dimanche 21 août 2011

Nouvelles considérations sur les techniques de construction des anciennes pyramides égyptiennes, par le Professeur Frank Müller-Römer - IV

Première partie

Deuxième partie

Troisième partie
Calculating the building time of the pyramid of Menkaure

"In calculating the efforts made in building the pyramid for Menkaure, and the resulting building time, we must use rough estimates in some points, especially as far as the block sizes are concerned, which are different in each layer. However, as the burial chambers of the pyramid of Menkaure are located below the core steps, estimating these numbers is comparatively straightforward.

– The measures of core and backing blocks are approx. 1.3 m x 1.3 m x 0.7 m (1.2 m³), thus their weight amounts to approx. 3 tons, as the specific weight of limestone is 2.4 tons per m³. (62) This calculation accounts for the fact that the core stones have slightly smaller dimensions than those of the outer walls of the core. These numbers serve the purpose of transparent reckoning, as the average block size will be used in the calculations of sledge loads and towing below.
– For the stones of the casing, the mean measure is 0.7 m x 0.7 m x 1.1 m (0.54 m³, approx. 1.5 tons). Two of these blocks would form a sledge or hauling load.
– Transport efforts as calculated here account for the building itself with core and casing, as well as the attached working platform. Moving of the winches would only become necessary occasionally, depending on the number of transport units. Therefore, I have excluded this task from my calculations.
– Also, the flat rate stone volume assumed for spaces filled with mortar, sand and so forth is negligible.

Throughout the construction process, a broad range of transport and building tasks was employed. After upward transport, the stones had to be moved horizontally into their definite position. Due to the limited number of ramps, upward transport required far more time than horizontal transport within one layer, as a greater number of workmen could be set to the latter task simultaneously. Notably, horizontal transport may therefore be regarded as comparatively irrelevant to the building schedule, compared to ramp-dependent transport. Consequently, I have only taken into account ramp-dependent transport.
My model of transport of a sledge load, or a hauling process, up to a height of one step, is based on the following practice-based assumptions : the duration of transport up to the next step amounts to approx. 5 min, that is a towing speed of approx. 4 m per minute. For fastening and releasing of the ropes from the sledge or the block, the shifting of the sledge or respectively the load, and the back transport of the empty vehicle, rope and so forth, 5 minutes should suffice. Therefore, a unit duration of 10 minutes is realistic.
Further to the schedule calculation, we need to keep in mind that the blocks for the upper steps had to be moved over several ramps, which was a more time-consuming procedure. On the other hand, if we start from the assumption that a block was conveyed continuously from one ramp to the next to its final destination, it follows that a new block would reach the construction level every 10 min, be it via sledge or hauling. The additional effort necessitated by the greater height that had to be overcome could be compensated by an increased workforce. The assumed average values for block weight and unit duration directly influence the estimated building time. If these sizes are altered, a new schedule will result. Obviously, calculations such as these are always more or less hypothetical, depending on the numbers inserted above.
The respective volumes of the six core steps of the pyramid of Menkaure add up to a volume of 140 736 m³, that is 117 280 stone blocks at a volume of 1.2 m³ each and an equivalent number of sledge loads. Taking into account the number of ramps, the total work units amount to 24 093 with redirection rollers. The top of the pyramid with its volume of 900 m³ was built at a later stage when the platform had been installed (see above). Further to a calculation of the building time of the core, we propose that a sufficient workforce was available on three hundred days of the year, ten hours per day (shift work), which means that under the circumstances outlined above (10 minutes per load), a total of 18 000 units were conveyed in a year. Therefore, the building of the core took 1.3 year. We still have to add construction and removal of the ramps on the first to sixth step, that is 28 ramps at a total volume of approx. 15 400m³, requiring a total of 0.3 year if work took place on all sides of the pyramid. Therefore the entire core could have been readily completed in just 1.6 year.
My calculation of transport volume and building schedule for the working platform attached to the outside, plus the pyramids’s top, backing and casing, covers a volume of 141 358 m³ or 22 458 work units. If again we base our calculations on the availability of the workforce as outlined above, a total 1.5 year results. Next, the ramps and the platform need to be removed, and the stones left in boss need to be dressed. The removal of said structures requires 0.6 year in keeping with the framework set out above.
Only a rough estimate can be offered for the dressing works on the facade. A time frame of
about 0.2 year seems realistic in the light of the following considerations. Firstly, the lowest sixteen layers were left partly unfinished (granite wainscot). Secondly, we may assume that on one day (10 hours of work in shifts) one team of workmen succeeded in dressing a facade area of 1.0 m x 1.0 m (Tura limestone from the seventeenth layer onwards). In the case of the granite wainscot, the area would have only been 0.3 m x 0.3 m. By employing a large number of teams, the same work process can be undertaken in the lower part of the pyramid, on all four sides at the same time. At a height of 65 m, excluding the eight lowest layers, an approximate duration of 434 days or 1,5 year results.
The dressing works would take slightly longer than the time needed for removal of the working platform (0.6 year). However, we must still keep in mind that the removal of the working platform and the ramps must be coordinated with the dressing and disruptions due to mutual impairment must be dealt with. To account for such unforeseeable circumstances, another 0,2 year is added.
The debris accumulated during removal of the ramps and the working platform, largely mud bricks, amounted to roughly 60 000 m³, that is approx. 25% of the pyramid’s volume. While mud bricks could be distributed for use on the surrounding agricultural domains, the rest of the debris was stored on the North Eastern slopes of the desert plateau and in the area towards the South of the Giza plateau.
The building of the pyramid of Menkaure, not counting the time spent on pre-construction preparations, therefore took approx. 4.8 years. All in all, an average volume of 202 m³ of stone or 186 blocks were moved per day.

Towards a comparison with the building times of the Red Pyramid and the Pyramid of Khufu
How does the schedule proposed for the pyramid of Menkaure compare to that of the other pyramids ? In the following chapter, I propose for comparison some calculations for two other
pyramids, for which a good enough time frame is widely acknowledged. Again, my calculations are based on estimates in some instances. I base my arguments on the points made above with reference to the core steps, ramps, and unit duration. Similarly, the construction of backing, casing, and working platform is assumed to follow the same principle as illustrated above.

Building time calculation for the Red Pyramid

In order to estimate the building time for the Red Pyramid, we need to first establish a framework of data that can only partly be verified by archaeological evidence due to the pyramid’s state of preservation.
• The height of the respective steps is assumed to be 10 m. Therefore, the pyramid has a total of nine steps.
• The average size of the backing and core blocks is approx. 1.1 m x 1.0 m x 0.7 m, (63) which is a volume of 0.77 m³, or a weight of 1.9 ton.
• Although the ramps are slightly longer, the duration of a single unit can be fixed at 10 min, because the average weight is lower in the Red Pyramid than in the one previously discussed.
• The number of working days per year and hours per day is equal to that discussed above.
• The building of the two burial chambers is not taken into account, as these are subterraneous structures.

Fig.28 Red Pyramid : Propos1 for Rampsystem

Our estimation of the building schedule of the Red Pyramid, not counting the preliminary preparations, will again start from the assumption that construction works took place on three hundred days of the year, on ten hours of each day, in shift work. It follows that a total of 18 000 transport units of 10 minutes each could be carried out per year. Thus, the construction of the core of the Red Pyramid took 11.6 years. The time needed for the construction of the working platform, backing and casing, adds up to 4.9 years. For subsequent removal of ramps and platform, and dressing of the blocks left in boss, 3 years should suffice.
The time frame for dressing comprises approx. 223 work days or 0.8 year, to which we must add half a year during which the removal of the working platform and dressing would need to be coordinated. If we assume that this time must be added to that needed for the removal of the working platform and so forth, the actual building time of the Red Pyramid results at 18.7
years, during which in total an average of 443 m³ of stone, or 576 blocks, were moved each day.

Building time calculation for the Pyramid of Khufu

Again, as in the case of the Red Pyramid, some assumptions have to be made with reference to the measurements and the number of the core steps that are partly impossible to corroborate by archaeological evidence as yet. However, the following assumptions can be made :

My starting point are my own observations and studies. In the tomb-robbers’ tunnel the structure of the core masonry changes abruptly in layer seven (lower edge at 7.06 m over foundation level), about 15 m behind the entrance. Considering the horizontal measures of the outer wall of the first core step (1 m) and of the casing which is not preserved (approx. 3 m), and also the inclination of the outer wall of the core steps (80°), there results a base length of the first step of the core of approx. 197 m (375 cubits). (64)
– The height of the steps is likely to be 11 m and the number of steps twelve. (65)
– The width of the steps accordingly is 5.75 m, if we take into account the alignment of 54°30’ along the steps’ edges on the lateral surface.
– The unit frequency differs the pyramid of Menkaure, because the ramps are significantly longer, thus approx. 12 min.
– The volume of the bed rock core in the base of the pyramid is assumed to amount to 7.9% by Haase (66), and 7.7% in more recent studies, i. e. 200 000 m³. (67) Goyon supposes a volume of 127 000 to 160 000 m³. (68) I shall use the volume of the rock core given by Haase.
– Further construction data as the number of working days per year and the size and average weight of the stone blocks and so forth are the same as for the pyramid of Menkaure. (69)

Fig.29 Pyramid of Khufu : Propos1 for Rampsystem

In order to estimate the building time of the pyramid of Khufu, not counting the preliminary preparations, we again assume that a sufficient workforce was available on three hundred days of the year, ten hours a day, in shift work, so that at a duration of each transport unit of 12 minutes a total of 15 000 rounds per year could be achieved. Accordingly, for the core of the pyramid of Khufu, the time needed for transport of blocks amounts to 13.4 years, or just 12.8 years if we subtract the volume of the bedrock core. The building time for the working platform, backing and casing amounts to a total of 6.9 years. Finally, the ramps and the platforms need to be removed, and the blocks left in boss require dressing, which all in all takes 2.8 years. The time spent on dressing amounts to approx. 286 work days, or one year, to which we may add another year as a time reserve for any necessary coordination procedures.
If we assume that this must be added to the removal of the working platform and so forth, the
actual building time of the pyramid of Khufu results in 22.5 years, during which an average amount of 393 m³, or 327 blocks per day were moved.
I should like to emphasize again that these observations are based on rough estimates for the block sizes and weights, and to be precise, they ought to be modified by adding some more detailed explanations, for example as to how the large limestone blocks over the entrance on
the North side of the pyramid, or the granite blocks of various shapes (Grand Gallery, burial chambers, stress-relieving chambers, plugging blocks) were moved. It may well be the case that in the lower area at pyramids’ East some tangential ramps with a less steep inclination were erected in order to convey the larger blocks with a weight of up to 60 tons, which were delivered along the causeway.
On the other hand, it is quite possible that the causeway was continued at an even inclination (10 : 1) towards the base of the pyramid or into the actual building site, reaching a height of 46 to 53 m (fig.30) above base level, at which height the ceiling slabs could be conveyed further upwards. These considerations, however, must be continued elsewhere.

Starting from his own observations on the topographical situation at the SW corner, Lehner shows that the ramp approaching the pyramid from the Southern quarry, would have met the pyramid at about 30 m above base level. (70) Also, we should not go without mentioning a solution for heavy load transports during the building of the pyramid of Khufu which Haase has published recently. (71)

Summary of results

 Several solutions that have so far been offered for issues of the construction of the Ancient Egyptian pyramids by Stadelmann, Arnold, Lehner, Goyon, G., Klemm and Klemm, Lattermann and Houdin, are based on the assumption that perpendicular or integrated ramps were used. Other explanations involve steep ramps running parallel to the core steps (Landt, Graefe). Still other authors suggest the use of winches, for example Riedl, Pitlik, and Keyssner. However, most of the authors assume that the core was built layer-wise ; moreover, many do not offer solutions as to how the pyramidion was set, and how safety was ensured when the casing was applied and dressed. Building time calculations are also only offered occasionally.
The observations which I have presented here and elsewhere, lead to the conclusion that difficulties encountered in building the Bent Pyramid prompted the architects of the Red Pyramid and later pyramids to revise their construction methods in order to control the threat to the pyramid through ground erosion and earth quakes. At the same time, it was crucial to keep the building time as short as possible to meet the king’s request for swift completion. As a result, a stepped core was erected, to offer more stability on the one hand, and on the other hand to enable ramps to be positioned on all four sides of the pyramid parallel to the core steps, in order to allow for material transports in a shorter time frame. Also, larger stone blocks were used. Subsequently, during the building of the pyramid of Khufu, the same methods were employed, and the size of the blocks was increased still further, in order to finish the vast building as soon as possible.
My hypothesis for the building of the Step Pyramids of the Old Kingdom involves combined use of ramps and redirectional rollers, which are both archeologically attested. Compared to previous suggestions involving transport of material by towing crews, the ramps were positioned in line with the steps of the core and to the working platform and were thus built with a steeper slope. As we have seen in the case of the pyramid of Menkaure, there is no longer a need for perpendicular ramps, or integrated ramps with less slope. Under these circumstances, we may dispense with any more complex solutions for the constructions. The use of several ramps with a steeper gradient ratio (26.5°, 2:1) on the core steps and the platform, on each side of the pyramid, leads to a steady and frequent influx of material, thus emerging as the best time-saving device that has been proposed so far.
Following the construction of the core and the removal of the respective ramps, backing and casing were applied layer-wise, after stepped working platforms with ramps had been installed around the pyramid, all of which were removed as the dressing of the casing proceded top down. This same platform also offers a convenient solution for works at the pyramid’s top including the pyramidion.
The calculations of the transport effort and the resulting building time are based on the least favorable circumstances in terms of block size and weight, the ramp surface’s friction, and transport unit duration. If any of these conditions are found to be more favorable, either the number of ramps could be reduced, or the building could be finished sooner. I have occasionally mentioned alternative solutions to some details, such as the ramp surfaces, but these do not have a significant effect on the overall construction method.
A calculation for the building times of the pyramids of Menkaure, Seneferu (Red Pyramid), and Khufu would realistically render the following results :

Pyramid of Menkaure :
- planning and preparation : 1 year
- building time, including dressing : approx. 4.8 years (72)
- total : 5.8 years.
Red Pyramid :
- planning and preparation : 2 years
- building time, including dressing : approx. 18.7 years
- total : 20.7 years.
Pyramid of Khufu :
- planning and preparation : 2 years
- building time, including dressing : approx. 22.5 years
- total 24.5 years.

These building times fit well within the accepted dating of the reign of Seneferu (thirty-five
years), (73) Khufu (twenty-three years), (74) and Menkaure (twenty-eight, (75) or six, (76) years).

The present article offers a solution that has been developed from archaeological evidence. It is centred around the idea of ‘pyramid building with ramps and redirection rollers’. The solution involves steep ramps positioned in line with the steps of the pyramid’s core, equipped with rollers in the form of redirection rollers for transport, and superseded by a working platform attached to the pyramid’s sides to enable safe setting of the pyramidion and dressing of the casing.

(62) Specific weight of limestone 2.4 tons per m³: Arnold, D. 1997. “Kalkstein” in Lexikon der ägyptischen Baukunst. Düsseldorf : Albatros, 119.
(63) Perring has measured the height of twenty-one steps of backing blocks starting from the bottom step (Perring, J.F. 1839. Pyramids vol. 1. London: Fraser). In total, their height is 21.4 m, which leads to an average step height of 1.02 m. The width is said to be 0.9 m. Maragioglio and Rinaldi suggest the heights of the stones are 0.5 to 0.7 m and their depth 0.9 to 1.2 m (Maragioglio, V. and Rinaldi, C.A. 1964. L’Architettura delle Pyramidi Menfite vol. 3. Torino : Officine Grafiche Canessa, 126, and also Addenda, pl. 19 fig. 6). Stadelmann writes that the height of the corner blocks of the backing, up to a point at two thirds of the height of the pyramid, was 1 to 1.3 m (Stadelmann, R. 1982. “Die Pyramiden des Snofru in Dahschur. Erster Bericht über die Ausgrabungen an der nördlichen Steinpyramide“. In MDAIK 38, 380-381). Lepsius describes the blocks of the core masonry to be altogether ‘… perhaps not quite as large as in Giza …’ (Lepsius, R. 1897. LD vol. 1: Texts. Berlin, 206).
(64) This is an assumption based on calculation, which does not include the streaks of solid rock that reach the backing. The same is true for the suggested ramp arrangement on the first step of the core.
(65) Graefe proposes eleven steps at a height of 13 m each for the core steps, at a width of 7.35 m (Graefe, E. Über die Determinanten des Pyramidenbaus bzw. Wie haben die Alten Ägypter die Pyramiden erbaut ? at http://www.uni-muenster.de/IAEK/org/WMA/graefe/pyr/index.html). Thus, the line of the steps’ edges on the lateral surface also reaches 54.5.
(66) Haase, M. 1993. “Der Felskern der Cheopspyramide.” In Zeitschrift für Archäologie und archäologische Grenzwissenschaften 1/1993, 5-13.
(67) Haase, M. 2004. Eine Stätte für die Ewigkeit. Der Pyramidenkomplex des Cheops. Mainz: von Zabern, 17.
(68) Goyon, G. 1990. Die Cheopspyramide. Augsburg, 117.
(69) In a number of publications, an average volume of 1m³ is assumed for each stone block, from which an hypothetical volume of the pyramid of 2.3 x 106 m³ results. The solid rock core, cavities, and the spaces filled with mortar and rubble are neglected.
(70) Lehner, M. 1985. “The Development of the Gisa Necropolis : The Khufu Project.” In MDAIK 1985, 109-143.
(71) Haase, M. 2007. “Eine Rampe für Schwertransporte beim Bau der Cheops-Pyramide,” In Sokar 15 (2), 48-49 and Haase, M. 2010. “Die megalithische Mauer westlich der Cheopspyramide,” In Sokar 20 (1), 22 sqq and Haase, M. 2010. Eine Transportrampe am Gisa-Plateau ?, In Sokar 21, 22 sqq.
(72) The dressing of the casing is not fully finished.
(73) According to new results, thirty-three years (Krauss, R. and D.A. Warburton, 2006. Ancient Egyptian Chronology. Leiden: Brill, 490). The building of the Red Pyramid was begun in year fifteen of the reign of Seneferu.
(74) According to new results, twenty-six years (Krauss, R. and D.A. Warburton, 2006. Ancient Egyptian Chronology. Leiden: Brill, 491).