
I think there is nothing wrong to
investigate this proposal. The fact that this is new material does not means
that is wrong. I could defend my theory before a Scholar's panel, as I did to
gain my Master's degree in Engineering. As a professional civil engineer,
professional surveyor, professional photographer, have patented several
inventions in the US Patent Office, which are being used over the world. My book
contains more than 400 pages, over 225 computer generated drawings from the
pyramids, a collection of photos taken by me inside and outside the pyramids. I
spent over 35 years in these studies. I have read and search (many books, in
different countries during this period of time) to create my theory and support
it. I have visited and examined carefully the Giza’s area pyramid complex, and
taken photos, inside and outside the pyramids. I say this to demonstrate that I
have done my assignment, and that my work deserves respect and consideration, if
not credit at all.
Andre Pochan: Exposed Great Pyramid's creases
Andre Pochan worked seven years studying the GP and measuring all its structural parts. He is the author of a book published in Spanish and translated to English "The Mysteries of the Great Pyramid". It was originally published as L'Enigme de la grande pyramide".
of the GP faces are explained and their measurements given in Mr. Pochan's book.
In his book, he states: " the most remarkable fact is the hollowing of the
faces, which obliged the architect to displace the Descending Passage's axis
7.29 meters (14 cubits) east of the north apothem. This displacement of the axis
was necessary in order to avoid inundating the Subterranean Chamber, as each
face's hollow constituted a vast gutter capable of draining more than 2,000
cubic meters of water during a rainstorm". Mr. Pochan also sketched this
hollowing dimension at the midpoint of the base, as 0.92 meters (3 feet) as
presented in a paper read to the Institut d' Egypte in September 1935.This paper
also credited Mr. Pochan for the discovery of the red paint that once covered
the pyramids. Besides, recognizes that he was the first to called the attention
to the curious irregularity of the hollowing of the faces. The angle of the
hollowing at the level of the bedrock, as he measured and stated, is
approximately 27 minutes (0.45º = 27'
THE PYRAMID'S DESCENDING PASSAGE
THE CONSTRUCTION - AS I WOULD WORK IT OUT
To work out the
descending passage, it had to be done in two sections for a distance of 345’ :
from the ground level to the horizontal corridor and from the ground level up,
to the exit (entrance). This last section, up the pyramid, should not be too
In order to further confuse the intruders, the inclined corridor, leading to point X, was changed to horizontal, leading to the subterranean chamber (unfinished, because it was a fake). But maybe you have noticed, that before the entrance to this chamber, in the horizontal section, there is a small niche, about 1 meter deep and 1.85 meters width, at the west wall.
To me, this could be
the entrance to the real corridor to enter the Khufu's chamber which was
completely sealed. This information, to my knowledge, has never been
investigated. This supposed corridor to the King's mortuary chamber should
descend at any time in its way, to the Pharaoh’s chamber, which should be
located about 12 feet below, at the center of the vertical axis of the pyramid.
Inside the subterranean chamber, if you take a look to the west side of it,
there are some excavations which apparently were abandoned, but that also could
lead into corridors to the west side, descending to Khufu's chamber. If I am
right or wrong in my analysis, time will tell.
Function of f
Many people are scary about f and how to establish geometrically its ratio. I will explain, in a simple method, how to draw it based on right triangle, and based on a square.
You have a line (a - b), of any length, and want to divide it at the f point (it means that the ratio of the longest side of the line, divided by the shortest, is equal to 1.6180339 = f. With length (a - b) construct a right triangle with sides having the ratio (1 : 2).
1. From point c and radius (c - b) draw an arc to intersect the diagonal (a - c), and mark point d at the intersection.
2. From point a, and radius (a - d), draw an arc to intercept line (a - b), mark point e at the intersection.
Point e is the f point. The section of the line (a - e) divided by the section (e - b) = f = 1.6180339. If you divide section (c - b) by (a - e) the result is the inverse of Phi, that is, (1 / f) = 0.6180339.
It is required that the line (d - c) = 1, be increased so that its length (d - g) is equal to Phi = 1.6180339.
Construct a square using the length (d - c) as one of its sides. Divide the square drawing lines from the center of the sides, that is, (a - b) and (f - e). With a compass or a chord, from e, draw an arc with (e - b) as radius to intercept the extension of the line (d - c), mark point g, at its intersection.
Since (d - c) = 1, (d - e) = 0.5,
(e - b) = square root of (0.5)2 + (1)2 = square root of 1.25 = 1.118033989
(e - b) = (e - g) Therefore, (d - g) = (d - e) + (e - g) = 0.5 + 1.118033989
= Phi = 1.618033989.
How to create f3 and the inverse (1 / f3 ).
Draw a circle in each end of the diagonal, using the shortest side as its radius. See in the figure the values created of f3 and the inverse (1 / f3).
After you kneel down to clear the 41 inches of the limiting cavity to enter the Antechamber, the space is only large enough to permit a thin person. He has to rise straight up in a space having 16 inches (distance between the granite leaf and the Grand Gallery’s wall). He can barely move inside this space.
At a point in its floor line, just north of the granite leafs, the floor rises 1/4 inch, and then, following this level, continues until it reaches the entrance to the King Chamber, where appears a second rise of 3/4 of an inch.
.J. P. Lepre, indicates in his book The Egyptian Pyramids, page 89, that although the Egyptologists do not have a position regarding the two rises in the floor, that the two rises are not attributed to any roughness in the floor itself - for that floor is very finely leveled and smooth; but to a deliberate adjustment by the architect
To me, the 1st low passage (underpass) was completely seal from the entrance to the ½ inch raised floor stone block. This rise in the floor apparently was set on purpose to stop the sealing block from moving far into the inside of the antechamber, when the sealing block was placed, besides, to avoid been pushed by any robbers.
The second lower passage to the King’s Chamber apparently was also completely seal from its entrance to its exit a the king Chamber. The raise of ¾ inch in the floor elevation at the entrance of the chamber, could serve the same purpose, that is, to hold or stop the sealing block from moving too far forward, if pushed inside the chamber. That’s my explanation for the two raised floor blocks. Observe that the floor line was raised exactly at the appropriate locations, this has to be the answered for their existence.
However, It appears that the smart robbers cut around the Grand Gallery's south wall corner, getting access to the sealing block. They could introduce metal dowels into the block; tied them with chords, allowing pulling the sealing block back to the outside, into the Great step platform.
Entrance to the Antechamber Entrance to the King Chamber
After accessing into the antechamber, they brake the three blocking stones (portcullises), and repeat the same task again to take out the 2nd lower sealing block. Cut around the antechamber south walls (entrance to the King Chamber) to get access to the sealing block. Introduce metal dowels, which could be tied with ropes, to pull it out of the passage. This is what it can be read from the marks left. The cuts of the south wall of the Grand Gallery around the underpass-sealing block can be seen at the site, as the cuts at the entrance to the second lower passage (In this place, the cut sections were rebuilt, but it can be noticed. The following photos shown this entrance condition.
Yet another curiosity about the Antechamber, which everyone has his own ideas about its existence is the so called the carved boss. It was carved in the top section of the granite leafs. As I have stated before, we need to know the existence of something to look out for it. You cannot search for something you do not even imagine exists. Nobody knows what the carved boss is, therefore, nobody is looking for it. It just appear or looks like a working or handling boss for the stone block. If I show to someone a casting of the object, he could read nothing from it, except to describe it.
My case is different. the carved boss I found was what I was looking for. It was exactly carved in the location I expected. The size and configuration was also as I was looking for, scaled and derived from the Perfect Symbol. I reproduced the same carved object using my design configuration. It meant something special and positive for me. I was pleased to know that my years of personal crusade for finding the geometrical solution of the Great Pyramid showed another step toward its conclusion.
There is no reason (apparently) for the carved boss for been at that location. If the builder wanted to raise the block (that has the boss), instead of carving 1 inch deep in the whole face of the two slabs to create a 1-inch protuberance, why not dig a cavity in the upper slab’s side, capable of holding a wooden thick board? They were used to do that type of construction, as shown in thousands of block stones in pyramids, temples and tombs. Another thing, why the one-inch thickness of it, that denies its use as a working boss? Additionally , the horizontal bottom tip for the working bosses are flat to received the ropes and avoid slippage. the horizontal bottom of this carved object is slanted and will no work in the desired fashion as expected. For these particular reasons this carved protuberance could have had a different reason for its existence, time will tell!
I do not want to leave without citing this other example. I am referring to the well known "Zodiac of Dendera" in Egypt. This circular zodiac shows around its periphery the distribution of a group of men, women, and a group of constellation symbols. However, if any one is called to delineate the exact location of the figures in order to carved or paint the circular zodiac, it will be very difficult to create an equal geometrical pattern. Using the same configuration as for the boss, the drawing can be easily reproduced. I can reproduce the pattern, the astrologists will have to set the constellations. It can be done at any desired size scale for the zodiac.
"Zodiac of Dendera"