8. THE BENT PYRAMID OF EGYPT
By: Samuel Laboy
The Bent Pyramid was built in Dahshur, one of the many archeological regions of Egypt. The construction of this pyramid is attributed to pharaoh Sneferu, father of Pharaoh Khufu. It was built at the initial times of the 4th dynasty. This means that the pyramid was built before Cheop’s Pyramid.
The Bent Pyramid is well known because it has two different angles in its faces (see figure 183). This curious configuration has led to many Egyptologists and Archeologists to believe and sustain that the original design plans of the Pyramid were changed during its construction phase to change the original, unsecured angle, used for the sides. To finish the Pyramid, they used a more secure angle for the upper section. It is believed that the builders changed the angle, at a certain elevation, as a means of correcting a failure developed in the structure and avoids a collapse of the structure.
This is the most accepted theory by today’s investigators in relation to the curious configuration of this Pyramid. However, in my opinion, this pyramid was designed and built in that manner, as I will demonstrate.
I analyzed this structure using the same principles I used for the analysis of the Great Pyramid. That is, I used the circle for the design (see figure 184), its radius to represent the pyramid’s height, the vertical diameter to identify the vertical axis, and the horizontal diameter to identify the surface of the terrain.
Remember this statement for the future, from me.
The two different angles in the Bent Pyramid were set in the Bent’s design, independently from the failure theory presented by the scholars. These angles come from two geometric configurations, superimposed one over the other. One of them is the simple and basic "square circumscribed by a circle" and the other, although it took me time to find, finally I did. This second geometric configuration comes from a rare an incredible configuration.
According to reference [Ref. #13, Edwards, I. E. S., page 78], the slope angle of the lower section is 54° 31' 13" (54.52°). After part of the pyramid was constructed, the slope angle of the faces was changed to 43° 21' (43.35°), up to its top. The height of the pyramid is 336 feet, and it measures 620 feet in each one of its four sides.
I noticed that the design appears to be composed of two different transversal sections of pyramids, one superimposed over the other, as shown in figure 184. I decided to investigate separately, both pyramids’ projections.
While working with the projections of the two pyramids, I noticed some additional important geometric relations that called my attention.
Figure 185 shows the projection of the vertical cross section corresponding to the lower section of the Bent Pyramid. I observed that this configuration was very similar to the configuration that is created when a circle circumscribes a square. If a pyramid is built with this characteristic using my procedures, with a radius equal to R = 1, then, the sides of the pyramid would be equal to b = (R)(Ö2).
As I have explained, the tangent of the slope angle for a pyramid is equal to (D/b). So, the tangent of the angle for this pyramid will be equal to (D/ b) = (2) (R) / (R)(Ö2) = 2 / Ö2 = Ö2, that is, the square root of two. The angle with a tangent equal to Ö2 = 1.414213562, is 54° 44' 8". Note that this angle is very close to the 54° 31' 13" angle indicated in the reference for the lower section of the Bent Pyramid.
As a matter of fact, in another reference [Ref. #49, Tompkins, Peter, page 136], the slope angle for this section in the Bent Pyramid is informed as 54° 41', which is closer to my suggested angle. However, it is known that although the exterior of the Bent Pyramid is significantly preserved, its construction is considered poor in accuracy. So, I considered that my angle of 54° 44' 8" was correct, and it was the angle intended to be used by the builders. I used it for the design of the lower section. With 620 feet as the side length, the height for this pyramid’s configuration would be 438.41 feet.
When I examined the projection for the pyramid in the upper section of the Bent Pyramid (figure 186), I found another interesting fact. I noticed from the reference angle of 43º 21', that if I add only 30 seconds to this angle, it will become 43º 21' 30" (43.3583°). This angle, I calculated, is formed when the function of the tangent (D / b) is equivalent to 4 times the inverse value of Phi cubed. In a mathematical form, (D / b) = 4 / f³. Since D = 2 (R), rearranging the equation in terms of the value of b, the formula is changed to b = (R / 2)(f³). Consequently, with a value of R = h= 336 feet, the length of the sides of the base of this pyramid can be calculated as b = (336 / 2)(f³) = 711.66 feet.
I will explain why I consider this is an interesting formula. If the formula (D / b) = 4 / f³ is rearranged to read D f³ = 4 (b), it can be observed that (4b) represents the perimeter (P) of the square created between points I, II, III, and IV in the configuration. Note that the formula can be also rearranged to read D f³ = 4b = P. It means that the diameter of the circle multiply by f³, is equivalent to the perimeter of the square. Besides, it is known that the diameter of the circle multiplied by p is equivalent to the circumference of the circle (D p = C).
The formulas, D f³ = P, and D p = C, are unique and interesting for the construction of a pyramid. From them, we can calculate that the ratio between the circumference of the circle, and the perimeter of its base, is equal to the ratio of (p/ f³). Or, in another form, twice the pyramid’s height (=D) multiplied by p is equal to the circumference of the circle, while twice the pyramid’s height (=D), multiplied by f³, is equal to the perimeter of the base.
In figure 186, the inclined line QH, which corresponds to the slope angle of the face, cuts the circumference of the circle at point H’. Then, continues to intersect the extension line of the horizontal diameter of the circle at point H. Similarly, the line QT cuts the circumference of the circle at point T’. Then the line extends to intercept the extension of the other side of the horizontal diameter at point T. This information is important in this configuration where the length of the base of the pyramid is larger than the diameter of the circle. Therefore, the base of the pyramid is located outside the circumference of the circle (see Appendix C).
This configuration have others important geometric relations that I will explain at the end of this Appendix. I believe these are the reasons why the ancient Egyptians used this configuration, superimposed over the previous one, to form the upper section of the Bent Pyramid. Besides, in relation to this mentioned configuration, it worth to take a short look to the geometrical design of the Red Pyramid, also in Dahshur.
The slope angle in the faces of the Red Pyramid, located about 1 mile to the north of the Bent Pyramid, [Ref. #13, Edwards, I. E. S., page 89], approximates very closely the angle used in the upper section of the Bent Pyramid. Although the slope angle of this Pyramid is indicated as 43° 36' 11", it should also be 43° 21' 30", equal to the angle used in the upper section of the Bent Pyramid. It is important to indicate that the angle of 43° 21' 30" is supposed to represent the angle design in the plans, and could vary slightly from the final angle built into the structure. It is extremely difficult to perfectly maintain a specific design angle during the construction phase of such huge constructions as the pyramids, especially with the limitations in surveying equipment at those times.
Some small tolerances have to be accepted between the design data and the finished work. In my opinion, it is more important to find what angle was intended to be, or the reason for the use of that particular angle, than the actual, exact measurement, of the angle of an old, deteriorated structure, built five centuries ago.
Therefore, using the slope angle of 43° 21' 30" in the configuration of the Red Pyramid, its design could be explained with the same drawing shown in figure 186. The same formulas will also apply to its configuration. According to my references, the Red Pyramid’s sides 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. The height of this pyramid is informed as about 343 feet [Ref. #13, Edwards, I. E. S., page 90]. Another reference gives 4,111 inches (342.58 feet) [Ref. #47, Smyth, Piazzi, page 65]. Nevertheless, due to the poor accuracy in its construction, and deterioration through the ages, I considered that my calculated height is correct.
Returning to the Bent Pyramid’s design, when the cross section of both projections of pyramids is superimposed, under equal scales as shown in figure 187, the cross sectional plan of the Bent Pyramid emerges. The lower section shows a configuration which corresponds to the figure of a square circumscribed by a circle, while in the upper section, its configuration shows a ratio between the circumference of the circle and the perimeter of the base, equal to (p / f³).
The Bent Pyramid has two entrances, both in the lower section [Ref. #13, Edwards, I. E. S., page 81]. One is at the center of the north’s side, at about 39 feet over the base. The descending passage, shown as a projection in the west-east vertical plane in figure 187, has an initial gradient of 28° 22', and then continues at 26° 20'. At a distance of 241.50 feet from the entrance, changes its direction to horizontal, leading through a short corridor to a chamber. The other entrance is located at the West Side, 45 feet to the south of its face’s center, and about 110 feet inclined distance from the base. This passage has an initial gradient of 30° 09', and then follows with a gradient of 24° 17'. It descends a total distance of 212 feet to the base ground level, and then continues horizontally for 66 feet to another chamber.
In figure 187 it is shown that the vertical distance, from the base of the Pyramid to the point in the face where the angle is changed, is equal to 130.23 feet. While the inclined distance to the same elevation at the face, projected from the base, is about 160 feet. It is important to state that according to the geometrical configuration, the vertical distance for the change in the face’s angle is forced. This implies that these are the only measurements that fit in the configuration for this location, that is, considering the pyramid’s height of 336 feet, side’s length of 620 feet, and the already mentioned angles. If the vertical distance of 130.23 feet does not agree with the site measurements, it means that the distances indicated by the references are not correct. However, in that case, since all measurements in the sketch’s configuration varies proportionally, they can be easily recalculated.
The dimensions given for the Bent Pyramid’s passages by I. E. S. Edwards [Ref. #13, Edwards, I. E. S., page 81] differ completely from those given by Peter Thompkins, in length and in angles [Ref. #49, Tompkins, Peter, page 136]. My opinion is that the intended gradient angle for both passages according to the design plans, was 26.56505° (26° 33' 54.18"), that is, the angle formed when taking two horizontal units and one vertical. As a matter of fact, I consider that this is the slope angle used in both, the horizontal and the vertical passages in the Great Pyramid [Ref. #45, Pochán, André, page 14]. I believe that the distances and angular changes in the alignment of the passages in the Bent Pyramid were done to correct its alignment to an already established finishing location specified in the plan’s drawings.
If the elevation of the entrance to the west descending passage is located at 95 feet over the base as indicated by Peter Thompkins’ reference, and the length of the passage is 212 feet as given by I. E. S. Edwards’, the alignment of the inclined passage will be toward the location of point X, as shown in figure 187. It will be noticed that the gradient of the corridor was changed to horizontal, when it reached ground level at the pyramid, then continued horizontally to the entrance of a chamber. I want to call the attention to this detail, since I have stressed the importance that seems to have the location of point X in the pyramid’s designs (see Appendix D, where I show my formula to calculate the location of point X in any pyramid). The mentioned situation is similar to the one in the Great Pyramid, where the descending passage is also aligned in direction to the location of point X, and after reaching a specified location, the gradient was changed to horizontal and the corridor continued horizontally to the entrance of a chamber. The same situation is also repeated in the Red Pyramid’s descending passage. If these are coincidences, surely they are good ones!
In summary, the descending passages in these pyramids seem to be aligned toward the location of point X in the different geometrical configurations, however, before reaching the location; its gradient is change to horizontal, leading to a chamber. It would be interesting to find what is located at point X!
From another point, it is clear that if the change in the slope angles of the Bent Pyramid was a result of a failure in the structure, the event could have occurred at any elevation point during the construction. It would be unbelievable that it happened at the same height calculated for the combined plan as presented in this analysis. Besides, that the angles used for the emergency construction correspond with the angles specified for the two superimposed pyramid’s configurations superimposed in the drawing. The analysis shows that it is most certainly that the design of the Bent Pyramid was designed and built in that manner, and not as a result of a change in its construction, as explained and accepted by many scholars. The two pyramid’s configurations superimposed to form the Bent Pyramid’s design are important enough to deserve being combined in one configuration. This, probably done motivated by their religion, or any other belief.
This analysis could indicate that the ancient Egyptians knew the science of Geometry very well and that the figure of the circle was indeed used for their pyramid’s designs, using different geometrical configurations. This analysis supports my theory about the Great Pyramid’s geometrical design.
From other point, there is so much reference to the value of (1/f³) and to the geometrical configuration of the upper section of the Bent Pyramid, and that of the Red Pyramid, that it deserves a more detail explanation on my part. The value of (1/f³) is present in the formula (D / b) = 4 / f³, which I have explained apply to the geometric configuration of those pyramids. It can be interpreted from the formula that 4 times (1/f³) corresponds to the function of the tangent of the angle of the faces (D/b).
The value of (1/f³) can be easily traced by means of a triangle with sides in the ratio 1:2, as shown in figure 188. For example, trace a right triangle as shown between points Q, K, and Y, having its sides opposite to the hypotenuse in the ratio 1:2. Set the short side (QK) equal to 1, and the long side (QY) 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 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, since (Ö5 -2) is equivalent to (1/f³).
Figure 189 shows a more detailed drawing to illustrate the relation between 1/f³ and f³. The figure is composed of a base circle with radius (R = 1), and center at O. In addition, the figure shows the Great Pyramid’s configuration between points P, Q, and N (face’s angles equal to 51.82729°. Besides, it shows four circles of equal radius and geometrically interlaced points, with centers at Q, Y, B and K. Side QY of the triangle KQY is equal to R =1, while side QK = 2. The hypotenuse KY, using the Pythagorean theorem is equal to Ö5.
This section of the drawing is similar to that shown in figure 188. Consequently, 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 two. That is, (Ö5 - 2) = (1/f³). To facilitate the identification of WS, I showed its length equivalent to the diameter of the small circles shown in the drawing.
If the length of the line ML is analyzed, it can be observed that its length is equivalent to 4 radius (2 diameters), plus its segment WS. In another manner, ML = 4 (R) + (1/f³) (R). When R = 1, the formula is reduced to ML = 4 + (1/f³) = f³. Therefore, in this configuration the length of line ML represents f³, while its segment WS represents the inverse value, that is, 1/f³. (Note that from the formulas, it can be established that (f³ - 1/ f³) = 4, which is correct.)
The slope of the Red Pyramid’s faces is given by the function of the tangent (D/b) = 4 /f³, and its angle is equal to 43° 21' 30". When R = 1 in the formula, the length of the sides is (b) = f³ / 2. This value represents (1/2) the length of the line ML, which is equal to f³. This means that the perimeter of the base of the Red Pyramid is represented by 2 times, the length of line ML. This is, P = (4b) = (2) ML. From another point, since the face’s angle is defined by the ratio (D/b), it can be interpreted that it is also defined by the value (4) times WS. That is, 4 multiplied by (1 /f³).
The following will be to demonstrate the geometric relation between the Red Pyramid, the segment WS and the Great Pyramid’s configuration. For that purpose, in figure 189, the length of WS was projected to the section of the drawing that shows the Great Pyramid’s configuration. The length of WS is equal to the vertical distance between the center O, in the pyramid’s base, and the location of point X. I have already shown in Appendix D, that the vertical distance OX is equal to (1/f³)(R). Therefore, OX = WS = (1/f³).
Figure 190, derived from 189, shows the geometric relation between the Great Pyramid and the Red Pyramid. Curious... between the one built by the father (Sneferu) and the other built by his son (Khufu). To establish the geometric relation, determine the point in elevation over point O, in the vertical axis, equal to (4) WS. To establish the elevation for this point, trace 4 small circles (diameter = 1/ f³), tangents, over point O. The top of the superior small circle (point Q’) at the vertical axis, establishes the proportional height in the drawing, corresponding to the Red Pyramid. Its height is established between points O and Q’.
Observe in the figure that the circumference of the circle with radius R = 1, corresponds to the Great Pyramid and passes through the pyramid’s apex (point Q). While the circumference of the circle with radius R = 0.944272 (shown darker) corresponds to the Red Pyramid and passes through point Q’. The exterior circle traced with broken lines and R = 1.058661 shows the location of point X in the Red Pyramid.
If the formula (D / b) = 4 / f³ for the Red Pyramid’s configuration is expressed in terms of its base length, it would be b = (2)(R)(f³) / 4. The radius for circle of the Red Pyramid would be equal to R = (4) (1/ f³) = 0.944272. Therefore, the side’s length of the base would be equivalent to (b) = 2 (0.944272) (f³) /4 = 2. In other words, the side’s length of the base of the Red Pyramid is equal to the diameter of the circle used to trace the configuration of the Great Pyramid (D = 2R).
To establish the alignment of the inclined faces for the Red Pyramid, draw a line from point Q’ to point H and another to point B (the horizontal diameter of the design circle for the Great Pyramid). The function of the tangent angle of the faces for the Red Pyramid would be equal to (D/b) = (2)(0.944272)/2 = 0.944272. The angle corresponding to this function is 43.358196° (43° 21' 30").
Each pyramid’s configuration will define its dimensions when a pyramid’s height is established. For example, for the Great Pyramid, with a height of 480.66 feet, the length of the sides would be equal to b = 2(R) / Öf = 2 (480.66) / Öf = 755.75 feet. While for the Red Pyramid, with a height of 341 feet, the side’s length would be it would be equal to b = 2 (R) (f³) / 4 = (2)(341)(f³) / 4 = 722.25 feet.
If pharaoh Sneferu would desire to built his pyramid with the same height as the Great Pyramid, its side’s length would be equivalent to b = (2)(480.66)(f³) / 4 = 1,018 feet.
My geometrical method indicates that the location of point X for any pyramid is established by the extension of the line of the faces to intercept the circle of the drawing corresponding to that particular pyramid. The point where the line joining the two-intersection point in the circumference, with the vertical axis, establishes the location of point X. In this case, the projection of the face’s lines intercept the circumference at points located over the base of the pyramid. Consequently, when the two points are joined together, the location of point X will be in the vertical axis, but over the location of point O. This situation occurs when the sides of the square base of the pyramid are longer than the diameter of the circle.
Note that the lines Q’ H and Q’ T which represents the extensions of the face’s lines, intercept the circumference of the circle (corresponding to the Red Pyramid), at point H’ and in point T’. This means that the horizontal line H’ T’ cuts the vertical axis at a point that is over the location of point O. This would be point X. This happens when the sides of the square of the base of the pyramid are larger than the diameter of the circle. However, a further analysis shows that it is situated at equal vertical distance, over and under the center of the base of the pyramid. This is demonstrated with the exterior circle traced with segmented lines, and that shows the position of point X over the base and point X’ underneath the base. In Appendix D, I show my mathematical formula to determine the location of point X for any pyramid configuration.
Before I finish the Appendix, I want to demonstrate another method, which I developed to trace geometrically the angle of 51° 49' 38.3", which corresponds to the slope angle of the faces of the Great Pyramid, according to my theory. This method is most interesting since it provides a simple geometrical means to establish the geometric relations between the golden number (f), its inverse (1/f), and of major interest, the square root of the golden number (f) and of its inverse (1 /Öf).
The procedure is as follows. Set a square A, B, C, and D, as shown in figure 191, with sides equal to one (1). From the midpoint of side DC (point E), trace a line to point B. From point E, with EB as radius, trace and arc to intercept the extension of the side DC of the square (intersection at point F). Next, from point D, with DF as radius, trace an arc to intercept the extension of the side CB of the square and set point G at the intersection. Trace line DG to form the triangle GDC. The angle GDC is 51° 49' 38.3", equal to the slope of the Great Pyramid’s faces
The side’s length of triangle GDC are: the hypotenuse DG is equal to f, side DC is equal to 1, and side GC is equal to Öf. The angle GDC is equal to 51° 49' 38.3", which is equivalent to the slope angle of the Great Pyramid.
Although the angle GDC is equal to the slope angle of the Great Pyramid’s faces, the sides are not proportional. The vertical side GC is equal to Öf, while in the Great Pyramid corresponds to R = 1, the base’s side DC that is equal to 1, corresponds to (1 / Öf) in the Pyramid’s base. The hypotenuse DG, with a value of f, corresponds to Öf in the Great Pyramid. Nevertheless, if the sides of this triangle are multiplied by (1 / Öf), they will become proportional to the faces and the base of the Great Pyramid. The result will be the triangle GHK, also shown in the figure, with its sides proportional to the vertical cross section of the Great Pyramid. Angle GHK will be equal to 51° 49' 38.3". The side CK, corresponding to the Pyramid’s height is equal to 1, half the base HK, is equal to (1 / Öf), and the hypotenuse corresponding to the apothem (HK), will be equal to Öf.
I have presented enough data in this Appendix to demonstrate that it is completely feasible and possible that the pyramids were designed using different geometrical configurations within the figure of a circle. The fact that my formulas, mathematical calculations, the geometric configurations, the angles and characteristics in all studied pyramids agreed supports my theory. Besides, I have also demonstrated in this book that this geometrical method facilitates the surveying work to establish the necessary control points for the construction of pyramids.