PYRAMIDS OF CHEPHREN, MICERINUS, BENT AND RED
Pyramid of Chephren Geometric Design
The pyramid of Chephren is the second in size of the Giza’s pyramid complex. Its height, when finished, is estimated as 471.24 feet, and its side’s lengths, as 705.76 feet. When I studied the relation of the geometrical design of this pyramid, in reference to its configurations with the circle, I found that it could be traced using the same geometric principles and rules I established for the configuration of the Great Pyramid.
The configuration of Chephren pyramid, as will be shown, is related to the geometry of the Bent Pyramid. Chephren pyramid’s slope shows a configuration that appears to be a combination of the two independent configurations that form the Bent Pyramid’s design. These two configurations are: that used in the lower section of the Bent Pyramid and described as (1) when the square is inscribed in a circle. The other configuration is the used in the upper section of the Bent Pyramid and described as (2) when the square (base) is completely located outside the circumference of the circle. In this last configuration the ratio between the perimeter of the base (P), and the diameter of the circle (D), is equal to f³, or expressed in other way, the ratio (P / D) = f³. The combined slopes of these two configurations create a third configuration, which seems to be equal to that of Chephren Pyramid.
The slope (D / b) of the first configuration is equal to Ö2 = 1.41421356. The angle that corresponds to this slope is 54.73561032° (54° 44' 8.2 ". The slope of the second configuration is (4 / f³) = 0.944271912. The angle corresponding to this slope is 43.3581975° (43° 21' 29.5"). For information regarding these two configurations, see the Bent Pyramid Section in this chapter.
The combined slopes of these two pyramid’s configurations, shown as the product of their slopes, appear to be used by the Egyptian engineers to set Chephren pyramid’s slope. The product of the first configuration (D / b) = (Ö 2), multiply by the slope of the second, that is, (4 / f³), creates a combined slope that would be equivalent to (Ö2)(4 / f³) = 1.335402142. The corresponding angle for this combined slope is 53.17273225° (53° 10' 21.8"). This angle is in agreement with the angle measured and reported for this pyramid by Sir W. M. Flinders Petrie’s in his book, "The Pyramids and Temples of Gizeh ( The angle is = 53° 10', more or less 4'). The suggested angle fits completely with Petrie's angle.
The geometrical process I developed to trace the geometric configuration of Chephren Pyramid, is as follows:
1. As shown in figure 185, set line A-B in the N-S direction. Using a point (O) in the line, trace the circumference of a circle with a unitary radius (R1). Circumscribe the circle with a square (shown between points I’, II’, III’, and IV’). Trace the vertical diameter (Q’N) to the circle and define its horizontal diameter (H’T’).
2. As shown in figure 186, circumscribe the square (I’, II’, III’, IV’) with another circle and defined its radius R2. Trace the diagonals to this square, and inscribe another square inside the circle with radius R1 (shown between points 1, 2, 3, and 4) and sides equal to (b1).
If R2 is considered as the height of a pyramid and (b2) the base length, its section would be as shown by triangle H’QT’. The slope of the pyramid will be equal to (D2 / b2). Using the Pythagorean theorem, the diameter (D2) ², = (b2 / 2)² + (b2 / 2)², is equivalent to (2)(b2 / 2)², consequently D2 = Ö2 (b2) . Therefore, the value of (D2 / b2) = Ö2 = 1.41421356. This slope corresponds to an angle of 54.73561° (54° 44' 8.2), which represents the configuration of a circle that circumscribes a square. This configuration, as it will be shown, represents the lower section of the Bent Pyramid.
The following mathematical relations can be established, corresponding to the pyramid’s heights (R) and its bases (b):
R1 = Ö2 (b1 / 2) D1 = Ö2 (b1) Slope1 = Ö2
R2 = Ö2 (b2 / 2) D2 = Ö2 (b2) Slope2 = Ö2
The next step is to set a configuration which corresponds to the upper section of the Bent Pyramid, which shows that the diameter of the circle (D1) multiplied by f³ is equivalent to the perimeter (P3) of the base. Showing this in a formula, we have (D1 ) (f³) = 4 (b3) = P3, where (P3) represents the perimeter of the new square's base and (b3) represents its base length. In this last configuration, the sides of the square fall completely outside the circumference of the circle. Therefore, the value of (b3) is larger than the diameter (D1). Setting the value of (k) to represent the difference in length and since b3 = (D1) (f³) / (4), the value of k = (b3 - D1). Therefore, (k) = [(D1) (f³) / (4) - D1] = 2 R1 [(f³ / 4) - 1)]. The result is (k) = (1 / 2) f³. This means that the length (b3) of the square will increase by (k) = (1 / 2 f³) / 2 , or, (1 / 4 f³) to each side of the horizontal diameter (D1).
To trace geometrically the additional length of (1 / 4 f³) to each side of the horizontal diameter (D1), note that this length can be expressed as (1/4) of (1 / f³). In the Bent Pyramid Section on this chapter, I explained and showed, how to trace geometrically the value of (1 / f³) using a right triangle, with sides in the ratio 1:2. This same procedure can be used to set the value of (1 / 4 f³) in figure 187 in order to set the value of (b3). This procedure is as follows:
Set the circumference of a circle with center at H’ and radius equal to H’O. Extend the horizontal diameter to intercept its circumference at point L. Traced the right triangle QOL, where QO represents R1, and OL = (2) R1. The diagonal Q’L of the triangle represents the value of (Ö5) R1. Consequently, the sides of the right triangle are set in the ratio 1:2. From point Q, set an arc with radius QO cutting the diagonal LQ at point (a). Using an equal radius, from point L, trace another arc and mark point (b) at the intersection.
The distance from point (a) to point (b) represents the value of (1 / f³). Now, from point L, project both points, (a) and (b) to the horizontal diameter. Divide the distance between these two points in 4 equal parts. Each unit will represent (1 / 4 f³). Project this distance to the left side of point H’, and identified the intersection as point H. The distance between point H and point H’ will be equivalent to (1 / 4 f³) and will make the perimeter of the new square (shown between points I, II, II, and IV), equal to D1 (f³). Consequently, the sides of the new square are equivalent to (b3) = (D1)(f³) / (4).
Notice from figure 187, that when the pyramid’s height is equal to R3 , or its diameter is D3 , The corresponding square is located between the points I, II, III, and IV, having a base length equal to (b3 ) and a slope of 53.17273225° (53° 10' 21.8"). This slope represents Chephren Pyramid’s slope.
Figure 188 shows an enlarged section of figure 187 to show the details. It can be clearly seen the two basic configurations, which I stated, composed the Bent Pyramid geometry. Notice from figure 189, that when the height R3 is maintained and the base length is changed to (b2), the slope angle changes to an angle of 54.73561032° (54° 44' 8.2 ") which corresponds to the lower section of the Bent Pyramid, that is, the configuration when the square is inscribed in the circle. On the contrary, with the same base length (b3 ), and the height set equal to R1, the slope changes to 43.3581975° (43° 21' 30"), the angle which corresponds to the upper section of the Bent Pyramid.
As it has been shown, the geometrical and mathematical relation between these three configurations has been defined. The initial radius R1, is considered unitary, that is, its value is equivalent to 1. The value of R2, as calculated, is equivalent to R2 = R1 (Ö 2), or (R2 / R1) = Ö 2. Note that the value of R3 = R2 , therefore, R3 is also equivalent to R1 (Ö 2) and D3 = D1 (Ö 2).
In relation to their bases, (b1) = 2 R1 / Ö 2 = D1 / (Ö 2). The value of (b2) = 2 R2 / Ö 2 = D2 / Ö 2. The value of (b3) comes from the slope (D3 /b3) = (Ö 2)(4 / f³). From this formula (b3) = D3 f³/ 4 Ö 2), but since D3 = D1 (Ö 2), its value is reduced to (b3) = D1 (f³) / 4.The use of numbers and unit of measurements can be better observed in these geometrical relations. Up to this time, only the geometrical configurations and formulas have been presented to show their mathematical and geometrical relations. In this way, we are leaving the configurations and formulas to do their work, since the initial radius is assumed as a unit. This means, that when a value number and a unit of measurement is set, all others dimensions will be proportionally and automatically set in the configuration.
At this moment, the numbers and units of measurements can be introduced in the pyramid’s drawings and formulas, to prove their mathematical relations. Now, it can be verified that Chephren Pyramid’s geometry, really fit in the developed configuration.
In figure 190, the triangle HQT represents the cross sectional view of Chephren Pyramid, as seen through the center of its faces. It shows the pyramid’s height as 471.24 feet and its base length as 705.76'. The slope of the Pyramid is (53.17273°). The perimeter of its base is 2,823.04 feet and the base diagonals measure 998.10 feet.
The figure also shows a projection of the three pyramid’s configurations. The triangle QH’T’ represents the configuration with a slope equal toÖ2, having an angle of 54.73561°, which represents the lower section of the Bent Pyramid. The triangle Q"HT, having the same base length (705.76") as Chephren Pyramid, but a height of 333.22 feet, represents the upper section configuration of the Bent Pyramid. Finally, the triangle between points QHT represents Chephren Pyramid configuration with a height of 471.24 feet, a base length of 705.76 feet, and a slope of 53.17273225° (53° 10' 21.8").
The length of the sides of Chephren Pyramid was measured and reported by Sir W. M. Flinders Petrie’s in his book, "The Pyramids and Temples of Gizeh, as 8,474.9 inches = 706.24 feet, while the pyramid’s height was reported as 5,664 inches, more or less 13 inches, it means that its height could be the correct one, within a height’s range of 470.92 to 473.08 feet. The measurements, as shown in the figure are:
R1 = 333.22 ft D1 = 666.44 feet b1 = 471.24 ft
R2 = 471.24 ft D2 = 942.48 feet b2 = 666.43 ft
R3 = 471.24 ft D3 = 942.48 feet b3 = 705.76 ft
R3 = R1 (Ö 2) D3 = D1 (Ö 2)
P3 = (D3 f³)/(Ö2) = (2) (471.24) f³/ Ö2 = P3 = 2,823.06 ft
Note that R2 = R3 = b1, D2 = D3, and b2 = D1,
The following formulas apply to Chephren Pyramid:
1. The height of the pyramid is equivalent to half the product of the perimeter of its base and the square root of two, divided by the Golden Number cubed. R = (1/2) P Ö2 / f³).
If the base length (b3) is 705.76 feet:
The pyramid’s height = R3 = (1/2) P3 Ö2 / f³ =
R3 = (1/2)(4)(705.76)(Ö2 ) / f³= 471.24 feet
2. The base of the pyramid is equivalent to one fourth of the Golden Number cubed, multiplied by (D1).
If the pyramid’s height is 471.24 feet:
Base length = b3 = (1/4) f³(D1) =
b3 = (1/4) (f³)(333.22)( 2) = 705.76 feet.
3. The diagonals of the base are equal to the square root of the base length.
If b3 = 705.76 feet:
The diagonals = (b3 Ö2) = 705.76 (Ö2) = 998.10 ft
4. The perimeter of the base is equivalent to the product of twice its height (R3) and the Golden Number cubed, divided by the square root of 2.
P3 = (2)(R3)(f³) / Ö2) = 2,823.06 feet
P3 = R1 (Ö 2) Ö2 (f³) = D1 (f³) = (2)(333.22) f³= 2,823.06 feet
5. The slope of the pyramid is equivalent to tangent of (D3 / b3) = (Ö 2)(4 / f³) = 1.335402142. The corresponding angle is 53.17273225° (53° 10' 21.8").
My calculated dimensions and angles as compared with those indicated in the survey report by Sir W. M. Flinders Petrie, completely agreed with his established limits. This indicates that my design could be the correct geometric solution for Chephren’s Pyramid. It is impossible to conceive all of these occurences as coincidences. It also reinforces my theory about the use of the circle, and geometric configurations as parameters for the design of the pyramids. This is not high-math, this is a perfect domain of geometry. Continuing the work:
b3 = (D1 f³)/(4 Ö2) = 705.76 feet
D2 = 2 (Ö2) R1 = 2 (Ö2) (333.22) = 942.49 feet
b3 = (f³/ 2) R1 = (f³/ 2) (333.22) = 705.76 feet
As a final note, the triangle with sides 3:4:5, having its vertical leg equal to 4, and 3 at the base, shows a slope of 4 : 3 = 1.33333, which corresponds to an angle of 53.13010235° (53° 7' 48.4"). The similarity between these slopes, in my opinion, has induced many investigators to attribute the triangle (3:4:5) configurations to Chephren Pyramid geometry, something I believe, is incorrect. The angle does not fit Petrie's.
2. Pyramid of Mycerinus
The Pyramid of Mycerinus is the smallest of the three great pyramids. It is 213.90 feet height and side’s length about 343 feet. The geometrical configuration of this pyramid seems to be the most simple of all. Nevertheless, the architect or engineer gave all of us a lesson in Geometry with his design. The pyramid’s configuration is based on the ratio of its side’s length (b) and its height (R), which is equal to Phi (1.6180339), that is, b / R = f. In other words, the sides (b) of the pyramid are equal to the product of its height (R) and the Phi (f) value, b = Rf This geometric configuration was discussed and illustrated in Chapter 6. It is part of the drawing used by Leonardo da Vinci to illustrate the human’s body proportions by using the superimposed figures of a man inside the circumference of a circle and inside the perimeter of a square (also shown in figure 191).
In Leonardo’s drawing, the figure of the circle corresponds to the man when his hands are raised to the top of his crown and his legs are apart, while the square corresponds to his figure when the body is erected, his feet on the floor, and his hands horizontally stretched to the sides.
The base of the Pyramid of Mycerinus corresponds to the square shown in the figure between points C, D, E, and F. The tangent of the slope of the pyramid is (D / b) = D / (R) f = (2)R / (R) f = 2 / f =1.236067978. The angle corresponding to this slope is 51.026555266° (51° 01' 35.59".
The process to trace the geometrical configuration corresponding to the Pyramid of Mycerinus, using this mentioned configuration is as follows:
1. Trace a line of length A’ B’, oriented north. Select a point (O’) in the line A’ B’, as shown in figure 192. From point O’, trace a circle of any convenient radius for the drawing, and circumscribe a square (abcd) to the circle. Mark points A and B at the intersection of the vertical sides of the square with the horizontal line A’ B’. Trace the vertical line (fe) from the midpoint of sides (ab) to the midpoint of the line (dc) of the square. The distance between point A and B corresponds to the circle’s horizontal diameter, and the vertical line (fe) to the vertical diameter.
2. As shown in figure 193, with point (e) as center, and (eb) as radius, trace an arc to cut the extension of the side (dc) of the square. Mark the intersection as point (g).
3. Then, as shown in figure 194, with point (d) as center, and (dg) as the radius, trace an arc to intercept the extension of the side (da) of the square. Mark point (h) at the intersection. Then, from point (g), and radius (gd) trace another arc to intercept the vertical line through point (g). Mark point (j) at the intersection. Join points (h) and (j) to complete another larger square (hjgd), which will represent the square base of the Pyramid.
4. Following with figure 195, trace the diagonals (dj) and (hg) to the square (hjgd), and mark point O at their intersection. With point O as center, trace the circumference of another circle, using as its radius an equivalent length to one of the sides of the square (abcd).
Trace the vertical diameter QK and the horizontal AB, to this circle. Establish point H at the intersection of the horizontal diameter and side (dh) of the square (hjgd). Established point T, at its intersection with side (gj), of the same square.
5. Trace a line from point Q to point H and extend it to intersect the circumference, as shown in figure 196. Define point P at the intersection. Trace another line from point Q to point T, and extend it to intersect the circumference. Mark point N at its intersection. Trace a line to join points P and N. Mark point X at the intersection of line PN with the vertical diameter.
In figure 197, the triangle HQT represents the vertical cross section of the pyramid structure as seen through the center of its faces. The distance from point H to point T represents the length of the pyramid’s sides. Line HT, a segment of the horizontal diameter AB, also identifies the base of the pyramid and the surface of the terrain where it will be built. The inclined distances HQ and QT represent the apothem of the pyramid.
The four intersection points identified as (h) (j), (g), and (d) establish the location of the projection of the four corners of the square base of the pyramid. The big triangle formed between points Q, P, and N, defines the vertical cross section of the pyramid, as seen through the center of their faces, including its underground section. The section PHTN represents the area, underneath the base of the pyramid, where the underground corridors and chambers will be constructed.
After the geometric design is finished, the value and unit of measurement for its height will be set. All other dimensions will be proportional to the established pyramid’s height.
As I already stated, the ratio between the length’s side and the height of the Pyramid of Mycerinus, represents the Golden Number. That is, b / R = f. This formula can be re-arranged to R = b / f. This means that with a side length equivalent to 343 feet its height would be R= h = 343.17 / f = 212.09 feet. Since the side’s length and the height of the pyramid are known, the pyramid is defined. Figure 197 shows the calculated dimensions for the design.
The data reported by Sir W. M. Flinders Petrie for this Pyramid, shows that the results of my calculations for the shown design are within acceptable limits, and could be considered correct.
Pyramid’s height = R = h = 212.09 feet Length of the sides = b =f (R) = 343.17 feet
Tangent of angle q = (D/b) =2 (R) / f (R) =2 / f= (Ö5-1) = 1.23607.
Angle of slope= q = 51.02655° = 51° 01' 35.59"
tan q = (D/b)= (212.09) (2)/343.17 = = 1.23607 = (Ö5-1)
Dimensions of the Pyramid of Mycerinus: According to Sir W. M. Flinders Petrie’s Survey (1881-82)
Pyramid’s height = R = h = 2,564 inches (213.67') - (more or less 15 inches)
Correct value is between = 212.42 to 214.92 feet
Length of the sides = b = 4,153.6 inches = 346.13 feet
Angle of slope = 51° 00' (more or less 10')
THE BENT PYRAMID
The Bent Pyramid was built in Dahshur, Egypt, by pharaoh Sneferu, father of pharaoh Khufu, builder of the Great Pyramid. The Bent Pyramid is better known by its two different angles in its inclined sides. According to the book Secrets of the Pyramids, by I. E. S. Edwards, the slope angle of the lower section of this pyramid is 54° 31' 13" (54.52°). It indicates that after part of the pyramid was constructed, the slope angle was changed at some point in elevation, up to its top, to 43° 21' (43.° 35'). Egyptologists believe that the angle was changed because the Egyptian engineers thought that the steep angle cause a failure in the structure.
If you were ask to reverse-engineer the design of the Bent Pyramid, without having the opportunity to ever look at it. Just by reading books about its geometry, considering all measured dimensions at the site, which values depending on the surveyors who did the survey. With no assistance of any Egyptologist... or nobody. To the contrary, it has been already established in the Egyptology curriculum that this Pyramid was under construction but its angle had to be changed because it structure was deem to collapse.
The are few chances for someone to find these two geometrical configurations that would fit the design of two independently pyramids, and that when superimposed one over the other, the result give us the design of the Bent Pyramid.
If this is not correct, we may say that these are the highest number of coincidences before any investigation is considered true.
Anyway, all of these can be further supported by my detailed analysis of the Red Pyramid, and the three major pyramids in Giza.
If I am correct in my theory, the ancient Egyptians knew more about Math’s and Geometry than it is assumed based on the found evidence by Archeologists and Egyptologists. Besides, I suggest that the scholars ought to investigate further the timetable of the Red and Bent Pyramid construction. According to the geometrical configurations, it seems that the construction of the Red Pyramid preceded that of the Bent Pyramid, contrary to the opinion that indicates that the Bent Pyramid was built first. The fact that Red Pyramid’s configuration was used superimposed with another configuration in the Bent Pyramid, suggests that it was first used alone in the Red Pyramid, and afterwards, combined in the Bent Pyramid.
I do not believe the theory about the change in angle due to a failure. As a professional civil engineer, I think that the pyramid was built as it was planned, designed and built. The following is my conception as to how it was designed. The design follows a pattern composed of two simple, but completely different geometrical configurations.
My theory is based on the principle that the pyramids follow a geometric design, very simple to draw. I will show, in sequence, using drawings and calculations, how these two geometrical configurations could be combined to define the geometrical plan of the Bent Pyramid. I found the two geometrical configurations using the same principles and rules I developed to find the geometrical configurations for the Giza's pyramids, and other pyramid structures in Egypt.
The two geometric configurations, which I found, when superimposed, using an equal scale, create a combined configuration similar to that of the Bent Pyramid. The configurations are shown in figure 1, which represents the well-known configuration of a circumscribed square, and in figure 3, which represents the pyramid designed with the circumference of a circle inside a square (its base), but not circumscribed.. However, in this last configuration must exist a mathematical and geometrical relation between the ratio of the circle’s circumference, and the perimeter of the square. This relation is so important for geometry and math, that there is no doubt why the Egyptian engineers used it.
Figure 199 Top View
Configuration of a
In figure 1, the radius of the circle (= pyramid’s height) is set equivalent to one. The function of the tangent of the slope of the pyramid (using the concept of a circle) is equal to (D / b), where (D) represents the diameter of the circle and (b) represents the base length. For this configuration, the base length of the pyramid can be calculated, using the Pythagorean theorem. From figure 1, (R)² = (b / 2)² + (b / 2)², therefore, R = (b / Ö2), and b = (R) (Ö2) = (1.414213562) (R).
The function of the tangent of the angle for this pyramid would be equal to (D / b) = (2) (R) / (R)(Ö2) = 2 / Ö2 = Ö2, expressed in words, the tangent of the slope is equivalent to the square root of two. The angle with a tangent’s function equal to Ö2 = 1.414213562, is 54° 44' 8". (Note that this angle is a very close approximation to the angle of (55°- 00') measured by Dormer and (55° - 01') measured by Pietri for the lower section.
Other surveys of the Bent Pyramid show the following results for the angles and sides.
Lower Angle Upper angle base
Petrie (55° - 01' to 54°- 36') 621.57
Dormer (55° - 05') (43° - 01' - 30") 621.82
Mustapha (54° - 31' - 13)
Lauer (54° - 27' - 44 ideal)
The two angles, using my design are:
Lower section - 54° 44'
Upper section - 43° - 21'
The two angles: Measured "as built"
Lower Section -
Surveyors: Dormer (55° - 05'), Petrie (54°- 36'), Mustapha (54° - 31' ) = Average (54° - 24')
Lauer calculated as ideal angle (54° - 27' - 44")
It is important to mention that the angle of 54° 44' 8" shown in this configuration is supposed to represent the angle as designed in the plans. It is known that this angle could vary slightly from the final angle built into the structure. We must agree that it is extremely difficult to perfectly maintain a specific design angle during the construction phase of such huge constructions, such as the pyramids, specially, with the limitations in the surveying equipment available at those times.
It is understood that small tolerances have to be accepted between the design data and the finished work. To define geometrically the configuration of an old building structure, it is more pertinent to find what angle or distance was intended to be used by the designers, or the reason for their use in the structure, than the actual angle or measurement, as measured from the structure. Specially, when studying deteriorated structures built five millenniums ago.
Considering all these facts and the average of the measured angles, I considered that the angle of 54° 44' 8" was the angle intended to be used by the pyramid builders, and as shown in their plans. The difference from my design and the measured angle is about 20 seconds of a degree. My idea will be additionally supported with the accuracy found in the angle used in the upper section of the structure.
From the formula (D / b) = Ö2 the diameter of the circle can be easily determined since D= Ö2 (b). It means that for a pyramid with a 620 feet base (like the Bent Pyramid), the diameter of the design circle would be (D) = Ö2 (620) = 876.81 feet. Consequently, its radius, or pyramid’s height, would be equal to (D / 2) = 876.81 / 2 = 438.4062 feet.
Let’s consider the second pyramid's design. In this pyramid, the base (the square), falls outside of the circle’s circumference. This configuration shows that the length of the base is larger than the circle’s diameter. Consequently, the radius of the circle is less, or smaller, than half the length of the base. As mentioned earlier, it exists a mathematical and geometrical relation between the characteristics of this circle and the outside square.
The angle given by I. E. S. Edward’s reference for the upper section of the Bent Pyramid is 43º 21', while Dr. Mark Lehner in his book, The complete Pyramids, indicates 43° 22'. The average would be 43º 21' 30" (43.3583°). Let’s assume this average angle of 43º 21' 30" as the correct one.
Upper Section -
I did the calculation and I was amazed, surprised, to find that the calculated upper angle was 43º 21' 30", exactly the angle measured and reported by Egyptologists. The configuration of this pyramid satisfy the rare and unexpected math condition that can be resumed in a formula as (D / b) = 4 / f³. In other words, the function of the tangent (D / b) is equivalent to 4 times the inverse value of f³. Since D = 2 (R), rearranging this equation in terms of the value of (b), the formula could be changed to read b = (R / 2) (f³). Consequently, for a pyramid with a height of 344.10 feet (like the Bent Pyramid), the length of the base would be (b) = (344.10 / 2) (f³) = 728.81 feet. This is an important and interesting geometrical configuration, as it will be exposed.
1. The ratio (D / b) represents the function of the tangent of the upper 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.
2. If the formula (D / b) = 4 / f³ is rearranged as (D) (f³) = 4 (b), it can be observed that (4b) represents the perimeter (P) of the square shown between points I, II, III, and IV.
3. From the formula, it can be seen that the diameter of the circle multiply by f³ is equivalent to the perimeter of the square base, that is, D f³ = P.
P = 4 (b) = 4 (728.81) = 2,915.26 feet
(D) (f³) = 2 (R) (f³) = (344.10) (2) (f³) = 2,915.26 feet
4. As it has been shown, the product of (D) by p, represents the circumference (C) of the circle, that is, D p = C. Note the interesting fact of this configuration. The (D) multiplied by p represents the circumference of the circle, while multiplied by f³, represents the perimeter (P) of the base of the pyramid.
D p= C of the circle
D f³ = P of the base of the pyramid.
5. The ratio between the circumference of the circle and the perimeter of the square base, would be equivalent to C / P = D p / D f³= p / f³.
Do the ancient Egyptians knew that the angle of the slope they used in the upper section of this pyramid exhibits these extraordinary mathematical and geometrical relations? Or do we have to believe that this is also another coincidence?
If both geometrical designs are scaled and superimposed, the measurements and angles of the combined configurations
show to be equal to that of the Bent Pyramid. The base length of the combined pyramids would be 620 feet, while its height would be 344.10 feet.
When the cross sectional view of both pyramids are superimposed, using the height (344.10 feet) corresponding to the upper pyramid’s configuration, and base length of the lower pyramid’s configuration (620 feet) as its base length, under equal scales, a similar cross sectional view, as that of the Bent Pyramid, emerge.
Can we say that this is also just another coincidence? Is it a coincidence that two independent geometrical configurations of pyramids, when superimposed create the Bent Pyramid’s geometrical design?
It would be incredible to believe that the elevation calculated in my designed plan of the Bent Pyramid, for the change in angle happened to be exactly equal (154.60 ft) to the elevation that the construction phase of the Bent Pyramid had reached when the supposed failure event occurred.
As a summary, the lower section of this combined configuration shows sides equivalent to 620 feet, and its maximum radius is equal to 438.41 feet (projected height of the of the lower angle. The pyramid’s slope in this pyramid section is 54.735611° (54° 44' 8"). The radius of the circle is equivalent to 344.01 feet (= pyramid’s height), and the upper pyramid’s slope is 43.358196° (43º 21' 30"). This figure would represent the Bent Pyramid’s geometry.
Considering true the following statement, "The lower angle of the Bent pyramid is given by Petrie, using as reference his drawing identified as "The southern Pyramids and Peribolos" as 54º 44'. The base length = 620 ft represents an average of the length’s measurements taken to the base of this pyramid. This makes the calculated angle of the lower section of the Bent pyramid (54º 44') with my design, exactly equal to Pietri's angle".
Therefore, we can conclude that both
measured angles of the Bent, for the lower section and the upper section,
were compared exactly equal with my proposal of the combine design of two
different pyramids. The side lenght is also in agreement with 620 ft.
But there is more, the fact that there are two descending passages, one in the north side and the other in the west side, suggests that the arrangement of the two configurations were set independently in the north-south, and east-west axis. In my opinion, the geometrical configuration corresponding to the circumscribed square was considered in the north-south axis, while the configuration of the circumference of a circle located inside the square, was used in the east-west axis.
To observe the two descending passages projected in only one location, let’s think that the pyramid is located within a sphere, where the design circles represents sectional views in the north-south and east-west axis. Since the design of the pyramid is based on the figure of a circle, the north-south configuration could be rotated and shown projected in the east-west
This arrangement could be explained by means of the location of the point I called point X in the design configuration for the pyramids. This point X is located at the intersection point of the horizontal line joining the face line intercepts with the circle and the vertical axis. In the Great Pyramid, it divides the vertical diameter in two segments in the proportion of f= 1.6180339.
In the Bent Pyramid structure, apparently, the descending passage direction, and location of the entrance was established using the position of point X. An ascending inclined line, trace at an angle of 26.56505118° (26° 33' 54.2"), and set to intersect the pyramid north face, determines the location of the pyramid’s entrance. The mentioned angle is created when the slope is (2:1), that is, 2 horizontal units and 1 vertical. For the lower section of the Bent Pyramid, this point would be located at (D / f) = 876.81 / f = 541.89 feet from the pyramid’s top. If an inclined line is set from this point to intercept the pyramid’s face, the entrance would be located at 38 feet over the base. Fakhry’s reference book indicates 38.7 feet. Therefore, as I stated before, from this, it seems that the configuration for the lower section of the pyramid was applied to the north-south axis.
THE RED PYRAMID
Since the Red Pyramid has the same configuration used for the upper section of the Bent pyramid, we can take a good look to it. The Red Pyramid is located about 1 mile to the north of the Bent Pyramid. The slope angle for this Pyramid, although indicated as 43° 22' by the references, should be also 43° 21' 30". As already said, this angle is supposed to represent the angle from the designed plans, and could vary slightly from the angle built and measured in the structure. The references indicate that the Red Pyramid’s height is about 341 feet, and the base length about 722 feet.
Therefore, using the slope angle of 43° 21' 30" for the configuration of the Red Pyramid, its design could be explained showing the corresponding measurements. The same formulas apply to both configurations. Therefore, since the Red Pyramid’s side length are approximately 722 feet, then, the pyramid’s height, using my formula, would be equal to h = R = (2b) / f) = (2)(722) / f³ = 341 feet. Nevertheless, as explained, due to the poor accuracy in its construction, and deterioration through the ages, we might find a slight variation. In this case, the side's length is correct.
I believe and suggest that the scholars should further investigate the timetable of the Red and Bent Pyramid construction. According to the use of the geometrical configurations, it seems that the construction of the Red Pyramid preceded that of the Bent Pyramid, contrary to the opinion of scholars that indicate that the Bent Pyramid was built first. The fact that Red Pyramid’s configuration was used superimposed with another configuration to design the Bent Pyramid suggests that it was first used in the Red Pyramid, and afterwards, combined in the Bent Pyramid.
From another point, I made so much reference to the value of f³ and its inverse, (1 / f³), in the geometrical configuration formulas for the upper section of the Bent Pyramid, the Red Pyramid, and also present in the Great Pyramid’s design, that it deserves a more detail explanation on my part.
The value of (1/f³) is present in the formula (D/b) = 4/ f³ and in the ratio of the circumference of the circle to the perimeter of the square base, that is, C/P = p/f³. Its length can be easily traced geometrically using a triangle with sides in the proportion 1:2, as shown in the figure.
The figure shows a right triangle between points Q, K, and Y, having its sides opposite to the hypotenuse in the ratio 1 : 2. The short side (QK) is equal to 1, and the long side (QY) is equal to 2. With center at K, trace a circle with radius equivalent to QK (= 1). Then with center at Y, trace another circle with an equal radius.
The distance between the intersection points of the two circumferences with the hypotenuse (KY) represents the value of (1 / f³). This distance is identified between the points (W) and (S). Since QKY is a right triangle, having its short side KQ = 1, and the long side QY = 2, by means of the Pythagorean theorem, the hypotenuse KY is equal to Ö5. Therefore, segment (WS) of the hypotenuse will be equal to (Ö5 -2), that is, the length of the hypotenuse, less the radius of the circles with centers at K and Y. Consequently, the value of (Ö5 -2) = 0.236067978 = (1 / f³).
Figure 209 shows a detailed drawing illustrating both the relation between (1 / f³), f³, with the Red Pyramid, Bent Pyramid, and Great Pyramid’s configuration. The drawing is composed of a base circle with center O and radius R = 1. In addition, the drawing shows the basic Great Pyramid’s configuration as illustrated between points P, Q, and N (the slope angle of the pyramid is equal to 51.82729°).
The drawing also shows four geometrically interlaced circles of equal radius, with centers at Q, Y, B and K. The side QY of the triangle QYK is set equal to R = 1, while its side QK = 2 (R) = 2. The hypotenuse KY, using the Pythagorean theorem, is equal to Ö5 (R) = Ö5. \
From the formulas, the Red Pyramid side lengths are equivalent to (b) = f³/ 2. This (b) value represents half (= 1/2) the length of the line ML (= f³). This means that the perimeter of the base of the Red Pyramid is equivalent to the product of 2(R) multiplied by f³. In other form, P = (4b) = (2) f³. From another point, since the function of the function of the tangent of the slope is defined by the ratio (D / b) = 4 / f³, it means that the function of the angle is equally defined by the product of (4) multiplied by the length of WS.
The distance WS, between the points where the circumferences of circles with centers K and Y cut the line ML, is equal to the square root of 5, less 2, times (R). That is, (Ö5 - 2)(R) = (1 / f³). To facilitate the identification of WS, I showed its length, equivalent to the diameter of the small circles in the drawing.
If the length of the line ML is analyzed, it can be observed that it is equivalent to 4 radiuses (= 2 diameters), plus its segment WS. Expressed as a formula, the length of line ML = 4 (R) + (1/f³) (R). When R = 1, the formula is reduced to ML = 4 + (1/f³), value that is also equivalent to f³. Therefore, in this configuration the total length of the line ML represents f³, while its segment WS, represents the inverse value of f³, that is, 1 / f³. Observe that (f³-1 / f³) = 4, and by transposition, f³= 4 + (1 / f³).
If the radius is made equal to 480.6637 feet, that is, the Great Pyramid’s height, the length of the line ML = 480.6637 ( f³) = 2,036.12 feet. The value of WS = (480.6637) (1 / f³) = 113.47 feet. However, if the radius is 341 feet (the Red Pyramid’s height), ML = 341 ( f³) = 1,444.4992 feet and (b) = 1,444.4992 / 2 = 722.2496 feet. As shown in the figure, if the length WS = (1/ f³) (R), is projected into the section of the drawing that shows the Great Pyramid’s configuration between points P, Q, and N. It will be noticed that WS is equivalent to the vertical distance at which point X in the vertical axis, is located underneath the base of the pyramid, that is, 113.47 feet, when R = 480.6637 feet.
All of these calculations, geometrical configurations, mathematical and geometrical relations, ratios, and formulas, directly connect the geometry of the Red, the Bent and the Great Pyramid, as well other pyramids. They are mathematical evidence that the Bent Pyramid configuration is not product of a pyramid failure or change in plans because of its instability. This mathematical evidence will be here, as revealed, and when new archeological discoveries exposed the truth, I am sure there will be one flag waiving in the air.