Questions and Answers

Q: How was built the Cheops Pyramid ?
YM: The Egyptians built the pyramid of Cheops as a pyramid with degrees first, before being transformed into a smooth-faced pyramid.

Q: How do you say that?
YM: This deduction is based on the observation of the previous pyramids (Djoser pyramid with degrees, Meidoum and its collapse, the breach of Mykerinos, the satellite pyramids of Giza …)

Q: Where are these massive?
YM: The massive corresponding to the step pyramid are located every 9.1m (17.5RC), or twice.

Q: Where do these numbers come from?
YM: From the study of Flinders Petrie measurements and comparisons with laser measurements of Neubeuer or EDF microgravimetry.

Q: Back to the massive, how were they formed?
YM: The massive represent more than 84% of the total volume of the pyramid. The blocks used to build the walls are at most 60cm high and weight less than 1T. This is a deduction from Flinders Petrie’s measurements. If we look at Mykerinos breach, content of the massive should be similar, but we have to admit we are not sure, because Mykerinos pyramid is much smaller than Cheops pyramid: so it could be of rough blocks, banks made of smaller stones, soil, sands… Whatever the case, stones or materials were lifted along the walls of the massive through scaffolding or wooden crane. A possible scaffolding could be a mobile tower similar-to warfare tower described in Kaemheset tomb (IV-V dynasty).

Q: But everyone agrees that ramps have been used?
YM: The problem of ramps as they were considered, gently sloping, is their length that would exceed the kilometer to reach the top of the pyramid. A compromise would be to use gentle ramps for the first 54m then to lift blocks for the last 92m.

Q: And sloping ramps?
YM: It is not excluded that one or more access ramps of 26° slope belonging to the pyramid have been used for the first 30 meters at least.

Q: Where does this 26°  choice come from?
YM: This is a deduction made from the presence of the corridors and the great gallery. The angle 26° is easy to put in place, it is the diagonal of 2 squares put side by side.

Q: Yet it is difficult to move along a 26° slope
YM: Not if staircases are distributed on either side of the ramp, or if a counterweight system has been put in place.

Q: You say it’s better to lift loads rather than pulling them along a slope, so why have access ramps?
YM: These ramps could be either as the great gallery that would have been used mainly to mount the beams and rafters of the King’s chamber through a counterweight system, or surrounding stone ramps, part of the massive and not exceeding the angle of the slope.

Q: What about the remaining 16%?
YM: 14% are bigger stones used to constitute the bleachers that are currently known. They can reach 1.5m in height, weight from 1 to 3 tons or more, and come from two distinct quarries. This is a deduction from Flinders Petrie’s measurements. The other 2% used for smoothing the pyramid were either natural limestone from Tourah quarries, as « suggested » in Merer diary, or artificial limestone, as « demonstrated » by Michel Barsoum, starting from Joseph Davidovits work. Whatever the case, these limestones have almost disappeared for Cheops pyramid but remain on top of Chephren or on bent pyramid.

Q: Why do you say that the bleachers and the smoothing of the pyramid would have been done from top to bottom?
YM: For obvious reasons of accessibility, it is perilous to work on a cemented lining or fragile limestone. The stones of the bleachers are passed along the massive and placed by means of cranes associated with levers and wedges, from the bottom to the top but from the upper massive towards the lower massive. This is consistent with Herodotus.

Building steps

Lift or Pull stones

Is it better to lift or pull stones ?
In order to raise a mass at a given height H, is it more efficient to use a soft ramp (7°), a normal ramp (11°) or a strong ramp (26°) ? The answer to this question will be useful to explain building of the Giza pyramids.
Posing the problem
In order to raise a mass at an height H, the more the angle increases, the more the force to Apply is important and we need more workers to pull the mass with same effort. On the other hand, the raising distance is reduced. Then for a fixed number of workers, how should we use them to raise faster the mass at a given height ? Should we pull more masses on a longer distance or less masses on a shorter distance with go back ?
Solving the problem
Because of friction forces, the work W=F.D is non conservative. Then, in order to furnish the same power E whatever the path (a or b), we have: E=W/T=cte with T the time to raise mass.
E=Wa/Ta=Wb/Tb
Ta=Tb.Wa/Wb=Tb.FaDa/FbDb with Fa=P[sina+ucosa], Fb=P[sinb+ucosb] Da=H/sina, Db=H/sinb
Ta=zTb with z=[sinb/sina][sina+ucosa]/[sinb+ucosb]
Studying z
z represents the time ratio between cases angle a and angle b, it is too the ratio of masses and Energy: E=Wa/Ta=Wa/zTb=Wb/Tb then Wa=zWb
z>1 always, whatever the friction coefficient u and whatever the angle b, b>a
Concrete example
Realistic example obtained by comparing slopes a=7° and b=26° with u=0.5. The force or worker ratio is F(26°,0.5)/F(7°,0.5)=1.43. The distance ratio is sin(7)/sin(26)=1/3.6 and z=3.6/1.43=2.5. If we have 1430 workers, with 100 workers needed to raise a 5T block, in 10 travels we will raise 143 blocks of 5T that is 715T. In case of a 26° slope, 10 groups of 143 workers will be needed to raise 50T per travel. There will be 36 travels (10*3.6), the workers will raise finally 1800T (50*36).
Extreme case where b=90°
This is the shaft case studied by Peter C. Sundt in 2007 : he gives the formula of the hauling energy:
Wa=zWb=z.P.H where P is the weight and z=1+u/tana
Peter C. Sundt showed that in terms of energy it is better to use a shaft than a ramp. So why considering an elevator starting above 45m and not on ground level? He gave two questionable explanations : ramp lengths exceeding the km, and materials going into the ramp more than into the partial pyramid.
Conclusion
In 2007, Peter C. Sundt makes for the second time a compromise between ramps theories and lifting theories (before Jean Kerisel in 1991: « Les pyramides à travers les âges »). Concerning this aspect, I share same conclusion as him : use only ramps at lower levels and lift at upper levels. However Peter C. Sundt considers only one central elevator which is not enough, because there can’t be more than one load per elevator in contrary to ramps.

Following illustration considers massive with ~9 and ~18m height (35RC):

Uvö Holscher Ramps improved

Uvo Hölscher is an Egyptologist and German architect. He has stated for the first time the lateral ramp theory for explaining building construction such as Chephren pyramid. Unlike Flinders Petrie or Ludwig Borchardt who preconceive frontal ramps, Uvo seems to have imagined to put ramps along faces of internal massive of pyramids. Sadly it has been reproached two things to its lateral ramp theory:
* First reproach made to Uvo Hölscher are the strong slopes enforced to the ramps that can spread only along the faces. That is the reason why this theory has been accepted only to explain building of small degree pyramids. After that, new theories appeared such as lateral ramps with extensions, semi internal ramps, spiral ramps or internal ramps. These theories prefer gently slope ramps. As it has been demonstrated, it is more efficient to use strong slope ramps and even better to lift blocks along Walls, so first reproach made to Uvo Hölscher is no more valid.
* Second reproach made to Uvo Hölscher is that the presence of lateral ramps such as imagined by him forbid final coating installation, because its ramps are temporaries in bricks, since they exceed the angle of the slope.  Our solution: we consider surrounding ramps, part of the pyramid and not exceeding the angle of the slope. Unlike Uvo, these ramps should be not removable, made of stones and not of raw bricks.

Following illustration shows such kind of ramps… but keep in mind that the use of these ramps should be for the first meters (54m roughly)!

Block size measurements comparison

The ranks of the Cheops (Kheops or Khufu) pyramid were measured for the first time by the scholars who took part in the expedition of Egypt led by Bonaparte, namely Jomard & Cecile, and “Le Pere Coutelle”. The English Egyptologist Flinders Petrie carried out precise measurements at the north-west and south-east corners of the pyramid, published in 1883. The French Egyptologist Georges Goyon carried out new measurements, published in 1978.

Block size measurements study by Tarrel and Goyon

In 1924 an English engineer J. Tarrel had observed that the thick foundations at the beginning gradually decreased in height to a level from which there was an increase, followed by a further reduction in the thickness of the courses and so on. He thus counted 18 slices or groups of assizes to the summit (The Great Pyramid Races Anc Eg. 1925, II p.36-38). Goyon also found 18 groups of assizes. It should be noted, that the heights of the seats do not decrease regularly and that the slices are formed by an unequal number of seats. On the other hand, there has been no discovery of any mathematical order in the decrease of numbers: the heights of the seats do not diminish regularly, and the slices are formed of an unequal number of seats. Tarrel thought that, as preliminary work, the stones necessary for the construction of the pyramid had been accumulated for 10 years and spread over « hundreds of acres ». Goyon thinks that this mode of construction is a technique related to the nature of the soil from which stones of the pyramids were extracted. Tarrel and Goyon should be partially right.

Block size measurement first observation

First, let’s replace the coordinates of the foundation by the cumulated heights of the size of the measured blocks

Here is another point cloud representation of the measures of the size of the blocks:

The squares in the previous graph are the maximums extracted from Flinders Petrie measurement. Then let’s make a graph with these maximums:

We see clearly a sort of sinusoid made by the maximums, this is a first discovery of a mathematical order in the decrease of numbers:

Bloc size modelisation of maximum observed

The maximums extracted from Flinders Petrie measurements and other archeologist are modelised here (gray curve)

The fact that main and secondary maximums don’t follow same decrease law is related either to the way of building or the use of two distincs stone quarries (because Goyon and Tarrel explanations could be true both together). Looking at Merer papyrus, covering stones could be comming from two quarries like Tourah North and Tourah South; that is to say from one massive to another one, stones of the bleachers can be of two different types.

Flinders Petrie maximum coordinates

Measured	Extremums	Modelised	dy	dy/y (%)
1.488	0	0	/	/
17.919	17.8	18.25	0.33	1.8
30.158	29.5	27.375	-2.78	-9.2
38.151
40.711
48.334	47.6	45.625	-2.71	-5.6
54.528	53.4	54.75	0.22	0.4
59.636	62.65	63.875	4.24	7.1
65.882
69.83	72	73	3.17	4.5
75.799
83.315	82	82.125	-1.19	-1.4
89.669	88.7	91.25	1.58	1.8
97.525	100	100.375	2.85	2.9
102.23
105.87	105	109.5	3.63	3.4
109.436
116.107	115.7	118.625	2.52	2.2
125.928	125.4	127.75	1.82	1.4
134.623	135	136.875	2.25	1.7
146	146	146	0.00	0.0

Cheops pyramid levels

Now Flinders Petrie measurements can help us to differentiate the different levels of construction of the great pyramid. Indeed we have seen that there is a clear tendency to have maximas followed by a decrease to a minimum preceding the next maximum. This abrupt transition from a minimum to a maximum could mean the presence of one of the levels of a step pyramid. These levels should be found each 18,2 meters if we consider the main maximums only and 9,1 meters if we consider all the maximum observed*. The levels are thus theoricaly defined: (0), 18, 27, 54, 73, 91, 109 128, which remains close to the levels deduced from the measurements (0, (18), 30, 54, 70, 90, 106, 126).

The Egyptians most certainly used a map of the Pyramid of Cheops before building it. But on what scale have they worked? To answer this question, we will draw on the work of Ole Jorgen Bryn, and instead of considering the number 7 as a ‘magic’ divider, we will use the number 8 because there are probably 7 massives preceeding the smooth faced pyramid. The pyramid being 280 royal cubits high with a half base of 220 royal cubits, divided by 8, this made a grid made of rectangles of 27.5×35 royal cubits or 192.5×245 palms. And a ratio of 14/11 (35 / 27.5). 35 royal cubits corresponds roughly to 18m.

* In this case, the massives of the pyramid should be found theoricaly each 9,1 meters, that is at following heights: (0) 9 18 27 46 55 64 73 82 91 100 109 119 128 137 (or corresponding height given by Flinders Petrie that is for example 83m instead of 82,1… See Flinders Petrie maximum coordinates article)

Cheops Pyramid before smoothing in case of massives with ~9m height each