CHAPTER 5

Great Pyramid’s Geometrical Solution

It is not known how the geometrical design of the Great Pyramid was created. However, it is known that the structure has a pyramidal shape with four equal sides, built to certain dimensions. It is assumed that Pharaoh Khufu, from an infinite number of geometric forms and dimensions, chose this specific geometric design and established the top elevation to which his pyramid would be raised.

The process to trace the geometrical design of the Great Pyramid is one of the mysteries that many experts have tried, and continue trying to solve, during many years. Its geometrical configuration has led to many theories in the pyramid’s design field. Why the pyramidal form? What is the meaning of the pyramid? How was it designed? How were the design measurements set to perform the survey work over the original terrain? Why those dimensions? These questions and others are still waiting for the correct answer.

My purpose, after a research of so many years, is to establish the theory, and prove, that the geometrical design of this Egyptian pyramid has its origin in the geometric configuration I developed and exposed in Chapter 2. Besides, that when the radius of the circle used for the design is set equal to 480.6637 feet (also equal to the pyramid’s height), the dimensions of the pyramid delineated in the configuration will become equal to those corresponding dimensions as shown in the Great Pyramid’s structure. The geometrical specifications I set to trace the interior sections of the pyramid’s model, consisting of the location of horizontal and inclined passages, chambers, and other structural details in the pyramid, clearly shows equal or very similar measurements as those that have been measured in the corresponding parts in the Great Pyramid’s structure. We have to keep in mind that my pyramid’s model represents a design work, as shown in the plans. The Great pyramid’s dimensions show the actual measurements of and old structure built almost 5 centuries ago. Nobody would expect them to be exactly equal. Even if the work was carefully done, some measurements will slightly different. In this case, we have to evaluate if they are under an acceptable tolerance.

My theory concerning the geometric design of the Great Pyramid is based on simple geometric concepts. These make the theory easier to understand by the readers. The theory is not concerned with religion, Archeology, Egyptology, prophecies or extraterrestrial aspects. Not even with the measurement units used by the ancient Egyptians for its design and construction. The use of a circle with a unitary radius for the design permits us to do it in that way. Any measurement unit can be used in the system. To make it easier, the comparison between the dimensions of my pyramid’s model, and those of the Great Pyramid, were based on feet units.

The relation between the designs can be seen in another way. For example, in one side is my pyramid’s model, result of a design, independently created from the known Great Pyramid’s dimensions and characteristics, based on particularly fixed geometrical specifications and following a specific geometrical process. On the other side, we have the Great Pyramid’s structure, a stone millenary building, with internal and underground corridors and multiple chambers.

It is worth to question, what reasons exist for the geometrical projections of these two supposedly independent geometrical designs to be equal? Why then, when the height of the pyramid models is set to 480.6637 feet, the angles and dimensions in its design agree so precisely with those corresponding in the Great Pyramid’s structure? Besides, of most importance, although considering that both pyramid’s heights were set equal just by chance, this is not indicative that the slope angles, and all other dimensions and characteristics in the structures, have to be equivalent.

The results from the comparative analysis of both structure dimensions, presented in the previous chapter, showed that there is enough numeric evidence to establish the theory, and to confirm mathematically, that the designed model compares favorably with the Great Pyramid’s exterior and interior angles and dimensions for corridors and chambers. The accuracy in both designs is too much evident to be coincidental.

The following are some important items that strengthen my point of view of the equality between the two designs.

Figure 88 shows the specific area in the Great Pyramid’s drawing where the following reference numbers are located.

Figure 88

1. The inclined measurement of the Grand gallery, from north to south wall is 156.81 feet (equal to the measurement in the Great Pyramid).

2. The "great step", in my pyramid's model, is located in the same position as in the Great Pyramid. Its calculated vertical measurement is 2.71 feet (32.5 inches). Some investigators indicate that this step measures 36 inches. However, Sir W. M. Petrie states that the top corner of the step has been raised during reconstruction works, in relation to its level at the base of the south wall of the Grand Gallery. Therefore, its adjusted measurement should be 32.5 inches.

3. The downward step at the entrance of the Subterranean Chamber is also found in the Great Pyramid. My design shows its measurements as 32.5 feet. Actually is about 29 inches, however the chamber's floor was left unfinished.

4. The inclined descending passage angle (26.56551ş), and its length (344.91'), from the pyramid’s entrance to where it changes to horizontal, is evidently equal in both designs.

5. The length of the horizontal section of the descending passage leading to the subterranean chamber is 26.79', which is equal to this measurement at the same location in the Great Pyramid. In addition, both are located at 100' underneath the base.

6. The alignment of the ventilation shafts in the Queen’s chamber seems to fit almost exactly in reference to those on the Great Pyramid. It is said that these shafts do not have exits at the pyramid’s faces. However, in my model, the elevation angle is 38.17271ş and the projected length to their exits at the north and south faces, is 244.90'.

7. The alignment of the ventilation shafts in the King’s chamber, in the pyramid’s model, fit exactly the angles and measurements of the corresponding shaft in the King’s Chamber in the Great Pyramid. It should be noted that the length of the shafts in the King’s chamber, in their original position at the pyramid’s axis, did not fit in their length. However, when moved to the new axis location in the King chamber’s place, they fit accordingly with those of the Great

Pyramid. This is extremely important, since there are too many variables involved for getting the correct location.

8. The section of the design underneath the base of the pyramid provides a logical and reasonable explanation as to the existence and alignment of the underground passages and chambers, not found in other theories about this topic.

Coincidences are not unlimited, because if so... they would not be coincidences. It is evident that both geometrical configurations are identical. If both are identical, they must have a common origin. My theory can be sustained and proved with the data and the numeric evidence shown. Nevertheless, for the science of Egyptology, and other sciences involved, it will need confirmation. As I see it, this will be confirmed when ancient documents, or the original design’s plans of the Great Pyramid are found.

I think that those that believe that the passages and corridors were aligned with the stars, those that believe that extraterrestrial built the pyramid, and those that read prophesies for mankind, based on this Monument structure, and even archeologists, Egyptologists, will now think about the possibility that the origin of this structure was exclusively geometrical, although with theological motives.

Besides, it is also possible that the Egyptians designers prepared certain type of specifications for their Great Pyramid’s design and construction. My research included this phase. In Chapter 10, Criteria for pyramid’s designs, I presented the results of my investigation.

Although the geometrical process, in the design’s exercise, could be varied and simplified, I presented it in that way, which I considered easier to understand by all readers.

I. Advantages in the topographical survey

By means of my geometrical method, the Egyptian topographers could have easily established the necessary control points for the construction’s work. For example:

1. It allows the use of strings or ropes, to define the alignment for the use of long wooden mark rods, to establish and mark over the terrain the control points for the pyramid’s construction. The longest distance to measure would be equivalent to the circle radius (the pyramid’s height), that is, 480 feet and 7 inches. If the terrain for the construction was as perfectly level as it has been said, the strings and rods could be laid over the surface of the ground, increasing the measurement accuracy. They had to establish a good center point for the circle, which vertically would represent the pyramid’s axis.

2. The four points that define the corners of the base could be established measuring directly from the center of the circle. I am sure that the Egyptians would clear the circle’s center and necessary measuring routes to increase the accuracy of the base’s set up. If this topographical work was possible to perform over the terrain surface, there was no need to measure the base’s sides, except to confirm the measurements. Consequently, since the distances to be measured were shorter, this would increase their accuracy.

3. My geometrical method allows tracing, in an actual scale, over the surface ground, the projection and details of corridors and chambers to be constructed inside the pyramid, including the underground work to be done beneath the base. They could verify the angles and measurements of corridors and chambers directly over the construction site. The surface of the terrain inside the circle’s area could serve as a gigantic blackboard.

4. The geometrical method is also the ideal method to set the exact point over the surface terrain where the excavation for the descending passage, underneath the base, should begin and the angle to follow.

II. Characteristics of the Geometrical Design

The geometric property, which indicates that the circumference of the circle with a radius equivalent to the Great Pyramid’s height is equal to the perimeter of its base, has always been prominently exposed. The pyramid’s model design also responds to such a characteristic. The circumference of the circle is very close "although not equal" to the perimeter of its base. The ratio between those two values is almost equal to one, that is, 0.9990417.

Another interesting figure that I related with my geometrical design is that carved in the north’s side of the upper block of two suspended between the east and west wall’s inside the Antechamber (called granite leaf). This curious carving, although some investigators think represents a decorative sign of no importance, for me it appears to be something like the figure of a trademark left by the builders. Although I do not deny that other similar indentations could be made with other purposes. This indentation in the antechamber appears to me like a clue to the Great Pyramid’s design. It is actually too much deteriorated, but from its actual form, it is highly possible that the original form was as sketch in Chapter 8. In this chapter I show the relation I found between the geometry of this carving and my pyramid’s geometrical configuration. The fact that the form of this figure can be easily traced and scaled in its dimensions, using the same geometrical configuration I used for the Great Pyramid, denotes that there could be an important connection. While the carving of such a figure seems to be very complicated to delineate and carve, using my geometrical process it becomes a simple task.

The Bent Pyramid, built by pharaoh Sneferu in Dahshur, shows a special geometric configuration with two different angles in its sides. It has been said that the slope angle of the faces was changed during the construction’s stage when the structure developed cracks in the faces. At this time, the angle was supposedly changed to a less inclined slope in order to improve its structural stability.

When I examined the information and data about this pyramid (see Chapter 12), I had the feeling that the two different angles at the faces were the result of superimposing two different sections of pyramids, one over the other. Each one of the superimposed pyramid, have different geometrical characteristics. After my analysis, I concluded that the two different angles shown in the faces of the structure are the effect of a very good an interesting design made by the ancient Egyptians designers; contrary to the attributed failure in their construction work, as seen by today’s experts. This unique design leads to link, in a definite way, the pyramid’s design with the figure of the circle, as explained in this book. As far as I know, this type of analysis of the Bent Pyramid’s geometry has never been presented before.

III. Pharaoh Khufu - Mortuary Chamber

The mortuary chamber built for Pharaoh Khufu had to be an extraordinary one. A pharaoh with the power to put to work, each day, hundreds of thousands of men for 30 years (according to Herodotus), in order to build the dwelling place for his afterlife, surely have to construct a majestic mortuary chamber for himself. Neither the sarcophagus in the King’s Chamber, or other chamber’s properties found in the Great Pyramid, are representative of the ambition that a pharaoh of such power, ingenuity, and obsession for perfection, would show.

I believe his mortuary chamber was built near, or underneath the position established as point X, in my geometrical configuration’s design. It can be noticed that the inclined descending passage’s direction is toward the point X, shown in the configuration. At a certain location in its route, the descending passage becomes horizontal, apparently trying to conceal the direction of the real mortuary’s chamber. It is interesting to note that the change in the descending passage inclination angle takes place at the exact location calculated in my pyramid’s design model. The horizontal section leads to an unfinished Subterranean Chamber. It is possible that with the construction of this chamber, the designers wanted to hide, even more, the real pharaoh mortuary chamber’s location.

Some other indications of the possible presence of the pharaoh’s mortuary chamber at point X, are as follows’. The floor of the east side of the Subterranean Chamber shows very deep excavations done in the floor, but nothing of value has been reported as found. In the design of my pyramid’s model, the pharaoh mortuary chamber would be located to the west side wall of the Subterranean Chamber, not in the east side. It is exactly in the pyramid’s axis location or to its west. In the horizontal section of the descending passage, approaching the Subterranean Chamber exists an enlarged section, in height and width of the passage dimensions that could be an entrance to a corridor leading to an unknown chamber. Besides, also in the inside of the Subterranean Chamber, in the west wall side, there is a small excavation in the wall, showing no exit that might be an abandoned, or sealed entrance to another chamber. To these, also can be added the almost vertical corridor tunnel from the Grand Gallery to a place near the horizontal section in the descending passage. One ramification of this tunnel could lead to the suggested location of the pharaoh’s mortuary chamber.

The imaginary vertical line that establishes the center of the entrance to the pyramid and corridors is located about 23.85 feet to the west side of the center of the north’s face. Although there are many arguments in relation to the purpose of this displacement, an additional one could be, to make the pharaoh’s chamber more difficult to find.

It is important to mention here that although the entrance to the pyramid, passages and chambers were displaced to the west side of the vertical axis, their measurements and angles, as projected in the cross sectional view of the pyramid, from North to South, are not affected.

IV. Three-dimensional Design of the Great Pyramid.

My studies about the Great Pyramid are based on two dimensions. The figure of the triangle in the geometrical configuration represents the vertical sectional plane, north to south, cutting through the center of the pyramid’s face. Nevertheless, it is known that the pyramid is a three-dimensional structure and that the figure of the circle and the square represent the projection of a sphere and a cube, where the side of the cube is equivalent to the pyramid’s sides.

Pharaoh Khufu, with his initiative, gave the humanity a real model, a gigantic one, of the pyramid produced with this important geometrical configuration (figure 89). He succeeded in building the best proportioned and the most beautiful pyramid. In the upper hemisphere of this imaginary sphere, he shows to mankind the greatness of his work, the Great Pyramid. In the lower hemisphere, underneath its base, is the secret part... his mortuary chamber and treasures?

 

Figure 89

In reference to the circumference of the circle, it represents a projection of the plane cutting a sphere. Some people are surprise by the fact that the ratio between the height of the pyramid and the perimeter of its base is equal to the ratio between the radius of the Earth’s Globe and its circumference, as seen through the Earth’s Globe parallel where it is located.

If the design of the pyramid was established by means of the circle, as I have indicated, this should not be a surprise. Since the Earth’s Globe is almost spherical, all cutting planes through the center of the Globe will produce the projection of a circle. So, all cross sectional views of the planes cutting parallels and longitudes, produce the circumference of a circle.

The formula for the circumference of a circle is C = 2 p R, where C is the circumference and R is its radius. In figure 92, assuming the Earth’s Globe is a sphere, the circumference of the projection of a sectional view through the center would be equal to Ct = 2 p (Rt), where (Rt) is the Earth’s radius. As explained in my configuration, the pyramid’s design emerges from the figure of a circle. The circumference of this circle would be equal to Cp = 2 p (Rp), where (Rp) is the radius of the circle. The radius of the circle also represents the height of the pyramid (hp). The perimeter of the base of the pyramid (Pp) is equal to 4 times the length of it side (4b). That is, the value of Pp = 4b.

 

 

Figure 90  & Figure 91

I have explained in this chapter that the circumference of the circle is almost equal to the perimeter of the pyramid’s base. Their ratio is almost equivalent to one. Assuming this value is one, then Cp = Pp = 4b. Consequently, It can be substituted in the formula the value of (hp) by (Rp) and (Pp) by (Cp). Since both formulas are equivalent to 2p, they can be set as equivalent. Then it can be established that (hp) / Pp = (Rt) / Ct. This proportion shows that the ratio between the height of the pyramid and the perimeter of its base is equal to the ratio of the radius of the Earth’ Globe and its circumference. For example, for a terrestrial radius of 3,963 miles and a circumference of 24,900 miles the ratio would be Rt / Ct = 3,963 / 24,900 = 0.159. While for the pyramid, with a height of 480.66 feet and sides 755.75 feet, the ratio would be hp /Pp = 480.66 / (755.75)(4) = 0.159, That is, the same value obtained for the Earth’s Globe. In relation to the location of the pyramid, the parallel where it is located is irrelevant and has nothing to do with the formulas.

There is no doubt that the construction of the Great Pyramid was complicated, and required extensive and painful work. It is not easy to build a structure without machines to increase effort capacity and efficiency, and that requires three decades of labor by thousands of workers.

The situation becomes even more difficult to obtain such a high degree of precision and tolerance levels in their work. Surely these additional requirements will multiply the effort required. It is not a simple task to cut and place a huge stone block, weighing over 50 tons, and find lately that it was misplaced by several inches, and there is a need to reposition the block. Sometimes it’s not the human effort required, but the space limitations to do it. The Great Pyramid is not a work that will be duplicated in human history.

I remember, while standing alone observing the extraordinary architecture of the Grand Gallery, under dim electric lights, my meditation about the pleasure that I would experience in meeting personally the "genius" creator of the geometrical design of the Great Pyramid and its Grand Gallery. How did he lead his supervisors and laborers to move and place those huge stones according to his elaborated design plans? Most specially, using manual technology, that even at this time, the experts are not able to explain in an exact and convincing manner. At that instant, I had hundreds of questions to make, but nobody to answer. I never believed that human effort alone could produce a work of such magnitude. Probably many things were done and happened five millennia ago, that we are not even able to imagine.

The geometric configuration exhibited by the Great Pyramid is one where the circle, the triangle and the square (in representation of the sphere, the pyramid and the cube) are interlaced together in complete harmony and beauty. All of these in accordance with the Golden Section rules and the Golden Number presented in one of its better expressions. This geometric configuration is unique and has most interest (see Chapter 11). It deserves to be explained and mentioned in books, specially, about Geometry, Trigonometry and Arts. It is very simple to delineate, and contains a great philosophical explanation for the proportions of the parts of the human body (see Chapter 6, Leonardo da Vinci).

It is also believe by many that the Creator used the Golden Section to proportion the distances in the Universe, in the human body and his natural habitat, including plants and animals that are part of it. Assuming that my theory is correct, pharaoh Khufu, after examining all the configurations available for his pyramid (see Chapters 10 and 11), chose this geometric configuration because it shows the special characteristics, assumed as parameters by his God and Creator to built the World.

Figure 91

After determining the geometric configuration for his pyramid, Pharaoh Khufu had to establish its height. As mentioned before, the height of the Great Pyramid is 480.66' (= 153 p). Therefore, the circumference of the circle having the length of the Grand Gallery’s ceiling as its diameter (153 feet) is also equivalent to the height of the pyramid. Figure 92 shows also this relation in the Grand Gallery. If the idea of using the product of these two factors to establish the pyramid’s height is correct, then, a measurement equal to 153' (or its equivalent in any other unit of measurement) must have influenced the pharaoh decision (see Chapter13)

The present elevation of the Great Pyramid’s top platform is informed as 450.00', [Ref. # 13, I. E. S. Edwards, page 118 (height = 450 feet)]. As illustrated in figure 93, the elevation of the top of the platform for our pyramid's model is 450.06 feet. This elevation also represents the sum of 140.25 + 156.81 + 153 = 450.06 feet. The partial vertical distances from the base of the pyramid to its top’s platform is 140.25', which represents the elevation of the King’s Chamber over the pyramid’s base, 156.81' represents the distance between the north and south walls of the Grand gallery, and 153', the length of the Grand Gallery’s ceiling. Is it possible that the pyramid was built up to the top of this platform (actual elevation), as shown in figure 93? Or is it a coincidence that both platforms have equal elevations?

 

 

Figure 93

The dimensions and geometrical relations shown in figure 93 correspond to those calculated for my pyramid’s model. However, by comparison, it was established that the figure also represents the Great Pyramid's dimensions and characteristics.

Therefore, it can be stated that when the height of my geometrically designed pyramid's model is scaled to 480.6637 feet (= 153 p), its dimensions and characteristics represent the dimensions and characteristics as have been measured by surveyors at the Great Pyramid's structure, consequently, both design plans are equal.

 

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