1. | Layout of the Giza Pyramids |
2. | The Great Pyramid of Cheops |
3. | The Pyramid of Chephren |
4. | The Pyramid of Menkaure |
The layout of the Giza plateau and Pyramids. |
Introduction
There are three main pyramids on the Giza Plateau named after the pharaohs to whom they are claimed to be dedicated. In this work they are given reference numbers according to their size and position from northeast to southwest. They are the pyramids of Khufu/Cheops (G1, the Great Pyramid), Khafre/Chephren (G2) and Menkaure/Mykerinos (G3). The dimensions of the pyramids and of the plateau may be found to vary slightly in publications, but in this article the work of Sir Flinders Petrie is used as a target for the designs. An online source for Petrie's work can be found at: The Pyramids and Temples of Giza Online There have been many attempts to reproduce the layout of the three pyramids on the Giza plateau, but most are based on the assumption that a cubit of constant length was used. My analysis of the individual pyramids suggests that there may well have been two cubits, one used at G1 (the Great Pyramid) and one for both G2 and G3. I did fear that there may have been more cubit lengths and that it would therefore be nigh on impossible to determine the intent in placing the pyramid centres. However, led by the thought that the plan would very likely be fairly straightforward and might, more than likely, reflect pi, I continued my investigations. As a start, it may be worthwhile considering Petrie's plateau measurements in a slightly different light. It is a sensitive subject, but I have always assumed that, in addition to lengths that would have been physically measured, some dimensions on Petrie's plans would be based on theodolite sightings from baselines of given lengths. It would follow, then, that there are inherent assumptions in Petrie's measurements about what the Ancient Egyptians intended and achieved, particularly with respect to right angles. For example, I do not know if the builders placed a marker at the point of intersection of lines due west of the centre of G1 and due north of the centre of G3, or whether Petrie found one, but I take it that Petrie's dimensions assume that there is a precise right angle there whether or not this was achieved by the builders. Petrie's work and analysis supports the view that the builders used a cubit at the Great Pyramid (G1) of approximately 524mm (the Royal Cubit). It also suggests that the builders introduced a pi ratio into the construction, such that given pi to be equal to 22/7 then half the perimeter divided by pi equals the height. |
The Hypothesis
I assumed that the Giza layout was set out in whole numbers of cubits, and it was only at implementation that the different cubit lengths were introduced. My analysis of the G2 and G3 pyramids introduced a potential cubit of 528mm, so at that point I had two cubit lengths to work with. The G1 cubit would operate on a baseline at north and the G3 cubit would operate south of this. One of my thoughts was that the east-west plateau measurement might be 1100 cubits with the north-south dimension 1400 cubits, the reason being that the triangle would thus replicate the G1 pi ratio (11:14). Were the southward (G3) cubit 528mm as postulated, the length of the north-south line would actually be 1400 cubits (Petrie's 29,102.6 inches x 25.4 / 528). However, at the north, dividing 22,616.6 inches by 1100 derives a G1 cubit of 20.56 inches (522.2mm) whereas the cubit arising from the dimensions of G1 appears to be 20.61 inches (more or less the mean of the other two). Had the actual G1 cubit been used, the length of the baseline would be 1097 cubits. Were the length 1100 such cubits (considered unlikely) then the corner angle would have been some 91 degrees. While the cubit used at G2 appears as if it could well be 528mm it is difficult to determine whether the pyramid's centre was positioned using the assumed G1 or G3 cubit. By Petrie's measurements, the centre would lie approximately 640 G1 cubits west of the centre of G1 and 670 G3 cubits south. The task would be to explain this |
A Template for the layout of the Giza plateau and Pyramids. |
The Layout
The plan begins with an east-west baseline of 1100 cubits of 20.56 inches. The Great Pyramid is built at the eastern end using a cubit of 20.61 inches. Thereafter, for the construction of the other main pyramids (G2 and G3), a cubit of 528mm is used. I suggest that the plan shown at the link below may have been the model for the enterprise. This is a geometrical design with the northwest corner of G2 upon the arc of a circle passing through the centres of G1 and G3. Link to Giza Model The key to placing G2 is a G1-type pyramid formed at the centre of the design having a height of 294 cubits and a base of 462 cubits. This layout may then have been modified (see left) by extending the height of this pyramid to 550 cubits (half the width of the plateau), the southward distance to the centre of G2 thus being defined by where the slope meets the west face of the plateau triangle. The G2 pyramid may thus have been initially designed with its base defined by a G1-type pyramid. |
The finished template matches Petrie's measurements to within four inches north and seven inches south. If checking the calculations, bear in mind to use a cubit of 20.56 inches from east to west (as a true right angle at NW has to be assumed) and a cubit of 528mm from north to south. |
G1: The Great Pyramid of Cheops / Khufu | |
Petrie's plan suggests that the base of the Great Pyramid is 440 cubits, and from the extant casing blocks, he determined that the original height would have been 280 cubits. This leads readily to the suggestion that π as 22/7 is present in the design.
In the figure, AB = 2AC / π. |
An overall design for the Great Pyramid. |
A π ratio of 25/8 may also be present in G1. It should be appreciated that there are in fact eight faces to the Great Pyramid. The thought is that the central indentations on the faces might be a consequence of a second π ratio.
A sub-design for the Great Pyramid. The diagrams are exaggerated, but it may be appreciated that with a height of 280 cubits and with π as 25/8 there would be scope for an inner square with sides of 427/5 cubits that makes an appearance at the indentations, which are assumed to be 1.25 cubits (7 hands or 35 digits) which is about 655mm. If the builders divided the indentation into quarter cubits then the closest to π they would get conceptually is 1759/560, though I imagine it would be too much to expect them to know which division came closest. |
G2: The Pyramid of Chephren | |
The G2 pyramid is widely reported to have a height of 274 cubits and a base of 411 cubits (the cubit being that of G1). It should be noted that the height to base ratio is 4:6.
While considering the dimensions, it struck me that a π relationship would be present at G2 if using the diameter of the circumscribed circle and π as 25/8. This is because with a base of six units the radius would be 3.125 (25/8) units. |
A design for the Pyramid of Chephren. |
It is generally held that the dimensions of G1 make 'sense' with respect to the observed ratios. However, this does not apply to G2 as I have described. Should this specification be correct then it might be suggested that a height of 272 cubits and a base of 408 cubits would be more in keeping with the design. However, this would suggest a G2 cubit of 528mm.
Nevertheless, the revised dimensions would result in a diagonal of 576 cubits (cos45=17/24) and a corner edge slope of ATN(17/18). The dimensions may thus begin to be more understandable and integrated - more so than 274, 411 and 581 cubits. The internal dimensions of G1 are generally consistent with the external (they use the same cubit.) Similarly, there is evidence within G2 that a cubit of about 528mm is viable, and this also applies to G3. |
G3: The Pyramid of Menkaure | |
Petrie gives the base of G3 as 4163.6 inches and estimates the height as having been 2580.8 inches. In this paper, G3 is deemed to have a base of 200 cubits and a height of 128 cubits with π as 25/8.
The base is given as 4157.5 inches, so the suggested height would be 2661 inches, that is Arc Tangent (32/25) representing an angle of 52 degrees. This would be in accordance with Petrie's measurements and the angle of the limestone casing blocks as reported. |
A design for the Pyramid of Menkaure. |
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