Giza Area - Layout of the pyramids
The layout of the three major pyramids at Giza has always been a matter of discussion. Many scholars believe that each pyramid was built independently from the other. However, other scholars consider that the existence of a general plan for the area at the time of construction is possible. As yet, any plan or theory in this respect has never been accepted. A recent theory indicates that the arrangement of the pyramids in the Giza area follow a pattern similar to the configuration of a group of stars in the firmament. Although this theory had been debated lengthily, it does not have the endorsement of the Egyptologists.
In 1881, Sir W. Flinders Petrie performed a comprehensive survey of the Giza area. His report was presented in 1883 and has been the basic reference for the Gizaís pyramid correct location. Figure 216 shows a detail diagram traced following Petrieís Report measurements. As it can be noticed from the figure, it comprises a rectangular area with a longitudinal size of 2,976.15 feet and a width of 2,435.74 feet. The figure shows the distance between the pyramids sides and center-to-center measurements. This rectangular area called my attention.
My theory about the Great Pyramidís geometrical design included the circle as the basis for its design. I suspected that the area where the three great pyramids were built was, in the same way, related to the circle, although the area was rectangular.
After some time devoted to this task, I found a geometrical method and a configuration with the circle, which can be used to trace the rectangular area. The sides of the rectangle are proportional to the rectangular area that enclosed the three great pyramids. I used a circle with a unitary radius to trace the configuration. The procedure is illustrated in the following figures and is as follows:
1. As shown in figure 217, trace a line Aí Bí, aligned to the north. Using a center O, in the line AíBí, trace a circle with a unitary radius (Rí = 1). Trace the vertical diameter of the circle as QK. Mark its horizontal diameter as AB. Set the diameters as Dí, which will be equivalent to (2)(Rí).
2. From point Q, trace and arc using QK as its radius, as shown in figure 217. From point K, trace another arc using an equal radius. Mark points S and M, at the point of intersection of the circumference of the two arcs, which will be located on the line AíBí. The distance SM will represent the length of the rectangle.
3. As illustrated in figure 218, from point Q trace an arc using QO as its radius. Then, from point B, trace another arc with BO as radius. Mark point N at the intersection of the arcs.
4. As shown in figure 219, with center at O, and ON as its radius, trace the circumference of another circle (shown in hidden lines). Mark points H and T, at its circumferenceís intersection with line AíBí. Mark point W and point V at their circumferenceís intersection with the vertical axis. The distance WV will represent the width of the rectangle.
5. Using SM as its length, and WV as its width, construct the rectangle (shown between points I, II, III and IV.) The rectangle constructed using this geometrical procedure as will be demonstrated, will have an equal configuration, and will be proportional to the sides of the rectangle that actually encloses the three great pyramids.
To determine the numerical values and unit of measurements for the length and width of the rectangular area, it is only needed to set the value of the diameter (Dí) of the circle. The diameter (Dí) is equivalent to (Rí) multiply by (2). The sides of the rectangle are link to the value of (Dí). The mathematical relations between (Dí) and the sides of the rectangle can be expressed using the following formulas.
Observe that the length of the rectangle is equivalent to the distance between point S and point M. The distance between points S and M can be obtained adding the length of segments SO and OM, First, it will be calculated the segment SO (refer to the right-triangle SOK in figure 219). Using the Pythagorean theorem in the calculation:
Segment SO (squared) = (SK)≤ - (0.5Dí)≤ = (Dí)≤ - (0.5 Dí)≤
Segment SO (squared) = (Dí≤) - (0.25Dí≤)
Segment SO (squared) = (0.75Dí≤) = 3 Dí≤ /4
Segment SO =÷3 (Dí/2)
The segment OM (see right triangle KOM) can be also calculated using the Pythagorean theorem.
Segment OM (squared) = (KM)≤ - (0.5Dí)≤ =(Dí)≤- (0.5 Dí)≤
Segment OM (squared) = (Dí≤) - (0.25Dí≤)
Segment OM (squared) = (0.75Dí≤) = (3)(Dí≤) / 4
Segment OM =÷3 (Dí/2)
Therefore, the distance SM = ÷3 (Dí/2) + ÷3 (Dí/2) = ÷3 (Dí), consequently, the length of the rectangle is equal to ÷3 (Dí).
The width of the rectangle is equivalent to twice the length between points W and O, which represents the diameter of the circle shown with radius ON and in hidden lines. The radius ON is equivalent to segment WO and OV, since they are radiuses of the same circle. Using the Pythagorean theorem to calculate the length ON:
ON (squared) = (0.5 Dí)≤ + (0.5 Dí)≤
ON (squared) = (0.5Dí≤)
ON = WO =(÷0.5) (Dí)
Since WO = OV (radius of he same circle)
WV = width of the rectangle =(÷0.5)(Dí)(2) = ÷2 Dí
Therefore: The sides of the rectangle will have the following measurements:
Length =÷3 (Dí).
Width = ÷2 (Dí)
The diagonals of the rectangle, line (I-III) and line (II-IV), using the Pythagorean theorem:
Diagonal (I-III) =(÷2 Dí)≤ + (÷3 Dí)≤ = ÷5Dí
Diagonal (II-IV) = (÷2 Dí)≤ + (÷3 Dí)≤= ÷5Dí
The length of the diagonals are equivalent to ÷5 Dí
As it has been shown, the geometrical configuration of the designed rectangle depends on the value of Dí. Consequently, this diameter determines the dimensions of the length and the width of the rectangle. We can represent the value of Dí using any number with any unit of measurement, possibly used by the ancient Egyptians. We can use meters, feet, cubits, inches, or any other unit of measurement. The width and length of the rectangle will vary in proportion to the number and unit of measurement assigned to the diameter Dí.
How can we determine if really the Egyptian engineers used this method? Do they leave some hidden imprints in their design to certify that this is the method and configuration they used? I believe they did, and I will show why.
As already stated, my method to design the rectangular area depends on the diameter Dí of the circle. Any number and measurement unit can produce a determined rectangle. In order to arrive to the correct length and width used for the three great pyramids, the exact value of Dí has to be used in the formula. Letís say that I assign a value to Dí, and by chance it is correct. It will be just a coincidence, it could happen.
But if the number to be assigned to Dí represents the sum of two fix, related, and exact measurements directly associated to the Giza area, that could be another thing. As I already stated the design of the Great Pyramid was based, according to my theory, in the figure of a circle, where its radius is equivalent to the pyramidís height, therefore, the diameter represents twice its height. The design of the rectangular area is also based on the circle. Now, what do you think if I select the measurement of the diameter of the circle used to trace the geometrical configuration of the Great Pyramid as the value of Dí and it completely fits the calculation for the rectangular area? Interesting? However, letís add up to the measurement of the Great Pyramidís circle diameter, the length of the base of the Pyramid, to eliminate the guessing chances and coincidences. Therefore, the value of Dí would be equivalent to twice the Great Pyramidís height, plus the length of its base. Not using approximates measurements numbers, but the exact numbers. The pyramidís height can be assumed correctly equal to 480.6637 feet; its base is 755.7488 feet. Letís make the value of Dí = 2(480.6637) + (755.7488) = 1,717.076 feet.
In other words, to trace the layout of the rectangular area which enclosed the three pyramids, I will used a Dí value equivalent to 1,717.076 feet. Letís see this carefully. I will prove my numbers are correct by means of the following calculations. For a short illustration in the calculations, letís make a quick check using the Great Pyramidís height and base measurements, expressed in cubits.
It is generally accepted that the Great Pyramidís height is 280 cubits, while its side length is 440 cubits. Using these numbers, the value of Dí would be:
Dí = 2 (R) + (b) = 2(280) + 440 = 1,000 cubits
With the value of Dí established, the sides of the rectangle shown between points I, II, III and IV in the figures, can be easily calculated:
Length of the rectangle =÷3 Dí = ÷3 (1000) = 1,732.0508 cubits
Width of the rectangle = ÷2 Dí = ÷2 (1000) = 1,414.2136 cubits
The diagonals of the rectangle would be = ÷5 Dí= ÷5 (1000) = 2,236.0679 cubits. As it will be noticed, the rectangle consists of two congruent triangles, each with sides ÷2Dí, ÷3Dí and ÷5Dí.
Now, to compare the measurements, we have to convert Petrieís measurements to cubits. For this, I will use his defined cubit of 20.63 inches for the cubit. The length and width of the rectangle, in cubits units, would be:
Length = 2,976.15 feet / (20.63/12) = 1,731.16 cubits
Width = 2,435.74 feet / (20.63/12) = 1,416.81 cubits
If these measurements are compared with those in our designed dimensions:
Calculated Petrieís Survey Difference
Length = 1,732.05 cubits 1,731.16 cubits [0.89']
Width = 1,414.21 cubits 1,416.81 cubits [2.60']
The difference in the length of the rectangles is 0.89 cubits, while the difference in the width is 2.6 cubits. This represents a difference of 0.0514 percent in the length, and 0.184 percent in the width.
Nevertheless, letís examine the same calculations used for the cubits, using feet units. The height of the Great Pyramid will be set equivalent to 480.6637 feet. Since the Great Pyramidís height is proportional to its base, when one dimension is established, the other is also set. The pyramidís base length (b) = (2) (R) / ÷f, and also equivalent to (b) = D / ÷f. Consequently, if the value of R is known, the value of (b) can be easily calculated. The diameter of the Great Pyramidís circle would be equivalent to D = (2)(R) = (2)(480.6637) = 961.3274 feet. The side length of the base would be (b) = (D / ÷f), that is, (b)= 961.3274 / (÷f) = 755.7488 feet.
Since I had set Dí = 2 (R) + b, its value would be: (961.3274) + (755.7488) = 1,717.0762 feet. By working the formulas, the value of Dí can also be expressed as Dí = D (1 + 1 / ÷ f) and also, Dí= (b) (1 + ÷f). It means that Dí can be calculated either using the pyramidí height, or the length of its base.
Now, using Dí = 1,717.0762 feet, the rectangular area for the three pyramids can be calculated and compared with Petrieís rectangular area.
Calculated (Petrieís Survey) Difference
Length =÷3 (1,717.0762) = 2,974.06' 2,976.15' [2.09']
Width = ÷2 (1,717.0762) = 2,428.31' 2,435.74' [7.43']
The measurement of the diagonal of the rectangle would be equivalent to ÷5 (1,717.0762) = 3,839.50 feet. This is a curious rectangle, since its diagonal is of the same length of the diagonal of the right triangle with one leg equal to the diameter Dí and the other leg equal to (2Dí), that is, in the ratio 1:2. The diagonal would be equivalent to ÷5(Dí) = 3,839.50 feet.
Although substantially accurate, the results still show very small differences in the length and width of the designed rectangular area and the rectangular area used by the ancient engineers, as measured by Petrie.
However, new information from Dr. Mark Lehner, shows that the correct base measurements of Micerynus Pyramid are 335 by 343 feet (Ref: The complete Pyramids, page 134). Therefore, the dimensions of the base of the Pyramid of Micerynus, informed by Sir W. F. Petrie as 346.24 feet, have to be corrected. Therefore, the rectangular dimensions of length and width set by Petrieís Report need also to be corrected.
The correction for the width would be: (346.24' - 335.00') / 2 = 5.62 feet, while the correction for the length would be (346.24' - 343') / 2 = 1.62 feet.
Consequently, the real measurements of the rectangular area used by the Egyptian surveyors would be less than those indicated in Petrieís Report. However, I considered that Petrieís Report, regarding the measurements between the centers of the pyramids, is completely correct, since their centers are easier to detect than their original corners. Due to this fact, the size the rectangular area needs to be adjusted accordingly. The corrections to the length and width of the rectangular area would be:
Length of the rectangle = L = (2,976.15' - 1.62') = 2,974.53 feet. This length is in agreement with the 2,974.06 feet shown in my calculations.
Width of the rectangle = W = (2,435.74' - 5.62') = 2,430.12 feet. This width dimension differs by merely 1.81 feet (approximately 22 inches), from my calculation of 2,428.31 feet. This insignificant difference represents (0.0007448) of a total width of 2,430.12 feet. Now, the comparison between my designed rectangle versus the actual site rectangle shows:
Calculated (Survey Rectangle) Difference
Length =÷3 (1,717.0762) = 2,974.06' 2,974.53' [0.47']
Width = ÷2 (1,717.0762) = 2,428.31' 2,430.12' [1.81']
These results are extremely accurate, considering the huge size of the rectangular area. It is clear, from these results that both rectangles, the designed and the laid out in the field, were originally intended to be equal.
The chances of these two rectangles to have equal length and width, just by coincidence, are almost none.
Since the calculation of the designed rectangular area is based on the Great Pyramidís height, (or its base length), it clearly indicates that the general layout of the pyramids was a planned job.
Figure 221 shows the layout of my calculated dimensions for the rectangle, using the height of the Great Pyramid as the basis for the calculations. The Great Pyramidís height was set as 480.6637 feet. The distances between the centers of the pyramids correspond to those indicated in Petrieís Survey Report.
Letís examine carefully figure 222. The rectangular area that it shows was established using the diameter (D = QK). The drawing shows that the composition of diameter (D) represents the sum of the diameter of the inner circle (with diameter QKí and equal to twice the pyramid height) plus the length of the base of the pyramid (shown between points K and Kí). The figure also shows Chephren and Khufuís Pyramid location.
The results of the measurements show that the Egyptians engineers could have set up the rectangle measurements based on the Great Pyramid dimensions, as indicated. The circle generates the complete and accurate rectangular area for the location of the three pyramids.
As shown in figure 223, the location of point A, identifies the center of the Pyramid of Chephren. As it is known, the noticeable displacement of its center from the rectangular area, have been the basis for some theories about the pyramidís location arrangement in relation to some firmament stars. The pyramidís center is shown in the figure displaced 51.81 feet to the left and 260.29 feet upward, measured from the center of the rectangle.
The configuration of the Great Pyramid was laid-out in the circle with diameter Dí, and is shown between points HQT. Since the circle with Dí has a greater scale than the circle used to create the Great Pyramid configuration, I reduced proportionally the scale.
The diameters ratio is Dí / D = 1.7861514. Consequently the vertical displacement of 260.29 feet in the drawing scale, represents 260.29 / 1.7861 = 145.73 feet in the Great Pyramid scale, while 51.81 feet represents 51.81 / 1.7861 = 29 feet, as shown in figure 224. The position of this point in the Great Pyramid drawing is 145.73 feet over the base line, and 29 feet horizontally from the vertical axis. Another description would 5.5 feet over the Kingís Chamber floor and 18 inches in front of the north wall. Just like the position of a standing person inside the Kingís Chamber, in front of the sarcophagus.
In figure 222, note that the east side of the projection of the pyramidís base (PU) is displaced 22.26 feet east, in relation with the position corresponding to the center axis of the Great Pyramid, placed in its true location in the lower right corner of the rectangle.
It looks like this difference in distance, or displacement, represents the famous displacement to the east side in the entrance to the Great Pyramid, from the center of its north face. The reasons for this displacement have never been satisfactory explained. This distance was previously been informed as 7.20 m (23.62'), however, according to new measurements, the correct distance is 6.82 m (22.37 feet). As shown in the figure, the displacement shows the same relative position, and a similar value, to the displacement at the entrance to the Great Pyramid. Although only speculation, it worth to note.
In summary, I have presented sufficient mathematical data to show that the layout rectangular area where the three great pyramids were built followed a well-designed plan by the ancient Egyptians engineers. The drawings and calculated dimensions herein presented support my theory that indicates that the rectangular area formed by the position of the three great pyramids is related mathematically and geometrically to the figure of a circle and that the Great Pyramidís dimensions were the basis to delineate the rectangular area.
The probability of finding the proportional length and width of the rectangular area used by the Egyptians to build the three pyramids by means of using the figure of a circle, are almost null, The probability of finding a mathematical expression comprising the length and with of the Gizaís pyramids rectangular area with the height and baseís length of the Great Pyramid, are also practically none. Besides, the fact that the sum of twice the height of the pyramid, plus the length of its base, determines the length and with of the rectangular area where the three great pyramids were built, is just amazing.
The ancient Egyptian civilization transmitted to humanity, through paintings, hieroglyphics, drawings, and statues, information as to their daily work, their beliefs, food, their believe in the Creation, afterlife and Gods, but nothing about their pyramidís design and construction. It seems that the pharaohs ordered to keep secret all activities and records related to their pyramid construction.
However, there is enough evidence, particularly in the Great Pyramidís structure, to demonstrate to all, that the Egyptian designers and builders had an excellent knowledge of Mathematics and Geometry. A building of such size, with such accuracy in its construction, requires a well-designed plan, with complicated calculations. Why scholars, just observing their stonework, recognized their extraordinary skills in stone construction techniques, while observing the design, almost perfect, and its exceptional structure, do not recognize their extraordinary mathematical and geometrical skills? Why the scholars believe that their mathematical science was basic and elemental? It is known that there are no papyrus or Egyptian paintings showing that they had and advance knowledge in these sciences. However, in the same manner, there is no evidence to show how they really did their precision stone work, that even today, it is still under debate by scholars.
Pharaoh Khufu, now resting in his afterlife, had to be very satisfied that his engineers successfully achieved the well-designed geometric plans for his Pyramid. Besides, he should be very pleased to know that his work had been, and will continue been for the ages, of admiration to all mankind. On the other side, the mankind is grateful to Pharaoh Khufu and the members of his family, Chephren and Micerynus, for such a wonderful legacy, the three great pyramids and the Sphinx.
Egyptologists, topographers, engineers, technicians, specialists, Authors, can be sure that they will have plenty of work to do in the matter of pyramids for many years to come. The modern human beings, with their advanced technology, computers, space travels, will not rest until they discover the well-kept secrets of their construction. They will continue excavating deep in the earth, and looking through satellites from the sky, until they find the way in which the Egyptians accomplished such a huge work using the limited instrumentation and technology available five centuries ago. When that happens, at that time, the theories developed through the ages to explain their construction and design could be evaluated, judged, and the corresponding merits awarded.
The results of my investigations presented in this book, and my theory about the geometrical solution of the Great Pyramid, and other Egyptian pyramids is my modest and humble collaboration to the science of Egyptology. I present them with the purpose of opening new ways of approaching the main avenues to the knowledge.