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Calculation of the Number of Stones in the Great Pyramid

This page was originally written as a post to the Egyptian Electronic Forum Discussion List, June 4, 2003

I have since modified it, based on further discussion and calculation.

 

Rife in Egyptology literature is the myth that the number of stones in the Great Pyramid is 2.3 million, more or less. Mark Lehner in The Complete Pyramids, Thames and Hudson, 1997, has 2.3, pg 108. I. E. S. Edwards in The Pyramids of Egypt, Viking Press, 1972, has the same number. Countless other authors could be cited.

In a recent post to the EEF discussion list Jay Enoch mentioned a paper, "Building of the Great Pyramid," History of Technology and Culture, Vol. 44-2, 2003, pgs 340-354, in which James Frederick Edwards proposed another method for lifting the stones up onto the pyramid that provides a far more sensible solution to the problem than hitherto devised. He suggests the sides of the pyramid directly provided a steep ramp up which, according to his calculations, the stones could be pulled. He also uses the time-honored 2.3 million stones.

The Great Pyramid is built over a small mound, which reduces the actual number of stones necessary to build the structure. We are unsure of the exact size and shape of the mound but estimates have been made from knowledge of where interior passages meet the bed rock. See illustration by I. E. S. Edwards, pg 86, and various other sources.

Sir Flinders Petrie, from his working notes, (Pyramids and Temples of Gizeh, Field and Tuer, London, 1883), calculated the height of the mound at 25 feet, pg 211, rising in steps. See his Plates VII and IX.

The total volume of the pyramid is 1/3 X Base Area X Height.

Petrie measured the mean base length at 230.348 meters. From projected measurements up the slopes on the four sides, and using the transit at the top of the pyramid placed there in 1874 to locate the original capstone, he calculated the height at 146.71 meters.

Calculating the volume we obtain 53059.98 X 146.71 X 1/3 = 2,594,816.96 cu meters. (Lehner gives 2.7 million cu met.)

From this must be deducted the volume of the mound, and the volume of the interior passages and chambers.

In order to be conservative I calculated the mound without the supposed steps at 7.6 meters height and taking up 7/8 of the interior area. This would make the volume of the mound 7.6 X 46,427 = 352,849 cu met.

From Petrie's published measurements I calculated the total volume of the passages, Queen's Chamber, Grand Gallery, Antechamber, King's Chamber, and Relieving Chambers at approximately 1300 cu met.

We can see that this is a miniscule amount.

Thus the estimated total volume of cut stones is approximately 2,241,000 cu met. (The volume of cut stone may be somewhat larger depending on the size of the mound.)

The stone courses vary in thickness, from as much as 1.476 meters on the first course to very nearly one royal cubit or 0.524 meters for many of the courses near the top.

The stones at the very top of the pyramid were removed centuries ago. The total number of extant courses is 203 on the northeast and 201 on the southwest. From levels of all courses Petrie measured this extant top height at 138.478 and 137.394 meters respectively.

Thus the mean thickness of all courses is 0.683 meters.

If we assume that each stone is a cube the volume of a typical stone would be 0.3194 cu meters. When this is divided into the total pyramid volume the result is 7,016,280 + stones.

We know that many of the stones are not cubic. Although they are held to a very tight tolerance on course thickness they vary considerably in width and depth.

Curiously, one can see from the photograph that many of the stones are not simply thrown against one another, but carved to tight non-linear fits. One can also see that some stones are wider than the thickness and some more narrow. Refer also to picture at the top of the page. These pictures show that the carving to tight fit would increase the time necessary to build the pyramid, since it seems necessary to make those fits as the stones were placed next to one another, and not done off site.

Also, from views published by PBS

http://www.pbs.org/wgbh/nova/pyramid/explore/khufutoplo.html

one can see that many of the stones are oblong, with geometric ratios in dimension of perhaps 1:1:2. Seldom, if at all, do the stones exceed this ratio; many tend more toward cubes.

If we assume a universal geometric ratio of 1:1:2 the number of stones would be 1/2 the above calculated value, or 3.5 million. Since many of the stones are more narrow this assumption may not be correct and the actual number of stones may be greater than this value.

Further, we assume, as has everyone in the past, and with lack of measures, that the stones interior to the structure follow the same thickness and lateral dimensions as those now visible on the exposed surface.

We could take Petrie's measure of each course height, and perform a more rigorous estimate, but we probably would not be any closer on the estimate. This is especially true in light of the great variability of stone lateral dimensions.

From this estimate we can safely conclude that the number of stones in the Great Pyramid is far greater than the 2.3 million assumed throughout the Egyptology community, and probably is more than 3.5 million.

This has a profound impact on our estimates of the schedule necessary to build this great structure. The assumption of 2.3 million is used universally for that calculation. One of the notorious difficulties, discussed at length by Petrie, and many others, is the fortuitous nature of events. A Pharaoh comes to power, he begins his construction project, and completes his massive enterprise just coincidental with his demise. Many in the past have spoken against such unrealistic schema of reality.

In his paper James Edwards uses these assumed values, calculated the rates at which the stones could be pulled up the sloping sides of the Pyramid, and again comes to the same fortuitous estimate, again coincidental with the demise of Khufu.

Unfortunately, if the number of stones is nearly double that assumed in all previous calculations, the time necessary to construct the pyramid would be nearly double also, more like 40 years rather than 20. This throws the entire schema into disarray. It magnifies the incredible, and now impossible, coincidence of completion with the death of Khufu.

An Addendum to the above post.

This Addendum was inspired by a discussion that the width and length of the stones could proportionately reduce the number required. Members of the list posted the fact that Anthony Sakovich had published a similar number in the KMT Journal. For example:

From: Jon Bodsworth

Anthony Sakovich wrote an article in KMT Volume 13 Number 3 (2002) "Counting the Stones" where he came to a figure of around 4,000,000 for the number of blocks.

I then responded:

Regarding the ratios of the thickness-width-length it seems to me we have a debate that is easily verified, or denied, by real evidence.

I cannot travel to the Great Pyramid and make personal measurements. But many photographs exist. The GP may be the most photographed object on earth.

I offered two web URLs that help define this debate.

We can see from the stones at the top of the pyramid that they are kept in "neat" geometric proportions. That is, they are not wildly "flat" with reduced thickness, but are, at the most, in ratios of 1:1:2. There is no visible ratio of 1:5:5 at the top.

http://www.pbs.org/wgbh/nova/pyramid/explore/khufutoplo.html

If we use the photograph by Jon Bodsworth I earlier mentioned we can estimate how the lateral dimensions compare to the thickness. Of approximately fifty easily visible stones only five stones are in excess of 2:1 in lateral dimension to the thickness. Near the bottom of the picture, (and near the bottom of the pyramid) are two very large stones. The one appears to be 4:1, (50/12 arbitrary units), while the other is uncertain because we may be looking at more than one stone not distinguishable by visible vertical lines. However, on courses above those two we do not find severe ratios, such as 5:1.

Using an arbitrary graphics scale I found that the number of stones with ratio of the width to the thickness group as follows:

0-0.5  (3)
0.6 - 1.0 (16)
1.0 - 1.5 (13)
1.6-2.0 (16)
2.1-2.5 (5)
2.5-3.0 (1)

 

We should note that the "flat" stones are at the corners of the structure; we do not know how they may have carried through the entire pyramid. Also, we do not know how the stone dimensions may have followed one another from structure to structure. Until we perform a thorough survey we may debate this endlessly.

The mean ratio of width to thickness from the above numbers is 1.33.

If we were to assume the length of each stone at 2:1 of the individual width (which procedure seems unsound, given the evidence from the top of Khufu), we could more closely estimate the number of stones. This would make the ratios 1:1.33:2.66. In my earlier post I assumed a universal ratio of 1:1:2. Sakovich also made similar assumptions about the ratios.

This multiplies out to a value of 3.54 vs. 2.0 I assumed.

Hence the number of stones would be proportionately reduced. In this illustration about 2.3 million reduced from my initial estimate of 4.0.

Trust good old Petrie. I deeply respect his acute practical sense and admirable analytical mind.

I am convinced.

Ernest

 

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