Characteristics and Dimensions - Model Pyramid
After finishing the geometrical design for the pyramid’s model in the previous chapter, its height will be established. Since a circle with a unitary radius was used for the design, this means than an infinite numbers of values and units of measurements are available to set its height. It should be noticed that when a value is selected for its height, all other dimensions in the configuration would be proportionally set. Besides, the geometrical configuration will keep its shape, because it is independent from the pyramid’s height. Therefore, the pyramid’s dimensions will adapt, proportionally, to any scale used to enlarge or reduce the pyramid’s size. However, it is also known that the determination of the pyramid’s height depends on the availability of equipment, materials, labor force, and other related aspects.
As I already stated, when the measurement calculation for the pyramid’s model is finished, a comparison will be made with those dimensions taken and reported by the reference books about the Great Pyramid. At this time, it can be assumed that if the general layout, and angles and dimensions in the design model are equal to those corresponding angles and dimensions as measured and reported in the Great Pyramid, both compared under the same unit of measurements, the two designs should be equal. Consequently, the geometric procedure used to create both designs should be also similar or equal. This would be in this way, no matter the unit of measurement used by the ancient Egyptians to design and build the Great Pyramid.
In Addition, the Egyptians designers could use cubits, pyramidal inches, primitive inches, or maybe any other unknown unit of measurement not identified to this time to build the pyramid, it does not matter. The distance between two points can be expressed in any unit of measurement, in accordance with the preference or convenience of the surveyor. Nevertheless, the same distance can be compared to the distance between two other points, if using the same units of measurement. This means that if the distance value calculated between two points in the pyramid’s model is equal to the corresponding distance in the Great Pyramid, using the same units of measurement, both distances are equal.
I decided to set a height equivalent to 480.66 feet for the pyramid’s model. This height is equivalent to 480 feet and 7.96 inches (equal to the product of 153 times the value ofp), expressed in feet unit. As explained before, the relation between the number 153 and the constant p inspired me, 30 years ago, to undertake this study and research about the geometry of the Great Pyramid. The height is the only numerical value to be introduced in the model design. Base on it, all the dimensions for the pyramid’s model will be calculated.
The dimensions shown in this Chapter and illustrated in the drawings were as calculated, and correspond only to the pyramid’s model, with the exception of those measurements enclosed in parenthesis and written in italic type, which refer to the dimensions of the Great Pyramid as indicated in the references. To make easier the comparison, I will use the unit of feet expressed to two decimal places. I believe that going further in the dimension accuracy would imply that human beings did not build the Great Pyramid.
Figure 58 shows the dimensions calculated for the exterior structure of the pyramid’s model. Since the radius of the circle is equal to 480.66 feet, its diameter is twice, that is, 961.33 feet. The calculated side length (shown as distance HT) of the pyramid’s model is 755.75 feet [Reference43, Petrie Flinders W. M., [the average of the pyramid’s base is 9,068.8 inches.] This number can be assumed as 9,069 inches that represents 755.75 feet. Note that using the formula I developed, the base length (b) = 2 / (Öf) R = 755.75 feet. The apothem, the incline distance from the center of the base to the projected top of the pyramid, is equal to 611.41 feet [Ref. #44, Pochán, André, page 42, (indicates that the GP apothem = 611.41 feet)].
It is interesting and calls the attention immediately, that the inclined sides of the triangle PQN, that is, sides PQ and QN, each one measures 755.75 feet, which is equal to the measurement of the sides of the pyramid. This means that the square of the base of the pyramid (between points I, II III, and IV) is equal to the square that can be traced using the sides PQ, and QN, of the triangle PQN. In Chapter 11, there is more information regarding the properties of this geometrical configuration.
The base of the triangle PQN, that is, side PN, is calculated as113.47 feet underneath the base of the pyramid. Therefore, the location of point X, the place selected in the geometrical exercise to located the pharaoh’s mortuary chamber and shown at the intersection of the pyramid’s axis and the center of line PN, is also located at 113.47 feet below the center of the pyramid. It will be noticed that this point is a control point in the design and construction of the pyramid.
In figure 59, it can be observed that the angle or slope of the line (Xm) set to establish the descending passage, is 26º 33' 54" (26.56505º) [Ref. #44, Pochán, André, page 42, (indicates = 26.56505º)]. This is the same slope angle of the line KL, as was established. The calculated angle for the faces of the pyramid, is 51º 49' 38.26" (51.82729º) [Ref. #44, Pochán, André, page 40, (indicates = 51º 51' more or less 1 minute)].
In the pyramid’s model, the entrance to the pyramid is located at a vertical distance from the ground surface equivalent to 54.17 feet. See details in figure 60, [Ref. #13, Edwards, I. E. S., page 119, (indicates = 55 feet, more or less)].
The section of the projection of the descending passage from point X to point S, measures 253.73 feet. From point S to point (m) (the entrance to the pyramid) it measures 121.14 feet. That is, the total distance from point X to point (m) is 374.87 feet.
Point S, over the surface ground, represents the place where the excavation for the descending passage leading to the pharaoh’s mortuary chamber, has to be initiated.
Figure 61 shows the horizontal plane section of the pyramid square base, superimposed over the pyramid’s vertical cross section.
The length of the line (Xm) is equal to 374.87 feet, as it had been shown. The line (Xm’), is also equal to 374.87 feet. Line (Ot) has a length also equivalent to 374.87 feet. It means that the concavity of the face is equal to the difference between OT, and Ot, that is, equivalent to 3 feet [Ref. #44, Pochán, André, page 38, (indicates that the GP face concavity = 3 feet)].
Figure 62 shows the location of the king’s chamber and queen’s chamber in relation to point X. Observe that the elevation of point Z (king’s chamber floor elevation) over the base of the pyramid is equal to 140.26 feet [Ref. #44, Pochán, André, page 150, (indicates that the GP King’s Chamber floor elevation = 140.88 feet)]. Point V (the end of the inclined floor of the grand gallery at the vertical axis) is located at 137.54 feet over the base of the pyramid, while point W (elevation of the queen chamber) is located at 70.13 feet [Ref. #44, Pochán, André, page 150, (indicates = 70.44 feet)].
As required by the geometrical specifications, the vertical distance from point X to point Z is equal to the distance from point X to point S. This distance is 253.73 feet. The elevation of the queen chamber’s floor (point W) is located 183.60 feet over point X.
In figure 63, it can be observed that the elevation angle of the ascending passage is equal to 26º 33' 54.2" (26.56505º) [Ref. #44, Pochán, André, page 47, (indicates = 26.56505º)].
The floor of the subterranean chamber (point Y) is located 102.79 feet under the base of the pyramid. This vertical distance is equal to the distance between DT and TB, as required by the specifications.
The elevation of point V, at the end of the grand gallery’s floor, is 240.33 feet (half the pyramid’s height) over the floor of the subterranean chamber, and 251.01 feet over point X. The vertical distance from point X to point Y is equivalent to 10.68 feet.
The difference in elevation between point Z (253.72') and point V (251.01'), over point X, is equal to 2.71', that is, 32.5 inches. This difference in elevation represents the vertical distance of the so-called "great step" leading to the top of the platform at the south end of the Grand Gallery. Figure 64 is an enlarged view of this section [Ref. #43, sir W.M. Petrie, (indicates step = 32.5 inches, measured from its base, to the horizontal elevation of its mark at the south wall of Grand Gallery.)
According to Petrie, the top corner of the step is at a high level, sloping downward to its projection in the south wall of the Grand Gallery. Therefore, the height of the step should be considered from its base, to the level line from the wall.
Figure 65 shows that the inclined length of the descending passage, from the entrance to the pyramid (point m), to the location where it changes the direction to horizontal (at point S), is equivalent to 344.91 feet [Ref. #13, Edwards, I. E. S., page 120, (indicates that this distance is closed to 345 feet)].
While the calculated horizontal section’s length is 26.79 feet [Ref. #47, Smyth, Piazzi, page 221, (indicates this horizontal section is about 27 feet)]. Nevertheless, the intersection between the sloping descending passage and the horizontal section, shows a change in the passages cross sections. Therefore, it depends on the points where the measurements were read.
The floor of the subterranean chamber is located 10.68 feet above point X. Since the elevation of the floor of the horizontal section of the descending passage is 13.39 feet, also over point X, there will be a vertical step (13.39 - 10.68 = 2.71 feet = 32.5 inches) to go inside the chamber. This step is similar to the "great step" located in the grand gallery (see figure 66) [Ref. #44, Pochán, André, page 42, (indicates = 28 inches)]. We should remember that the subterranean floor in the Great Pyramid was left unfinished.
In figure 67, the inclined distance from point V to point D, the construction line of the floor of the ascending passage (including the grand gallery), from its limit in the pyramid’s vertical axis, up to its intersection with the base line OT, is equal to 307.55'.
The section of the descending passage from point D to point p has a length of 156.81'. While from point I to point p, the distance is equal to 129.90' [Ref. #13, Edwards, I. E. S., page 121, (indicates = 129 feet)].
Figure 68 shows the details, and calculated measurements, for the subterranean chamber’s width and the dead end corridor.
The length of the corridor to the queen chamber from point p’, to the pyramid’s vertical axis (point W) is equal to 140.26'. While from point p, to point W, it is equal to 134.83'. Line ZD’ is an imaginary line running from the top of the platform (point Z) to point D’ located in the horizontal radius OB corresponding to the circle. This is the construction line to build the grand gallery. Remember that this line was displaced from point p’ to point p, in order to make the floor line of the grand gallery coincide with the floor line of the ascending passage in the pyramid’s model.
Figure 70 shows the longitudinal dimension of the Great Gallery, as it has been reported. The purpose of the figure is to verify if its measurements correspond with those calculated in the pyramid’s design for its location. Its length, from the north wall to the south wall is given as 156.81'. It can be observe in figure 71, corresponding to the model design that the longitudinal dimension coincides exactly in the space set for the grand gallery. The length of 153 feet in the ceiling was created with the seven corbels sections built at the north and south walls. The difference from the inclined measurement between walls at the floor line, and at the inclined ceiling is 156.81 - 153.00 = 3.81 feet, or 45.72 inches. If this difference is divided in 7 corbels in the north wall and 7 in the south wall, the walls were displaced inward about 3 and one-quarter inch in each one of the seven corbels located at each side, which agrees with the width of each corbel width.
In figure 72 it is illustrated how the longitudinal section of the Grand Gallery looks after being superimposed over the space determined in the geometrical exercise for the grand gallery. Except for its height, the diagram just fit exactly in the space. The gallery’s height at its south wall shows a height of 26.79', while the height of the Grand Gallery in the Great Pyramid is reported as 28 feet. Nevertheless, it does not explain at which point the measurement was taken. [Sir W.M. Flinders Petrie Ref.# 43], indicates that the length from the north wall of the Grand Gallery to the center of the niche in the Queen's Chamber is 137.40 feet. Note in the drawing that it is exactly the distance we obtained.
Figure 73 shows the dimensions of the Queen’s Chamber in the Great Pyramid. It shows the cross sectional view from north to south.
Figure 74 shows the measurements of the cross sectional view of the Queen’s Chamber when they are set in the pyramid’s model drawing. The dimensions were set using the elevation of the floor line and that of the vertical axis as a reference point. In this way, the location of point G (the intersection point for the ventilation shafts), and the shaft angles and measurements, could be verified.
Note in the figure that the outlets of the shafts are located a 5.63', or 5 feet and 8" over the floor line.
The shaft measurement of 5.63 feet above the floor, allow us to calculate that its horizontal section measures 6.98'. This measurement is reported as 7 feet [Ref. #47, Smyth, Piazzi, page 428 (indicates = 7 feet)].
The elevation angle for the north and south shaft is 38º 10' 21.8" (38.17271º) as shown in the figure [Ref. #44, Pochán, André, page 53, 56, (38º 28' = indicates the angles = 38.46666º)].
Figure 75 shows that their exits at the faces of the pyramid, if they have them, should be located, vertically, at 222.95' over the base of the pyramid.
The total length of the shaft would be 244.90', that is, 6.98' from the horizontal section, and 237.92' from the inclined section. In addition, the figure shows that the inclined distance from the base line projection, to their exits, would be 283.60'.
Figure 76 shows the calculated location and measurements of the ventilation shafts at the King's Chamber. The angle of the inclined shaft at the north section is 31º 43' 3" (31.71747º) [Ref. #44, Pochán, André, page 57 (indicates = 31º 33')].
The inclined distance for the north shaft, following the face line from the base to its exit in that face is 332.16'. The exit is also located at 261.13', measured vertically over the base. The total length of the shaft would be equal to 233.77'. That is, the sum of its 10.37' corresponding to the horizontal section, plus 223.40' for the inclined section. [Ref. #47, Smyth, Piazzi, page 223 (indicates = 233.23')].
The elevation angle of the south shaft is 45º. It has a total length of 175.74'. That is, the sum of 6.59' corresponding to the horizontal section, plus the 169.15' for its inclined section[Ref. #47, Smyth, Piazzi, page 223 (indicates = 174.42')].
The inclined measurement from the base line projection to its exit in the face was calculated as 335.19'. The vertical distance of its exit, measure from the base line, is 262.83'.
Figure 78 shows a detailed view of the longitudinal cross section, north to south, from the platform (point Z), to the south wall of the King’s Chamber. The cross sectional view was set using the reported dimensions for this area inside the Great Pyramid. The purpose of the illustration is to compare these dimensions with the measurements and location of the ventilation shafts, as set by the pyramid’s model geometrical design.
The longitudinal length of the rectangular King’s Chamber, from east to west, is 34 feet and 4 inches. The width, from north to south, is 17 feet and 2 inches (17.17'). The chamber’s height as reported, varies from 19 feet 1 inch (19.083'), to 19 feet and 2.75 inches (19.229'). These variations are attributed to the unevenness of the floor slabs, probably due to seismic movements.
However, it has been found that the original chamber’s height dimension is equivalent to half the diagonal measurements between its floors opposite corners. If that were the case, using the Pythagorean theorem, the diagonal would be equal to the square root of (17.17 squared plus 34.33 squared), which is 38.38'. Half this measurement would be 19.19', that is, 19 feet and 2.31 inches.
Figure 79 shows the measurements as calculated in the pyramid’s model. It includes the measurement between the pyramid’s vertical axis and the ventilation shafts vertical axis (38.38'). Note in figure 78 that the horizontal distance from the south wall of the King’s Chamber, in the Great Pyramid, to the pyramid’s vertical axis is 44.70'. Note also, in the pyramid’s model design in figure 79, that this distance is exactly equal [Ref. #44, Pochán, André, page 61, (indicates = 44.70')].
Observe that the inclined lines for the shafts intersect the horizontal lines for the shafts, creating the horizontal section of the shafts. These horizontal sections are located 3.44' (41.28") over the chamber’s floor. The horizontal section, for the south shaft, measures 6.59' [Ref. #43, Flinders, Petrie, page 24, (indicates = 6.66')].
A small difference could be noted if the measurements were taken over the base of the shaft, in its center, or over it’s ceiling. It should be noted that the only way, in which the measurements should be equal, is when the convergence point (M) is placed at the correct position.
Another important detail is the height of the King’s Chamber. Observe that when from point u, it is projected to the vertical plane; the distance from point u to point c (= 19.19') the height (uh) of the chamber could also be established. Point c, besides being the center of the two vertical axes, defines the limits between the Antechamber and the King’s Chamber.
Initially, as it will be remembered, in the pyramid’s model design, the vertical axis of the ventilation shafts coincided with the Great Pyramid’s vertical axis. Afterwards, it was displaced horizontally to the south. Figure 80 shows all the measurements that have been established with its new location.
The vertical elevation between points Z and d was used to shift the vertical shaft axis from point M’ to point M. Point d represents the middle elevation point between point Z and point G. Point G is the convergence point of the two Queen Chambers’ shafts. The vertical distance ZG, as shown in the figure, is 76.75’; consequently, Zd is equal to 38.38'. It means that the vertical shaft axis was displaced 38.38' to the south of the pyramid’s axis.
Figure 81 shows the plan view of the section comprising the King’s Chamber, the Antechamber and part of the Grand Gallery. It shows the relation between the dimensions shown in figure 79.
The dimensions shown correspond to the model pyramid design, and the space fill-up with those of the Great Pyramid to demonstrated that they perfectly agree with my model.
There is no doubt that pharaoh Khufu had to be completely satisfied with his architect (and genius), that with a high degree of excellence, accomplished his assigned job of the well conceived and designed pyramid plans. In the next chapter I will compared the dimensions of the pyramid’s model and the corresponding values of the Great Pyramid to prove their extreme similarities.