
CHAPTER 2 Geometrical Method In my initial studies about the Great Pyramid, I placed myself in the position of the engineer, or architect, at the time of receiving the instructions from Pharaoh Khufu to build his pyramid. I thought about his knowledge, his tools to perform mathematical calculations, for tracing the drawings, and also the possibilities of undertaking, and satisfactory finishing, such a big and complicated task. The job was not easy. The structure to be built would take 30 years of work (according to Herodotus) Many engineers, supervisors, and laborers will participate in the construction during this time. The designed plans needed to be available for all concern people during the time of construction. Probably, a special place was designated to safeguard the plans, perhaps in a special chamber, somewhere around the pyramid. For this reason, I believe that the original plans of the pyramid were secured and saved during, and after the completion of the Great Pyramid. These plans were necessary for future repairs, or to make changes and additions during the construction. Besides, I do not believe that a designer or engineer will destroy the plans prepared to construct the biggest and most important structure of his time. As it is known, architects, engineers, designers, all have the tendency to keep their plans as evidence of their creation. There are many Egyptian structures, statues, paintings, and monuments, full of hieroglyphics and drawings that show names, tools, and their daily scenes, in those times. I do not think that either pharaoh Khufu, or his designers, failed us. I am certain that finding the original plans of the Great Pyramid is a matter of time. Until they are found, speculations and theories will continue to emerge about its geometrical design and characteristics. There are numerous theories about the construction of the Great Pyramid, and its geometrical characteristics. Among the theories related to the geometrical design, those that sustain that the figure of the circle, the triangle and the square form the basis of its geometrical design attracted me. I examined these theories, but I found that none of them establish the relation of the measurements and the harmonic arrangement of the three figures. In addition, they do not make reference to the underground work beneath the pyramid, and most important, the pyramid’s internal geometric configuration. My idea was to develop a geometrical method, or process, to produce a design, equal or similar, to the geometrical design and characteristics of the Great Pyramid. I wanted to develop the method without having to measure angles, distances, and without calculations. I decided to explore the idea of using the figure of the circle for the design. Besides, I had as reference the fact that the perimeter of its base is "almost" equal to the circumference of the circle, if traced using the height of the pyramid as its radius. It took me some years of research and investigation, but I finally solved my problem. I used a circle with a unitary radius to trace the design. That means that the circle’s radius is equal to one (1). In addition, I designated the radius to symbolize the pyramid’s height, the vertical diameter to represent the vertical axis, and the horizontal diameter, the base line of the pyramid. With this basic setup, I developed a geometrical series of steps to be followed to create the figure of a triangle, which represents the vertical section of a pyramid as seen through the center of its faces. I also created the outline, or projection, of the square base of the pyramid, as projected in the vertical plane, using some additional lines. The configuration of the three figures, the circle, the triangle and the square, as created, is illustrated in figure 12.
Figure 12 As stated, the radius of the circle (OQ) represents the pyramid’s height and the horizontal diameter AB represents the surface of the terrain where the pyramid will be built. The triangle HQT represents the cross sectional view of the pyramid, as seen through the center of the faces. The square defined by points I, II, III, and IV, stands for the projection of the base of the pyramid, as seen in the vertical plane. Note that the section of the triangle between points P, H, T and N, is located below the base of the pyramid, that is, below the surface of the terrain. This finding was very interesting and important for me. I knew that chambers and corridors were constructed under the pyramid. I had to investigate if the alignments and distances of corridors and chambers, underneath the base of the Great Pyramid, coincide within the geometry of this section. It is also important to point out and demonstrate that although my basic configuration looks similar to those displayed by some authors displaying the circle, the triangle and the square to identify the Great Pyramid’s geometry, they are not equal. There are many differences in the way in which they are traced, and also in their composition. The drawings shown by these authors represent just an idea, a concept, developed with the only purpose of explaining the geometrical design of the Great Pyramid. On the other side, the configuration of my three figures, although similar in their geometrical arrangement, is established following a geometrical process. My configuration can be traced independently from the knowledge that with could have about the Great Pyramid's design. The configuration shows unique characteristics and I have never seen it explained or shown in any book. In Chapter 11 you will find a description of its most important and unique characteristics. To my understanding, the reasons for the use of this configuration to design the Great Pyramid is that it is the most intriguing, beautiful, and harmonic, of the many geometrical configurations available to the ancient Egyptian engineers for the design of a pyramid. Other geometrical configurations that could be used for the design of pyramids were: The configuration of a circle inscribed in a square, the configuration of a square inscribed in a circle, an equilateral triangle inscribed in a circle, and many others. In Chapter 10, I explained my idea as to how the pyramids were design and the parameters, or variables, I believe, were used with that purpose. In figure 12, the figure of the triangle and that of the square are completely defined. I mean that the angles do not change and the whole design is proportional to the radius, or to the pyramid’s height. The angles of the faces of the pyramid (QHO and QTO) are fixed and set independently from any pyramid’s height. In my geometrical configuration, the ratio between the circumference of the circle and the perimeter of the designed square is almost equal to one (1), no matter the length of the radius (or the height of the pyramid). In other words, the circumference of the circle is almost equal to the perimeter of the square. It is less than one tenth of one percent. As already explained, this is one of the characteristics attributed to the Great Pyramid’s design. After studying the outside measurements of the pyramid, I needed to establish the length and alignments of the inside and underground corridors and chambers that exist in the Great Pyramid. I noted that these alignments and measurements seem to be the result, or the application of some type of geometrical specifications. I continued tracing and verifying my alignments with those in the Great Pyramid until I investigated the whole internal structure. Even I considered the location of the four (usually called) ventilation shafts. Finally, everything seemed to fit in my model, as if it were the solution of a geometrical puzzle. The section of the pyramid model, located below the surface ground, provided me a reasonable explanation for the underground work performed under the Great Pyramid. Nevertheless, I had to investigate further this situation. I had to establish the height of the pyramid (the value and the unit of measurement). I had an infinite numbers and units of measurements to select from. For some special reason (see the Introduction), I selected a height of 480.6637 feet (number that represents the product of 153 and Pi (p = 3.14159...), expressed in feet). With this number as the pyramid’s height, I calculated all the relevant measurements of my design in order to compare them with those shown in the reference books for the Great Pyramid. I found that the measurements of my angles, distances of corridors, and the general geometrical characteristics, were equal or fall under certain very small tolerances to those corresponding dimensions exhibited by the Great Pyramid. Design of a Model Pyramid I decided that the better way to demonstrate my theory is by designing an example of the geometrical design of a pyramid. With this purpose, I will present a geometrical exercise to design the vertical cross section of a pyramid as seen through the center of its faces, the projection of the square of the base, as projected in the vertical cross sectional view, and all internal and subterranean corridors and chambers as shown in the Great Pyramid’s structure. In the exercise, no mathematical calculations, formulas, or numerical values will be used for the distances. No special use of drawing equipment will be made, except a ruler to draw the lines and a compass to trace the circles. For the exercise, the Great Pyramid will be considered as nonexistent. No use will be made of its known angles, dimensions and characteristics. However, as it will be seen, at the end, the pyramid’s model will result with equal characteristics, angles and dimensions, as shown in the Great Monument. As explained, the design will be base on the figure of a circle, having a unitary radius; that is, the radius is considered equal to one (1). The radius also will symbolize the pyramid’s height. Initially, the figure of a triangle will be defined. The triangle symbolizes the projection of the vertical cross section of the pyramid, as seen on the vertical plane that cuts through its axis and the center of its sides. Then, the four corners of the base will be established and drawn as a projection in the vertical plane. In other words, it is like if the horizontal square base of the pyramid was rotated from the horizontal plane, to the vertical plane. After the triangle that symbolizes the cross section of the pyramid is set, the internal sections of the pyramid will be traced. That includes the location of chambers, corridors, ventilation shafts, etc. This will conclude the first phase of the geometrical design without the use of numbers and calculations. The complete pyramid’s geometrical design configuration will be finished. Then, will come the second phase, which is explained in Chapter 3. After the design is completed, the pyramid’s height will be set. Since the circle’s radius is equal to one, all the dimensions in the designed structure will vary under a constant ratio, as the height of the pyramid is changed. In other words, at any pyramid’s height, a scale model of the pyramid will be created. We will set a value, and unit of measurement for our pyramid model. The angles, distances, and all relevant measurements corresponding to the exterior and interior of the structure will be calculated. The angles and dimensions, as calculated, will be compared with those corresponding values found in the reference books, for the dimensions and characteristics corresponding to the Great Pyramid’s structure. The comparison has to be made using the same type of unit of measurement. However, it is important that the measurements of the chambers, to be located inside our pyramid model, be of the same dimensions as those found in the Great Pyramid. In this manner, the distances between the corridors, chambers, and shaft locations of the model, can be verified for their correctness. Since the figure of the circle also represents the cross sectional plane of a sphere cut through its center, virtually the whole pyramid design can be rotated in the vertical plane to any angle inside the sphere’s projection. Consequently, if the vertical plane drawing is rotated to a horizontal plane, the vertical cross section of the pyramid will be projected over the surface of the terrain where it will be built. This means that the pyramid’s crosssection; corridors, chambers, etc. could be traced over the surface where the pyramid will be built, as if it were a giant blackboard. As in all construction work, we will set the specifications for the construction, in this case, geometrical specifications. Please note, the specifications I developed and presented here were set up with the only purpose of adding interest to the exercise, and to call the attention of the reader toward some important and unique details I discovered in the design of the Great Pyramid. Corridors, Chambers and structural Details The corridors, chambers, and structural sections found inside the Great Pyramid are as numbered and illustrated in Fig. 13, with the exception of the real mortuary chamber of pharaoh Khufu, which has never been found. In the exercise, it is required that the location of the corridors, chambers, and structural details, should be established in accordance with the specifications’ requirements. 1. Mortuary chamber for the pharaoh 8. Ascending passage 2. Descending passage 9. Corridor to the Queen’s Chamber 3. Original entrance to the pyramid 10. Grand Gallery 4. Concavity of the pyramid faces 11. Descending passage 5. King’s Chamber (KC) 12. Deadend corridor 6. Queen’s Chamber (QC) 13. Ventilation shafts (Lower) 7. Subterranean Chamber 14. Ventilation shafts (Upper)
Figure 13 Having set the guidelines, we can start the exercise. Geometric Specifications: It is the desire and order of Pharaoh Khufu to create the design, and build, the most beautiful, well proportioned, and most lasting of all the pyramids in Egypt. This pyramid structure will be the restingplace for his eternal afterlife. The pyramid’s design will be based on the parameters the Creator of the Universe used to proportion the living organisms, and their habitat. For the design, the architects shall use the unique geometric configuration, where the figure of the circle, the triangle and the square, representing the sphere, the pyramid and the cube, show the maximum expression of harmony and beauty. The same geometric configuration and parameter used by the Creator to proportion the human body*. After the geometric design of the pyramid is completed and approved, the pharaoh himself, will indicate the height to which his pyramid will be raised. Pharaoh Khufu wants to present to the future generations a real model, a gigantic, and a colossal structure of this important geometric configuration. The upper hemisphere shows his huge, immense pyramid, and the lower hemisphere, the secret parts below the pyramid, that is, the location of his mortuary chamber and treasures. * See Chapter 6, Leonardo Da Vinci Besides the specifications set by Pharaoh Khufu, the following geometrical specifications shall apply: 1. The mortuary chamber of the pharaoh shall be placed in the most important location of the design configuration. The chamber location should be secured from robbers, and offer the peace and rest that he will need in his eternal afterlife. 2. The descending passage leading to this mortuary chamber shall be straight from the pyramid’s entrance to this mortuary chamber. In this way, his spirit will get in and out freely. 3. The entrance to the pyramid shall be placed at the north side, high enough over the terrain, to make it difficult to detect by the mortal intruders. 4. The faces of the pyramid will be concave to the inner side of the pyramid. This concavity in the faces will gradually increase from the top of the pyramid, to its base, where it will have the maximum displacement. The displacement at the base will represent the difference in length, between the inclined distance of the descending passage from the pyramid’s entrance to his mortuary chamber, and half the length of the pyramid’s sides. 5. A second chamber (King’s Chamber) will be placed high in the pyramid structure. It will have a platform entrance and a following antechamber. The vertical distance from the floor of this chamber to the floor of the pharaoh mortuary chamber will be equal to the incline distance from the entrance to his mortuary chamber, and the pyramid’s base line. 6. A third chamber (Queen’s Chamber) shall be built, at the midpoint vertical distance from the base of the pyramid and the floor of the second chamber. 7. To mislead the mortal intruders regarding the exact location of pharaoh’s mortuary chamber, a fake chamber (Subterranean Chamber) will be built underground the pyramid. The floor of this chamber will be located under the base of the pyramid, at a vertical distance equivalent to the difference between the height of the pyramid, and half the length of its sides. 8. To access the second chamber (KC), an inclined passage will be built. The extreme south side of this passage will reach the pyramid’s vertical axis. The elevation of the floor of this chamber, over the floor of the subterranean chamber, will be equivalent to half the pyramid’s height. The angle of elevation of this passage will be equal to the angle of inclination of the descending passage. 9. The corridor to the second chamber (QC) will be horizontal and straight, running from its intersection with the ascending passage, to the entrance to the chamber. 10. An inclined gallery (Grand Gallery) will be built following the same gradient of the ascending passage. The limits of this gallery will be; at the south, the vertical pyramid’s axis, at the north, the point of intersection between the corridor leading to the third chamber (QC), and the ascending passage. 11. To confuse the intruders as to the true location of the pharaoh’s mortuary chamber, the inclined alignment of the descending passage will be changed to horizontal and will lead to the fake subterranean chamber (Subterranean Chamber). The change in direction will correspond to an inclined distance, measured from the intersection between the inclined floor of the descending passage and that of an imaginary line parallel to the floor of the ascending passage, equal to the horizontal distance from the mentioned intersection, to the pyramid’s vertical axis. The imaginary line should be traced from the top of the platform at the end of the gallery, up to intersect the descending passage floor. The length of the horizontal passage section of the descending passage should be equal to the subterranean chamber’s width. 12. There will be a downward step to enter into the fake chamber. This step will be identical to the step in the upper section of the gallery (Grand Gallery). A horizontal "dead end" corridor will be constructed at the south wall to make the intruders think that the underground construction was abandoned. The length of this corridor shall be equal to the total distance of the horizontal section of the descending passage and the fake chamber width. 13. Two communication shafts will be constructed in the third chamber north and south walls (QC). The alignment of these conduits will be perpendicular to the pyramid’s faces and will be used for communication between the labors working inside the chamber, and those working in placing the outside layers of rocks over the chamber. When the chamber is finished and the construction of the second chamber (KC) begins, this shaft will be closed. 14. Two communication shafts will be also constructed in the second chamber (KC), at the north and south walls, with exits in the pyramid’s faces. These new shafts will have their exits at the faces of the pyramid, and will be kept open for the proper communication between the labors working in the second chamber and inside the pyramid, and those working in the outside in the outer casings. 15. The slope angles, and alignments of the communication shafts, will be a key to identify the original design of the pyramid. To make it clear, the alignment of the communication shafts in the third chamber (QC) will be set in reference to the vertical diameter of the design circle. While the alignment of the communication shafts in the second chamber (KC) will be fixed in reference to its horizontal diameter. 16. The vertical axis of the second chamber communication shafts (KC), initially designed in the vertical axis of the pyramid, will be displaced horizontally to the south, to the construction location of the second chamber. The displacement of this axis to the south will be equal to half the vertical distance from the second chamber’s floor, and the point of convergence corresponding to the shafts of the third chamber (QC) at the pyramid’s vertical axis. 17. The entrance to the second chamber (KC) will be located at the horizontal midpoint distance between the final pyramid’s vertical shafts axis location, and the pyramid’s vertical axis. The height of the second chamber (KC) will be equal to half of the horizontal distance between the two vertical axes. 18. It is the desire and order of the pharaoh Khufu, that each one of these requirements be completely satisfied. No units of measurements will be used for the initial design, no numbers, no mathematical calculations will be allowed. When the design is completed, the pharaoh will give you the height for his Pyramid, and then, at that time, you will calculate and establish all the required dimensions to build his pyramid. (End of the geometrical Specifications to build the pyramid) At this time, begins the process to design geometrically, the pyramid’s model. No calculations will be used until the whole pyramid is completely designed according to the specifications. It should be made clear that for laying the control points over the terrain, the topographical work should be limited to the minimum steps required to obtain the necessary reference points. For example, it is not necessary to delineate over the terrain the whole figure of the circle as done in the design paper, only it is required to set the control points to get the appropriate measurements. Model Pyramid  External Geometrical Design To trace the figure of the triangle that represents the vertical cross section of the pyramid, and the square, which corresponds to its base, follow this procedure. 1. Trace a straight line from point A to point B. If the layout is going to be done over the terrain, set this line directly aligned to the North.
Figure 14 2. Select a point in the line and trace a circle (any convenient size, adequate for the drawing). The radius (R) represents the pyramid’s height. Establish point A and B in the circle to define the horizontal diameter.
Figure 15 3. Trace the vertical diameter of the circle. This can be done geometrically. From point A, using a longer measurement than the radius, trace an arc. Do the same, with an equal radius, from point B. Mark the two points of intersection between the two arcs (points a and point b). The line joining point (a) and point (b), at its point of intersection with the horizontal diameter, will be perpendicular to the horizontal diameter and pass through the center of the circle. Establish the points Q and K in the circumference, to define the vertical diameter.
Figure 16 4. Using point B as center and radius BO, trace a semicircle to establish point L in the line A’B’. With center L trace an arc tangent to point B, in the circle. Trace a line from point L to point K and mark point M at its intersection with the arc.
Figure 17 5. With point K as center and KM as the radius, trace an arc to intersect the circumference of the circle. Mark point N and P at the intersection points. Although this is the most effective way to explain how to define points N and P, They can be established following other geometrical procedures. For example, with a radius equal to that of the circle, using the center points in Q and B, trace arcs from O to intersect at point C. Then, with the same radius and center at C, trace another arc from point Q to point B. Set a line from point C to point K and mark point M’ at its intersection with the circumference of arc QB. Finally, using KM’ as radius and center K, trace an arc to intersect the circumference of the circle. The intersection points will correspond to point N and P.
Figure 18 6. Join points P, Q, and N, to create the triangle PQN. Mark points H and T, at the intersection of lines PQ and QN with the horizontal diameter AB. Mark point X, at the intersection of line PN with the vertical diameter QK.
Figure 19 7. The triangle HQT represents the vertical cross section through the center of the faces of the pyramid. Line HT, in the horizontal diameter AB, identifies the base of the pyramid and the surface of the terrain where the pyramid will be built.
Figure 20 8. The distance from point H to point T also represents the length of the pyramid’s sides. Using a measure equal to OH, or OT, from the center O, mark the distances OS and OU in the vertical diameter. Now, the lines of the base of the pyramid will cross through the points H, S, T and U.
Figure 21 9. With points H, S, T and U already defined, the corners of the base of the pyramid can be set. From point H, using HO as radius, trace an arc as shown. With an equal radius and using S as center, trace another arc. Do the same at points T and U. The four intersection points between the arcs, identified as I, II, III, and IV, identify the location of the four corners of the square base of the pyramid.
Figure 22 10. Join points I, II, III and IV to create a square. The projection of the base of the pyramid corresponds to the horizontal plane at the level of the surface of the terrain. The traced square represents a projection of the base of the pyramid, as seen in the vertical cross section plane.
Figure 23 Note: Figure 23 represents the basic geometric configuration created with this geometrical procedure. The section PHTN of the triangle PQN is located below the pyramid’s base. This is the area used for the underground work of corridors and chambers in the Great Pyramid. Point X is the balance point for the PQN triangle. It divides the vertical diameter in two sections, that is, QX and XK. The ratio of QX and XK is equal to Phi, (= 1.618033). Point X is the most important location in the design. It is located below the base of the Pyramid, at a vertical distance equal to the pyramid’s height divided by Phi cubed. The distance from point X to the top of the pyramid is equivalent to the diameter of the circle divided by Phi. And the section below point X, to the base of the circle, is equal to the diameter of the circle divided by Phi squared. These important relations, and others, will be discussed in other chapters.
Figure 24 Top view of the pyramid Model Pyramid  Internal Geometrical Design After setting the outside design of the model pyramid, the internal corridors and chambers, will be established in accordance with the geometrical specifications. 1. Pharaoh Khufu Mortuary Chamber The specifications required that the mortuary chamber be located in an important and secured place. Analyzing figure 25, it can be observed that the location of point X, just at center of the base of the QPN triangle, and the pyramid’s vertical axis, could be considered the ideal place. Point X represents the balance point of the base’s triangle, and its location is well protected and safe below the base of the pyramid. Consequently, the place located exactly below point X will be selected to construct the mortuary chamber.
Figure 25 2. Descending passage In figure 26, the inclined line KL was traced to set the pyramid external structure (see figure 19). The slope or gradient of this line suggests the use of a parallel line to construct the descending passage. So, from point X, trace a parallel line to KL, to intersect the inclined face of the pyramid. The straightline alignment requirement in the specifications, from point X to the entrance of the pyramid, is also met. Mark point (m), at its intersection with the face. In addition, mark point S at the location where the line Xm cuts the base line or horizontal diameter AB. Point S establishes the place over the terrain where the excavations leading to the pharaoh mortuary chamber will be initiated.
Figure 26 3. Entrance to the Model Pyramid The entrance to the model pyramid will be safe, if placed at a high elevation over the terrain, such as set at point (m). It will be difficult to find by intruders (see figure 27).
Figure 27
4. Concavity of the faces To establish the concavity of the sides of the base, from point X, project the distance from point X to the entrance to the pyramid (point m), to intersect the line XN, and mark point m’ at the intersection. Then, project the length of line Xm’ to the radius OT. With point O as the starting point. Mark the end point of the projected line, as point t. Line Ot, by construction is equal to the line Xm, and to the line Xm’. The difference between the radius OT, and its section Ot, represents the concavity of the base, at the midpoint of the base line. In other words, the concavity is equal to the distance between point t, and point T.
Figure 28
Figure 29 Figure 29 shows the relation between point t, and point T, that establishes the concavity of the faces. The imaginary line from the top of the pyramid to point t defines the concavity of the faces, at its center, from the top of the pyramid to its base. The concavity in the face is produced when the construction’s horizontal lines of the faces are displaced to intersect line Qt, instead of QT. This procedure applies to the other faces. 5. King’s Chamber From point X, and radius equal to XS, trace an arc to intersect the pyramid’s vertical axis. Identify the point of intersection as Z. Point Z represents the elevation of the platform at the south end of the Grand Gallery, the floor of the antechamber, and the floor of the King’s Chamber. The vertical distance from point Z to point X is equal to the incline distance from X to S, in accordance with the geometrical specifications.
Figure 30 6. Queen’s Chamber From point O and radius OZ, trace an arc as shown in figure 31. Trace another arc, with the same radius, from point Z. Trace a horizontal line from the intersection points of the two arcs (point a to point b). The intersection of the line ab with the pyramid’s vertical axis, identified as point W, defines the elevation of the Queen’s Chamber’s floor. So, the elevation of the floor of this chamber is located at the midpoint vertical distance between the center of the pyramid at the base, and the floor of the King’s Chamber, in accordance with the specifications.
Figure 31 7. Subterranean Chamber Using point T as the center and TB as the radius, trace a semicircle to intercept line OB, and mark the point D at the intersection. Mark point E in the intersection of the circumference and the vertical line traced at point T, then, from point E, trace a horizontal line to the pyramid’s vertical axis. Mark the intersection at the axis, as point Y. Point Y represents the elevation of the subterranean chamber. The underground vertical distance from center point O, to the floor of the subterranean chamber (point Y), will be equal to the difference between the pyramid’s height (radius OB) less half length of the base (OT), in accordance with specifications.
Figure 32 8. Ascending Passage As it should be remembered, the gradient of the descending was set tracing a line parallel to line KL. To be consistent, in this occasion, to trace the line corresponding to the ascending passage, we will use the same ascending gradient as the line LQ (see figure 33). As it is shown in the figure, from point D, trace a line parallel to line LQ until it intercepts the pyramid’s vertical axis. Mark point V at this intersection. In addition, mark point I, at the intersection of the line DV corresponding to the ascending passage, with the descending passage line (mX). Notice that the position of point V is very near, but slightly below point Z in the pyramid’s vertical axis (the elevation of the platform in the grand gallery).
Figure 33 9. Corridor to the Queen’s Chamber As illustrated in figure 34, from point W, that represents the elevation of the Queen’s Chamber’s floor, trace a horizontal line to intercept the ascending passage construction’s line, that is, the line from point I to point V. Mark point p in the intersection. The line from point W to point p defines the horizontal passage leading to the Queen’s Chamber.
Figure 34 10. Grand Gallery The Grand Gallery will be design over the inclined section of the ascending passage, from point p, to point V. As already noted, there is a difference in elevation between point V and point Z. Remember that point V is located below point Z. The difference in elevation will become the great step, at the south end of the Grand Gallery.
Figure 35
Figure 36 To design the Grand Gallery location, as shown in figure 36, trace a line from the top of the platform (point Z) and parallel to line LQ (used for the ascending passage) to intercept the horizontal diameter AB. Mark point D’ at the intersection. From point D’, trace another line parallel to the line LK (used for the descending passage) to intercept the pyramid’s vertical axis. Mark point M at the intersection. The triangle ZD’M has been created, as shown (on a larger scale) in figure 37.
Figure 37 A triangle equal to ZD’M will be traced, but making point M to coincide with point X (see figure 38). In other words, the triangle ZD’M will be transported vertically, to a new position in the pyramid’s vertical axis, where the point M will be placed at the same position of point X.
Figure 38 One way to trace it, is by setting the vertical distance from M to X, vertically over points Z (to create point k), and over point D’ (to create point D). Joining points X, k, and D’’, will create the triangle XkD’’, that will be equal to triangle MZD’.
Figure 39
Figure 40 As shown in figure 40, extend horizontally the projection of the line Wp (Queen’s Chamber floor line), to intercept line ZD’, mark point p’ at the intersection. Trace a vertical line at point p’ to intersect line kD’’. Mark point u, at the intersection. You will see a parallelogram between points k, u, p’, and Z. These four points will define the longitudinal limits of the Grand Gallery (see details in figure 41).
Figure 41 The alignment of the ascending passage was set from points D (in the horizontal diameter of the circle), to point V (in the pyramid’s vertical axis). Then, in order that the baseline of the parallelogram (Zp’) coincide with the construction line established for the floor line of the ascending passage (line VD), the parallelogram have to be transported horizontally, to the south side, to make its baseline coincide with the ascending passage’s floorline. In other words, point p’ to point p has to be moved as shown in figure 42. This will change the location of the other points of the parallelogram. That is, point Z will become point Z’, point k will become point k’, and point u will become point u’. The horizontal displacement of the parallelogram will be equal to the horizontal distance between point p’ and point p. With this new location for the parallelogram, the vertical side k’Z’ will be located at the left (south) of the pyramid’s vertical axis. A new horizontal dimension will be created, that is, Z’Z. This new dimension defines the platform’s length at the entrance of the King’s Chamber.
Figure 42 Note that the difference in elevation between points Z and V forms a step. Figure 43 shows an enlarged section of these details. The final Grand Gallery limits are established between points Z’, k’, u’ and p.
Figure 43 11. Horizontal Section in Descending Passage To locate the horizontal section of the descending passage (the place where it changes the gradient to horizontal), use the following procedure. From point I, trace a horizontal line (as shown in figure 44), to intersect the pyramid’s vertical axis. Mark point i, at the intersection. With center I, using the distance from point I to point i as its radius, trace an arc to intersect the descending passage’s line (Xm). Mark this intersection as point s. Finally; trace a horizontal line from point s, to the pyramid’s vertical axis, and mark point y. The line from point s to point y establishes the length of the floor of the horizontal section corresponding to the descending passage. Since the floor elevation of the subterranean chamber has been set at point Y, and it lies below point y, it means that the difference creates a downward step to enter into the subterranean chamber. This downward step to enter into the subterranean chamber is similar to the great step at the south of the grand gallery.
Figure 44 Figure 45 represents an enlarged section of the abovementioned step, to enter into the subterranean chamber.
Figure 45 12. Deadend corridor  Subterranean Chamber As shown in figure 46, with center in point y, trace a circle with radius ys. Extend line ys to intersect the circumference of the circle, and mark point k at the intersection. Point k establishes the location of the subterranean south wall. Extend the subterranean chamber floor’s line to intersect the south wall vertical line.
Figure 46 Trace another circle using point k as center, and the distance from point k to point s as its radius. Extend the line sk to intersect the circumference of the circle. Mark point n, at the intersection. The line from point n to point k represents the horizontal floor line of the dead end corridor.
Figure 47 13. Communication Shafts  Queen’s Chamber The specifications required the use of the vertical diameter of the circle to establish the length and elevation of the Queen’s Chamber’s shafts, From center O of the circle, trace a perpendicular line that intersects the inclined line PQ (face of the pyramid), and identify point O’ at the intersection.
Figure 48 With center O, and OO’ as the radius, trace another circle. Mark point Q’ at the intersection of the circumference of this circle and the vertical diameter (QK), in the upper section of the vertical diameter. As shown in figure 49, from point K, and radius KH, trace an arc from point H to point T. Mark point G’ at the intersection of the circumference of the arc with OO’ and point M as its intersection with the vertical diameter.
Figure 49 With point Q’ as center and Q’G’ as the radius, trace another arc to intersect the vertical diameter. Mark point G at the intersection. Point G will establish the convergence point of the two inclined shafts for the Queen’s Chamber, and point M’, will established the convergence point for those in the King’s Chamber. To set the alignment of the two shafts for the Queen’s Chamber, as shown in figure 50, from point G, trace a line perpendicular to line QN. Identify the intersection point t. Trace another line from point G, perpendicular to line PQ and establish point h, at the intersection. Line Gt and line Gh (figure 51), set the elevation and length of the two communication shafts in the Queen’s Camber. Point t, represents the exit of the north shaft in the north face, and point h, the exit of the south shaft in the south face.
Figure 50
Figure 51 14. Communication shafts  King’s Chamber The alignment of the communication shafts for the King’s Chamber will be established using the horizontal diameter, as set by the specifications. As shown in figure 52, trace a line from point B to point Q. Trace another line from point A to point Q’.
Figure 52 Then, from point M’, trace a line parallel to AQ’ and that intersects line QB, mark point k at its intersection. Then, set point b’, at the intersection of line M’k with the pyramid’s face, that is, line QT. From Point M’, trace another line, parallel to line QB and that intersects line PQ at point a’. Lines M’b’ and M’a’, showed in figure 53 represent the alignment and length of the King’s Chamber communication shafts. Point b’ identifies the exit of the north shaft in the north face and point a’ identifies the exit of the south shaft in the south face.
Figure 53 Since the specifications require that these shafts be constructed in the King’s chamber, it will be necessary to move this designed section to the location of the King’s Chamber. In other words, point M’, and the two projections of the shafts M’a’ and M’b’, will be transported to a new location. This movement will change the length of the shafts. Before setting the King’s Chamber location, the shafts will be transported to a new axis. As shown in figure 54 in an enlarged scale, establish point d (as explained before for figure 31) as the midpoint between point Z (top of the platform) and point G (convergent point of Queen’s Chamber shafts). From point Z, trace an arc with radius Zd, to cut the horizontal floor line of the King’s Chamber. Mark point u at the intersection. Point u will be the location of the new axis for the shafts. Trace the vertical axis of the shafts through the point u. Since the shafts axis have been located, transport point M’ horizontally to the south, and set it as point M, at the new shaft axis. Point M will be the new convergence point for the two shafts in the new axis.
Figure 54 The alignment of the two shafts will be the same, but their length to the exit in the faces of the pyramid will be dramatically changed (points a’ and b’ are changed to a and b). To establish the location of the King’s Chamber, set the midpoint (point c) between the corner of the platform (Z) and point u (vertical shafts axis). Point c identifies the division between the antechamber and the King’s Chamber. At point c is located the south wall of the antechamber and starts the short passage to the King’s Chambers.
Figure 55 Figure 55 shows the final position of the shafts in the King’s Chamber.
Figure 56 If from point u it is projected vertically, the distance from point u to point c, the King’s Chamber’s height will be established. See that in figure 56, the distance from point u to point c is equal to the distance from point u to point n. Figure 57 shows the complete configuration that it had been traced for the geometrical design of the model pyramid. As stated, all the design was performed without the use of mathematical calculations, formulas, the use of numbers, and without any measurements units.
Figure 57 With this chapter, the geometric exercise is completed. In the next chapter, the pyramid’s height will be set and all dimensions calculated in the computer and compared with all the dimensions corresponding to the Great Pyramid. You will be surprised!

