In full scale, on a leveled whitewashed rock surface near the entrance to the tomb of Ramesses VI in the Valley of the Kings, is the drawing of an ellipse intended to guide in the construction of the vault of the tomb. From this drawing the stonecutters could easily take the measures and transfer them to their working place at the roof of the burial chamber.
The photograph below is a view toward one end of the chamber, showing the elliptical curvature of the roof.
The tomb has been known since antiquity, attested by numerous graffiti. The Romans knew it as the tomb of Memnon; the scholars of the Napoleonic Expedition as La Tombe de la Metempsychose. It was cleared of debris by George Daressy in 1888. During excavations it was learned that the tomb was started by Ramesses V, the nephew of Ramesses VI. Egyptologists believe Ramesses VI usurped the throne of his nephew. He may also have taken the tomb for himself in the same action.
The decorations of the tomb nearly represent a treatise on Egyptian theology of that period around 1150 BC. In general, they provide the story of the origins of the heavens, earth, the creation of the sun, light, and life itself. The decorative plan for this tomb is one of the most sophisticated and complete in the Valley of the Kings.
Interestingly, this is the period of Moses and the people of Israel in their exodus from Egypt, within a few hundred years. This provides us with information on religious beliefs at the time, and the cultural milieu under which Moses was raised, as part of the royal household.
In the Annales du Service des Antiquites de l'Egypt, Cairo, Vol 8, 1907, pgs 237-241 Daressy provided a report on the ellipse found outside the tomb. Following is a copy of the drawing, as published by Dieter Arnold in Building In Egypt, Oxford University Press, 1991, pg 22.
I transferred the above drawing into QuickCad software to determine the measurements, the coordinate points, and elliptical properties. Following is a converted drawing.
I drew a mathematically correct ellipse and imposed it over the rock drawing. This shows the roughness of the original, not quite true, but probably good enough for a working drawing.
The tick marks of two cubits each show the actual length of the figure. They also display the mathematical knowledge of the Egyptians of that period.
They understood the design and construction of mathematical ellipses, with two foci.
They were able to construct a true ellipse (within the tolerance used for the tomb).
They used a mathematical design for the ceiling of the burial vault. They did much more than merely carve a roof out of the rock.
The foci are located at +/- 4 cubits along the major axis from the center of the drawing. They chose this value as the starting point of the mathematical design.
They knew the Pythagorean relationship of 3-4-5 right triangles. The vertical of the ellipse at 5 1/3 cubits length was designed to produce this relationship.
Their construction marks along the major axis demonstrate this knowledge.
They knew how to calculate mathematical ratios, illustrated with 3 is to 4 as 4 is to 5 1/3. They were not constrained by simple round numbers.
They could design with portions of cubits, here illustrated by unit distances of two palms.
They used a chord determined from the ellipse as the starting base point of the rock ceiling.
This horizontal distance was apparently determined by the desired width of the tomb.
They stylized the roof by flattening the top portion; they did not continue with the elliptical curvature. This is shown in the picture below.
This is a picture of the opposite end of the tomb. Please note that this comes from the Theban Mapping Project, with photo credit to Francis Dzikowski.
I shall now show some portion of their designs.
I have not attempted to show all aspects of the tomb design. I merely show those that are pertinent to our discussion.
First I fitted the ellipse by eye. I adjusted it according to best fit. This is a true ellipse, generated by my QuickCad software. Unfortunately, the photograph was not taken directly on center. Hence it shows a slight disparity from one side to the other.
I attempted to obtain dimensions from the Theban Mapping project, but their numbers do not reflect the actual dimensions of that part of the tomb designed around the ellipse. The width of the chamber is irregular. At the point where the chord of the ellipse turns into the vertical wall measures from 11.91 cubits to 12.09 cubits from one end of the chamber to the other.
After adjustment, I assumed the length of the measured major axis was 13.73 cubits, the same as that on the rock drawing. You can see how the major axis extends beyond the chamber width. On the one side this is carried through with the panel displays; on the other it stops due to the vertical wall. Refer to above photographs.
I then calculated the respective dimensions of the drawing from this reference. The measured height above the major axis is 5.34 cubits, almost exactly that calculated from the rock drawing. This is one-half the minor axis. The foci of the ellipse are at +/- 4.0 cubits, the same as on the rock drawing.
Note that the vertical minor axis falls in the center of a small display panel. This same vertical axis splits the Sun, and the Figure standing in the boat, presumably Ramesses VI being carried to heaven.
Next to the small panel on the right are two other panels. The 4.0 cubits mark falls directly on the right-most limit of the second panel.
In keeping with the odd number of cubits shown in the (1/2) minor axis of 5 1/3 cubits, the two rows of inscriptions and drawings are 1 2/3 and 2 1/4 cubits height respectively. Note that although the horizontal is marked off in two-palm widths on the working drawing, none of these numbers calculate to even multiples of palms or fingers. Obviously, the designer of the tomb drawings worked directly with cubits in calculating heights, and could switch back and forth between palms and cubits.
The mathematical ratio of the ellipse is 13.73 by10.66, or a ratio of 1.287. This was determined by the 3-4-5 triangle.