A while ago I posted a few 3D graphics that I'd done of the Great Pyramid. I had used Petrie's data to create a "virtual pyramid".



I used his core block heights to build up the pyramid and was then able to plot the internal elements using his measurements to link them where possible with the layer heights.



Petrie's data is ideal for such an exercise as he gives so many measurements as an offset from, for instance, the pavement or other internal elements. This is much easier to plot directly in 3D space than trying to construct angles which are more difficult for me and for Petrie to measure accurately in the first place. Petrie gives a beginning position and an end position for the Descending Passage, for instance, so I was able to fix them on a 3D grid and then join them up rather than construct a passage at a particular angle.

Romer in his recent book also refers to the advantages of using the core layers in the building of the actual pyramid and it's no different with a virtual one:
"Fundamentally these horizontal alignments underline how the pyramid's courses served in the manner of draftsman's graph paper, providing accurate horizontal planes of reference that projected the heights of some of carefully established points of the internal architecture right across the rising Pyramid. Cross-checked by a system of co-ordinates set up in design and field and against the polar stars, such horizontal controls, a series of 200 odd subsidiary baselines, would have provided the pyramid-makers with a constant and immediate control on building movement and inaccuracy."

Pilling up the core layers is easy using Petrie's measurements but putting the casing on is a different matter as you have to decide what measurement to use for the final height or conversely what angle to use.

Petrie says of obtaining an angle from the remaining casing stones at the North Face:
"The angle by means of the large steel square was vitiated by the concretion on the faces of the stones being thicker below than above, .1 inch of difference making an error of 6'. The small goniometer was applied to the clear patches of the stone, selected in nine different parts. These three casing stones in situ have not as much weight assigned to them as they would otherwise have, owing to their irregularities."

Petrie was aware of the casing stones on the East and West sides but they weren't fully uncovered until after his survey. Actually they are of little help as they are not as well preserved as those on the Northern face.

He eventually fixed on a angle of 51° 52' plus or minus 2' and "recreates" an ideal pyramid that he gives the measurements of 9068.8 inches plus or minus .5 inches x 5776.0 plus or minus 7.0 inches.

His drawing shows the casing stones at the North side and he draws the casing stones higher up as having a similar depth. However he acknowledges that this is unlikely in reality. I've followed his example though as I'm basically producing a 3D version of his measurements and drawings.



(Drawing reproduced from Petrie's "The Pyramids and temples of Gizeh" revised edition 1885: fold out plate number five.)

A 3D version of Petrie's drawing.



Petrie doesn't address the use of the "Seked" directly in his original survey but he seems to acknowledge its use in an article from 30 years later where he says "At every 14 cubits up, the face must be 11 cubits nearer to the cross axis".

The angle of other surviving casing stones all seem to be close to Petrie's figure of 51° 52'.

Smyth says "Mr Wayman Dixon... ...in 1869, discovered almost a whole casing-stone. Not a very large one, indeed, but with portions more or less of all its six, original, worked sides; or a completer example than is known at the present moment to exist anywhere else all the world over. This most unique specimen Mr Waynman Dixon graciously sent from Egypt as a present to me, and I have deposited it under a glass case in the official residence of the Astronomer Royal for Scotland, where it has been closely measured, and its ascending angle found to be certainly between 51° 53' 15" and 51° 49' 55";...".

The same casing stone is described by Rutherford as having an angle of exactly 51° 51' 14.3" (the so called Pi angle, it's no surprise that Rutherford supports pyramid Pi).

I've been unable to establish an angle for the casing stone in the British Museum. Unfortunately only the top and part of the front are the original worked faces and they are a little rough. However it's certainly around 52°. However all these detached stones have to be treated with caution as there's no way of knowing if the stone was laid exactly horizontally (Petrie suggests not all were).

As the Egyptians didn't use degrees etc quoting all these different angles to fractions of minutes and seconds is probably not very useful as there's no indication that the builders would have been able to distinguish between them. (Rossi suggests that a difference of .5° is probably not significant)

If we take Petrie's base measurement of 9068.8 inches all the "popular" suggestions for how the height was established fit within his tolerance of plus or minus 7 inches ie:

The seked ratio 5.5 palms (5 palms + 2 fingers) gives a height of 5771 inches.

An exact number of Cubits - 440 x 280 cubits - gives a height of 5771 inches.

The "Pi angle" 51° 51' 14.3" gives a height of 5773.377 inches.

Romer's diagonal of a grid square of 219.99 cubits by 279.99 cubits gives a height of 5770.6 inches. (Romer's method of establishing the angle is based on multiples of his "grid" of 17.33 cubits plus a bit).

Only Lauer, as far as I know, has claimed that it's the aris length and angle that's important. The aris is the corner edge which in the case of a pyramid of 440 x 280 cubits measures around 418.5 cubits.

As we don't know what tolerances the builders worked to or even the amount of settling etc that might have happened over thousands of years I think Petrie's stated tolerance of plus or minus 7 inches is still valid and as close as we'll ever get. Even if we knew for certain what method the Egyptians used to establish the height we will never know how close they actually got to it.

One extra piece of evidence has always intrigued me. In his four volume work on the Great Pyramid Rutherford refers to a correspondence he had with the French archaeologist Professor Jacques Vandier concerning his discovery in the 1950s of remains of casing stones still in place just below the pyramid summit. Maragioglio & Rinaldi also mention this with a reference to Vandier's "Manuel d'Archéologie Égyptienne". It's only recently that I've been able to read this passage for myself. Unfortunately it's no help as it adds nothing to Rutherford's or M & R's comments. It does no more than confirm that remains were found but there's no information on the size, exact position etc. So the attractive idea of being able to link casing stones at the base with some near the top and therefore measure an angle seems, at the moment at least, not to be possible.



Jon

www.egyptarchive.co.uk



Edited 2 time(s). Last edit at 07/12/2007 04:10PM by Hermione.