Lost Technologies of Ancient Egypt: Advanced Engineering in the Temples of the Pharaohs

by Christopher Dunn

  Figure 2.4. Left and right view of Ramses’ head

  The lines were applied to the photographs as references in order to size the two images of Ramses in the computer. I should stress here that the images were not distorted from their original aspect ratio during this process. I superimposed the Fibonacci spiral there to see if any correspondences occurred. Amazingly, there seems to be an uncanny and harmonious congruity between the spiral and the circle, as well as between the spiral and elements of both sides of the head. As I sized it to fit one side, then mirrored it for the other, the spiral was not changed in its size or aspect ratio.

  Fibonacci, also known as Leonardo Pisano, published his discovery of the properties of the series 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth, in his book Liber Abaci in 1202,1 but the mathematical construct was previously used in Indian mathematics by mathematician Virahanka (sixth century CE).2 It has also been said, not without controversy, to have been used by the Egyptians in their architecture—to define, for instance, the geometry of height and slope of the Great Pyramid.

  I should state here that my inclusion of Fibonacci spirals and circle geometry is not the result of a search for esoteric symbolism in the statue; it is simply a means to discern how the ancient Egyptians had created Ramses. I am not arguing that the ancient Egyptians intentionally encoded it into their statue. My investigation is intended more to illustrate the symmetry and exactness of the piece and explore the manufacturing implications than to argue for secret mystery schools, sacred science, and Leonardo da Vinci–style occult symbolism. Though I find these subjects fascinating, they are outside the scope of this book, and there are others who are far more knowledgeable than I on these subjects—so I will leave any such discussions to them.

  The line that follows the shape of the ear in figure 2.6 is not a Fibonacci spiral but was generated from one ear, then copied and mirrored on the other. From the photographs, it is clear that there are slight differences between the two ears—but one ear is damaged and the lighting on both varies such that we cannot state with confidence that they are identical within precise tolerances. What we can state, however, is that figures 2.3, 2.4, and 2.6 illustrate a nearly impossible task for a sculptor. The probability of the finger on your left ear and the finger on your right ear being positioned within the tolerances shown in this series of photographs is virtually zero. The possibility of a sculptor creating a head with a jaw line that is identical on both sides and two ears that are within the tolerances as shown in these photos is also vanishingly small.

  Figure 2.5. Fibonacci spiral

  Figure 2.6. Ramses’ ears

  To accomplish what we see here, one has no other option but to focus on precision. We are confronted here not with a coincidence, a stroke of luck impressed with crude, handheld tools, but a stark reality that the order of precision found on Ramses’ head demanded that the sculptor move into the realm of engineering and its essential science of measure: metrology.

  If we study the face further, it becomes obvious that the reverse image is not a perfect match. The nose, mouth, and eyes—all principle features of the face—do not match with the jaw in alignment. Nonetheless, moving the transparency over these features slightly brings them together, though not all at the same time.

  Figure 2.7. Reverse transparency matching the eyes and mouth

  Figure 2.8. Reverse transparency matching the nose

  RAMSES’ SHADOW

  Given the off-center alignment of the photographs of Ramses’ head, it became clear that I had to return to Egypt. When I examined the photographs I took of Ramses’ head in February and began to compare the symmetry from one side of the face to the other, I realized that in the photographs I took when my camera was handheld (as opposed to on a tripod), the central axis of the camera frame was not quite in alignment with the central axis of the statue. I knew that I could not capture an image that was perfectly in alignment without trial and error, and what I had produced had errors, so I determined that I could do better if I was able to use a tripod stand and take a series of photographs while moving the stand incrementally around the head and keeping the nose in the center of my viewfinder. To understand why this tripod setup is so important, we can consider the following series of sketches that represent a view from above looking down on the head.

  The ideal camera setup is illustrated in figure 2.9. To achieve this, the camera axis is oriented exactly along the central axis of the head, which is a theoretical line that bisects the features of the head. If an image of the head is then taken, flipping the image on its horizontal axis would make for an identical image.

  Figure 2.10 shows the head rotated 1 degree from center. What this means is that when the image is copied and flipped and compared to the original, some features will not match.

  The arrangement we see in figure 2.11 is the same one that was captured by my camera with my first set of photographs of Ramses’ head—when I copied, flipped, and overlaid a transparency onto the original. As we saw in figure 2.3, the mouth, nose, and eyes do not match, but the jaw outline matches perfectly. This is because the outline of the face is used to establish the central axis of the photograph and the nose is rotated slightly off-center.

  As depicted in figure 2.7, by making the mouth the center point, the outline of the face is thrown off axis and the nose is thrown off axis. The eyes, because they are approximately at the same distance from the camera as the mouth, then come into alignment.

  Even with a less than perfect alignment, however, I was elated by this discovery. The implications were immediately clear to me—and they were enormous. My preliminary studies indicated that the statue was crafted so that the left side was a mirror image of the right side. I realized then what was needed: I had to take another series of photographs and hope that one of them presented a closer alignment with the central axis of the head.

  The photographs I took in May 2006 were certainly an improvement upon those I took in February. However, I was still not quite satisfied with the results and returned in 2008 with a better camera that resulted in figure 2.12, which is photograph 70 of a series of 94 photographs that I took while moving the camera in action mode while panning in an arc around the statue, keeping the nose in the center of the viewfinder. I then duplicated the image and mirrored it and made a transparency, then positioned this over the original to match the features of the face.

  Figure 2.9. Top view of camera setup. The central axis is identified with an L overlapping a C, which is the standard notation engineers use in their drawings to denote a centerline.

  Figure 2.10. Rotated head

  Figure 2.11. Double image

  Figure 2.12. Ramses’ symmetry, taken in incremental steps while moving the camera in an arc around the statue

  As we can see, the match is much closer, but not perfect. Figure 2.12 shows that the jawline, mouth, and eyes match, but the nose is slightly off center. Figure 2.13 shows a close-up of the nose and mouth with the nose in alignment. It should be noted that the amount of error in the orientation of the camera is actually half of the width of the shadows revealed when different features are brought into alignment. If the camera was adjusted by half the amount of the error shown, then all shadows may disappear.

  Figure 2.13. Ramses’ nose symmetry

  As we can see from studying figure 2.13, the jawline, mouth, and eye from the right side appear to line up perfectly with the left, but it appears that the nose is slightly off center. In figure 2.14, the nose is brought into alignment and a shadow appears around the jaw and the lips. The shadow is quite useful for our study, because it actually provides a reference line with which to calculate the percentage of error from one side of the face to the other. These results are stunning—beyond anything I had imagined.

  Though visually the comparison between the two sides of the face is remarkably similar, dimensionally it requires further examination with metrology equipment to measure exactly the differences between the two sides. From a manufacturing p
erspective, the dimensional variation from a perfect form (i.e., the tolerance band) on the contoured surface of a form die depends on the ultimate function or purpose of the piece. Today’s machine tools can produce complex contoured surfaces to a level of accuracy that was not within the capability of machines forty years ago. When we compare Ramses’ head and modern machined surfaces, the analogy does not register as relevant or fitting, because the end products are created for completely different reasons, and a statue does not require the same precision as a contoured surface for, say, a rocker panel, a trunk lid, or a hood for a car body. Nevertheless, the comparison using a digital photograph compelled me to try to determine some kind of dimensional reference so that we can say with a reasonable amount of certainty that I was not influenced by an optical illusion.

  To this end, and in order to draw from the photograph a relative dimension from one cheek to the other, I enlarged the photograph to approximately five times that of a human head and applied dimensions (measured in inches) from a vertical centerline to the outline of the jaw. In this way, there was no interference with an overlying transparency and the resulting shadow; thereby there was less uncertainty as to exactly where the edge of the face was.

  The results in figure 2.14 show that the camera’s axis was shifted to the left of the axis of the head, and the dimensions of the nose and the ears indicate that a mere 0.140-inch (3.55-millimeter) rotation of the camera to the right would bring these dimensions closer to being the same. On a human scale, the amount of error in the orientation of the camera would be 0.028 inch (0.711 millimeter), just slightly more than the thickness of a thumbnail. The dimensions of the jaw line are within a tolerance band of plus or minus 0.010 inch (0.254 millimeter), which on a human scale is plus or minus 0.002 inch (0.0508 millimeter). Close to the ears the tolerance band increases to plus or minus 0.065 inch (1.65 millimeters), which on a human scale is plus or minus 0.013 inch (0.33 millimeter).

  Though they do not achieve perfection—but are much closer to the central axis of the head than those I had taken in February and May of 2006—the photographs featured in figures 2.12, 2.13, and 2.14 taken in November 2008 illustrate that slight variations in the camera angle can yield different results. Without specialized equipment and special permission, it is impossible to achieve the laboratory-type analysis that I am convinced must be achieved in order to quantify exactly the accuracy to which these statues were crafted. With the aid of two-dimensional computer software, though, we can extract some basic geometric information about the artifacts and compare one half of each face against the other. In this manner, we can glimpse, through the fog of millennia, the minds of the designers of the sculpture, and we can conclude that a sophisticated geometric protocol was used. Designers, engineers, and craftspeople in the modern era may relate to the complex sculpted surfaces that have been proven here to create both sides of the face in mirror image.

  Figure 2.14. Ramses precision

  The contoured surfaces of Ramses’ symmetrical face would be familiar to designers of everyday products that are created routinely today with computer algorithms known as non-uniform rational B splines (NURBS), which allow them to smoothly morph one shape to another with unbroken perfection. By using NURBS, computer-aided design programs create contours of airplane wings, turbine blades, and even the computer keyboard at your fingertips. Surfaces are now routinely designed and manufactured to the apparent precision of Ramses’ head. Incredibly, the ancient Egyptians were also able to routinely craft Ramses’ head and achieved the same results again and again from the north to the south of their linear, Nile-based empire.

  The stunning implications are analogous to looking through the static interference pattern of time and confusion and seeing the elegance and precision that is normally built into a Lexus in a place where only the most rudimentary techniques of manufacturing are thought to have existed. The techniques that the ancient Egyptians are supposed to have used—those taught us in school—would not produce the precision of a Model T Ford, let alone a Lexus or a Porsche.

  BACK TO THE DRAWING BOARD

  There should be no question in our minds now that Ramses’ face was carefully designed using a system of measure that was based on geometric proportions. But what geometric shapes did the ancient Egyptians use, and how were they applied in the design?

  We know that the ancient Egyptians used a grid in their designs,3 and that such a method or technique for design is intuitively self-evident. It does not require a quantum leap of an artisan’s imagination to arrive at what is today a common design method. In fact, it is used now not just for design, but also for describing organizational and conceptual methodology. Grids, graphs, and charts are used to convey information and to plot and organize work.

  With this in mind, therefore, I took the photograph of Ramses and laid a grid over it. Of course, my first task was to establish the size and number of the cells used in the grid. I assumed that the features of the face would lead me to the answer, and studied which features were most prominent. After musing over this question for a while, I took a chance on a grid that was based on the dimensions of the mouth. It seemed to me that the mouth had something to tell us due to its unnatural shape, so I placed a grid with cell dimensions that were the same height and half the width of the dimensions of the mouth. It was then a simple matter to generate circles based on the geometry of the facial features. I didn’t expect, though, that they would line up with grid lines in so many locations. In fact, I was astounded by this discovery. Going through my mind was: “Okay—now when does this cease to be a coincidence and become a reflection of truth?”

  PYTHAGORAS MEETS RAMSES

  Plumbing the grid for further information, I discovered that Ramses’ mouth had the same proportions as a classic 3-4-5 right triangle. The idea that the ancient Egyptians had known about the Pythagorean triangle before Pythagoras, and they may have even taught Pythagoras its concepts, has been discussed by scholars, though not without controversy.4 Ramses presented me with a grid based on the Pythagorean triangle, whether it was the ancient Egyptians’ intentions or not. As we can see in plate 4, the Pythagorean grid allows us to analyze the face as it has never been analyzed before.

  In a manner similar to that of the geometry of the crowns, as discussed in chapter 1, plate 4 shows that circle geometry was also used in the design of the face. The correspondences that appear between the grid and the circles that are generated by the facial features are numerous and noteworthy. This design scheme is fairly simple and elegantly harmonious, because all elements are interrelated and have connections to each other, whether crossing or touching their companion elements. For example, Circle A, the bottom eyelid, and Circle C, which describes the top of the upper eyelid, touch Circle F, which outlines the jaw. At the same time, Circle A is tangent to the grid and crosses the grid and Circle B at the same point. All the circles are tangent to the grid except circle C, which is tangent to the jawline. Circle G, which describes the arc of the lower lip, is tangent to the grid and Circle C. These elegant correspondences were created with full knowledge that minor changes in the circle diameters would provide different information, so it could be argued that a different geometric scheme could have been intended. However, the point in presenting it this way is to illustrate the geometric constructs that an artist might use if they were going to paint a portrait of a perfectly symmetrical head. What elevates the importance of this design is the fact that it was three dimensionally crafted, with elegance and precision, in hard granite.

  FIBONACCI MEETS RAMSES

  We may ask if there might be another way to describe the geometry of Ramses’ face. I pondered this question for a long time, and while examining the shape of the ear, I thought perhaps a Fibonacci spiral might have been employed in the design.

  A Fibonacci spiral is created by blending a series of arcs that are generated using three corners of each of the squares depicted in figure 2.5—with one corner as the center point of each radius. As it
turns out, the Fibonacci spiral did not match the geometry of the ear. Yet because I had already drawn the spiral in the computer and had the image of Ramses up on my computer screen, I tried to see if there were any correlations using a Fibonacci spiral with the geometry of the face and the Pythagorean grid. Once I established the grid pattern and the circles, I trimmed back the circles and drafted the outline of the nose to create figure 2.15.

  Using the outline of the Ramses face with just the Pythagorean angle grid, I applied a series of well-known geometric constructs to determine whether any correlation existed between Ramses’ head and more advanced geometry than what I had seen so far. Quite remarkably, the oval that frames Ramses face is based on the Pythagorean 3-4-5 triangle (its height is 1.333 times greater than its width).