THIS SIDE UP, ROLLING STONES, & BUILDING THE GREAT PYRAMID
A New Hypothesis
Roumen V. Mladjov, S.E. and Ian S. R. Mladjov, B.A.
Today, 4600 years after the Pyramids at Giza were completed, the mystery of how they were constructed remains unresolved. Although several hypotheses have been advanced, the Sphinx has successfully kept the secret throughout the millennia. That the pharaohs of the Old and Middle Kingdoms built the pyramids as tombs and memorials to themselves and their own divinity need not be proven. For more than forty centuries the height of the Great Pyramid of Khufu (Cheops) was the benchmark for tall building achievements. But among all the papyri, the inscriptions and reliefs on temple and tomb walls revealing the flow of Ancient Egyptian life and culture, there is no contemporary record of how the pyramids were built. The present study attempts to answer some of these questions in a more viable manner than the theories hitherto advanced. To obtain a more accurate perception of how the pyramids were built, we concentrated our efforts on exploring certain insufficiently explained factors in the existing theories on the construction of the Great Pyramid. Scrutinizing these hypotheses mathematically, we have found them to be untenable in their present forms. Utilizing the results of our study, we have advanced a new theory that would satisfy the required conditions.
The earliest pyramids, built for the 3rd Dynasty king Djoser and his immediate successors were made of stone blocks and are known as “step” pyramids due to their characteristic shape. At the beginning of the 4th Dynasty the reign of Snofru witnessed the fillingin of the pyramid’s “steps”, producing the “true” pyramid. Snofru’s son Khufu and grandson Khafre (Chephren) built the tallest and most enduring pyramids. During the succeeding reigns pyramid building continued on a smaller scale, and eventually with mudbricks rather than the much more durable stone. Over ninety in number, the pyramids constitute the chief monuments of the Egyptian rulers reigning circa 2630— 1630 B.C.
Sixteen of the extant pyramids are with a base exceeding 100 × 100 m (328 × 328 ft). The Great Pyramid is part of the 4th Dynasty necropolis at Giza, across the Nile from Cairo (Figures 1 and 2). The date of its construction is circa 2550—2530 B.C. and it is a “true pyramid” with “smooth” wall surfaces. The general dimensions of the Great Pyramid are:

side of the square base 230.36 m (755'9")

original height 146.7 m (481'4")

present height 137 m (449'6")

height = perimeter ÷ 2 π (4 × 230.36 ÷ 2 π) = 146.7

slope of walls 51º52'

volume 2, 594, 914 m^{3} (3, 394, 020 yards^{3})

weight 6, 550, 000 tons (14, 440, 000, 000 lbs)

average weight of blocks 2.85 tons (6280 lbs)

estimated number of stones 2, 300, 000
The stone structure is built with high accuracy. The difference between the length of the base sides and their average length is 109 mm (4 5/16"), which is less than 1/2000. The maximum deviation from the 90 degree angle between two adjacent sides is only 03'. Two different types of stone were used as material for the pyramid: local yellowish limestone for the core and fine white Turah limestone for the surface casing with estimated thickness of three meters.
The pyramid has some specific properties that make it different from most other solids used in construction. In particular, we should note the sharp decrease of the horizontal section from the base to the vertex. Thus the volume of the Great Pyramid significantly changes with the increase of the height. For the first 10 m (32'10") of height, the corresponding volume is 19% of the total. At 100 m (328') the volume reaches 96.8% of the total and for the remaining 46.7 m (153'2") the volume is only 3.2%. The center of gravity is located at 30.25 m (99'3") from the base, i.e. 50% of the pyramid volume corresponds to 1/5 of its height (Figure 3). This figure illustrates the volume distribution, a square function of the side of the horizontal sections, and the amounts of mechanical work.
Considering the characteristics of the pyramid from a construction point of view, the first zone of layers was considerably easier to build. As the amount of work (horizontal and vertical transportation) increases with the increase of the height, the accessible front decreases sharply. Figure 3 also represents the distribution of the accomplished work versus the whole work, measured in tons/meters. Only 11% of the mechanical work correspond to the first 50% of the volume. 89% of the work was accomplished by the height of 100 m (328'), amounting to 96.8% of the volume. The remaining 11% of the work account for the 3% of the volume located above 100 m (328').
In spite of the many studies, investigations, and hypotheses advanced, no satisfactory explanation of the building methods has appeared. Herodotus is the first to inform us about this vast construction project. According to Book Two of his Histories, the construction took thirty years, of which ten were spent in building the base and the ramp road to the pyramid and the remaining twenty years for building the pyramid itself.^{1} Herodotus’ figure is in obvious need for correction, as Khufu’s reign is agreed to have lasted only twentythree years.^{2} If Herodotus’ testimony is accurate to any degree, the decade spent in building the ramp must have coincided with the first decade of actual pyramid construction. The resulting twenty years would nicely fit within Khufu’s reign.
The modern hypotheses are based on two major theories: the stone blocks being hauled by groups of workers along huge ramps, or the blocks being consecutively raised up with the help of special complicated devices. Various scholars differ in their understanding how long, how steep and how many the ramps were, and in what arrangement.^{3} Thus, Dora J. Hamblin envisions a construction site where “broad delivery ramps led up from adjacent quarries to the growing pyramid, and 20 to 50man teams hauled at ropes strung over their shoulders to move 2.5ton blocks of local limestone slowly upward”.^{4} We cite D. J. Hamblin because her description corresponds well the most popular beliefs among the Egyptologists for the way the construction occurred four millennia ago.
The original assumption that the pyramids were built by slave labor has been rejected and it seems that the workers were recruited from among the peasantry. This new understanding is partly based on the quarry marks inscriptions found on some of the blocks as “The Craftsmen Gang,” “The Victorious Gang,” or “The Enduring Gang,” these names implying that the builders were proud of their work.^{5} Herodotus’ description of Egyptians being mobilized for the construction of the pyramids for three months at a time supports the same conclusion.^{6}
The “dragging” theory is based on artifacts showing scenes of fifty to hundreds of men hauling with ropes enormous stone sculptures (basreliefs in the Mastaba of Ti, 5th Dynasty, in the 12th Dynasty tomb of Djehutihotep, and of the Assyrians and Babylonians circa 700 to 600 B.C.), and also on the very general understanding that at the time of the pyramids, the Egyptians had not yet discovered the wheel.^{7} Additional support is the testimony by Diodorus of Sicily that the stones had been dragged over temporary ramps.^{8} A second theory, based on Herodotus’ description of machines used to raise the stones from step to step, has been developed by some authors, but as of yet no viable solution to that end has been proposed.
Apparently the theory of hauling/dragging the blocks does not entirely satisfy the scholars and they continue to search for something more acceptable. JeanPhilippe Lauer provides an elaborate review of the available theories on pyramid construction.^{9} The author concludes that none of the proposed theories provides a convincing answer to all the questions posed. He himself proposes an enormous frontal ramp growing continuously with the pyramid. Based on his calculations the ramp would have been 415 m (1362') long, 157 m (515') wide at the base and 146 m (479') in height by the end of construction.^{10} Such a ramp would almost rival the Pyramid itself in scale, and would have been an extremely expensive and timeconsuming undertaking.^{11}
Most scholars think that the Egyptians were using encircling ramps with dimensions that were similar to the dimensions of the pyramids themselves. To suppose a purely temporary construction on such a vast scale seems rather extravagant, even when talking about the Ancient Near East. Possibly the construction of the main mass (the core) of the pyramid went ahead of the exterior casing blocks with several layers, allowing for more modest ramps supported by the stone blocks already in place. The casing blocks were moved near their final locations with the same transportation system as for the main mass blocks and were then installed at their intended positions with lifting devices.
There is one major weakness in the hauling (dragging) theory. With all the precise calculations on how a single block can be lifted in position, few have noticed the discrepancy between the average speed of moving the blocks required to build the Great Pyramid upward and the presumed period of some twenty years. As Edwin Murphy states in a note to his translation of Diodorus’ description of the building of the pyramids, the Great Pyramid could only have been built at the impossible rate of some three hundred blocks per day, taking up all twentythree years of Khufu’s reign.^{12}
Our criteria for calculating the transportation characteristics at the site are as follows:

construction time  20 years, amending Herodotus’ statements and based on the accepted 23year reign of Khufu

working days per year  270 days (9 months x 30 days), excluding three months for the harvest and religious holidays, interruptions for extending the ramps, and assuming there were no reposing days

total number of stone blocks  2, 300, 000

duration of working day  10 hours

net working time  4 hours, excluding the time for breaks, lunch, returning to the starting position, etc. (The efficiency of labor at very heavy conditions was relatively low)

dragging team  25 to 35 men

average length of one team zone  60 m (197'), (net length of 25 to 30 m plus the intergroup distance of minimum of 30 to 35 m)

average number of dragging “lanes”  4, based on two maximum lanes per ramp, accounting the reduced number of ramps with the raising of the pyramid, three or four at the beginning, reduced to two and finally to one.
We focus on the transportation of the blocks from the base of the plateau to their installation place because this is the critical path for completing the project. We assume that the other important parts of the building process, such as the quarrying the blocks and the final masonry work of arranging the blocks together, also very labor and time consuming procedures, were less critical as the Egyptians had the opportunity to employ their large work force to its full potential and without the space constraints of the ramps.
The average daily number of stone blocks (n) transported from the base of the pyramid to the place of installation would be:
n = 2, 300, 000 blocks / 20 years x 9 months x 30 days = 426 blocks/day^{13}
The average speed (V) of the dragging team upward would be:
V = 426 blocks x 60 m / 4 hours x 4 lanes = 1596 m, say 1600 m/hour (5249')
The dragging teams could not have been able to obtain such a speed even for the transportation of a single block. To consider this required speed as the average speed for all the teams, for ten hours per day, every single day of the month is obviously impossible. We should not forget that we are trying to establish a reasonable average speed for the entire construction period of some twenty years, rather than the rate of a singular record effort. Apparently there is something fundamentally incorrect in the theories involving the hauling of the blocks. There must have been a different solution.
There could be little doubt that the officials responsible for the construction of Khufu’s pyramid realized the vastness of the work effort required for the completion of the project. We have to assume that ether the vizier Hemiunu, in charge of the construction, already had the answer to the problem available from previous experience, or that he had to find a viable, time and effortsaving solution.
Hemiunu must have chosen the location of the pyramid carefully, in a way that would be most favorable for a speedier and easier construction. Khufu’s pyramid apparently failed to exploit the higher location on which Khafre’s pyramid stands and which allows it to appear taller than the monument of his father.^{14} Hemiunu and his architects sought to minimize the work, incorporating an existing higher part of the plateau into the foundations of the pyramid.^{15} The size of this outcrop is unknown, but it may well have amounted to some 8% of the total volume, assuming 5 m (16') average thickness of the incorporated part.^{16} An additional opportunity must have presented itself when choosing the elevation of the ground level. Instead of leaving the existing level of the plateau, it was possible to lower the whole exterior base with some six to eight meters (20' to 26') by excavating the near vicinity of the foundations. Thus the existing surface could have been raised with another six to eight meters (an additional 12% of the volume), moreover providing a significant amount of the local stone blocks from these excavations. In this fashion some 20% of the volume of the structure could have been achieved, making the actual construction a relatively easier task. As suggested above, the ramps could have been largely incorporated in the future body of the pyramid in order to save a considerable amount of temporary work, rather than being built entirely outside the body of the pyramid (Figure 4).
Nevertheless the builders were still left with some 2, 100, 000 m^{3}, or 1,850, 000 stone blocks to move up to their places, or, using the same simple calculations above, an average speed of 1460 m/hour (4790 ft/ hr) is required. This task would have been impossible if the technology used is the one proposed by modern scholars’ dragging method for the millions of blocks. What other option do we have? Among the quarry marks and worker gang inscriptions found on some blocks, appears the intriguing phrase “this side up.”^{17} Why would someone bother to spend time and effort to inscribe the phrase if the 1.5 to 2.5ton (3300 to 5500 lbs) blocks were simply dragged upward? It is extremely unlikely that such heavy blocks would be rotated upside down. The only way to make sense of the curious inscription is to assume that the blocks were rotated, indeed rolled upward, rather than dragged.
Before rejecting this solution on the grounds that the Egyptians at this time were unfamiliar with the wheel proper, let us consider the possibilities. We are not suggesting that the Egyptians used carts to transport the blocks upward. To load the heavy blocks onto a cart and then to unload them would be by itself a very strenuous and timeconsuming task, especially considering the number of blocks involved. What the Egyptians needed were not the cart and the wheel, but the principle of rolling. A simple assembly of removable wooden parts can create a solid large type of wheel around the block (Figure 5). A pair of these devices, located near the edges of the block can convert it into a relatively easy to roll axis.^{18} These removable wooden parts may have been the prototype for the mysterious “rockers” known from New Kingdom models that scholars have long assumed to have formed a part of presumed lever devices supposedly employed in the movement of heavy stone blocks in pyramid construction.^{19} The technical advantages of such an assembly can be demonstrated with the following example.
For a stone block with weight of 3 tons (6.61 kips) and dimensions 0.75 × 0.75 × 2 m (2'6" × 2'6" × 6'6") at a slope of 1:10, the force necessary to drag the block is 12 to 15 kN (2.65 to 3.30 kips) at a friction coefficient of 0.3 to 0.4 (Figure 6). In the case of rolling, a force of only 3 to 3.5 kN (0.66 to 0.77 kips) would be sufficient. This method makes it possible to achieve the required average speed of 1460 m/hour (4790 ft/hr). This would also permit the team to be reduced to a more reasonable number of twelve to fifteen people. The smaller size of the team would then increase the number of teams that could simultaneously operate atop the limited ramp space.
Rolling the blocks with special devices, similar to solid wheels, gives us a viable answer to the questions of how so many blocks were transported upward in such a limited timespan and why the phrases “this side up” were inscribed. Herodotus’ testimony for the stones being raised up to their places by means of “machines formed of short wooden planks” may thus be accurate, though his conjecture of what these machines were and how they operated appears to be wrong.^{20} The assumption that the Egyptians were acquainted with the principle of the wheel remains to be proven.
A royal tomb discovered by Leonard Woolley at Ur has yielded the remains of heavy carts and battle chariots with solid wooden wheels. The funeral belongs to the period immediately before the First Dynasty of Ur, circa 2600 B.C. This makes the finds slightly older contemporaries of the Great Pyramid. Another royal tomb at Ur yielded a mosaicdecorated box, now known as the “Standard of Ur,” portraying a procession of war chariots with four solid wooden wheels made up of two joined semicircular components (Figure 7).^{21} This evidence makes it very probable that the solid wheel was already widely used in Sumer, and the Egyptians had ample chance to borrow, if not the wheel itself, then at least the principle of rolling. The cultural exchange between Egypt and Mesopotamia was considerable even at this very early period. Although the ziggurats of Sumer had a very different purpose than the Egyptian pyramids, it is likely that they were influenced by the “step” pyramid.^{22}
On a basrelief in the tomb of the 5^{th} Dynasty official Kaemheset in Saqqara there is a depiction of a siege ladder on wheels.^{23} The tomb dates only a century later than the Great Pyramid, and it appears to testify to the existence of the wheel in Egypt at that period. At any rate the hypothesis for rolling the stones up the ramps is based on a significantly simpler device than the wheel itself.^{24} The development of the wheel naturally went from a simpler form to a more complicated one, namely the proper wheel, such as the elegant chariot wheels ubiquitous in the New Kingdom reliefs of Sethos I, Ramesses II, and Ramesses III (circa 1291—1151 B.C.).
The elements of the rolling device, a prototype of the wheel, would be similar to the discovered models of “rockers,” representing a couple of thick wooden boards with curved bottom edges, similar to the supports used for rocking chairs, interconnected with round wooden bars (Figure 6). The real purpose of these “rockers” has never been discovered. A theory called the “oscillator elevator” has been advanced supposing the pyramids were built using the “rockers” for raising the blocks. We think that it is more probable that devices similar to the “rockers” were parts of the rolling device. Four (or two semicircular) interconnected rockers near the edges of both sides of the block would form, in effect, two solid wheels around the block (Figure 8). The only sensible explanation of the “this side up” inscriptions is the resulting “rolling” theory. Other theories fail to explain the necessity for this sort of inscription.
The most compelling argument for the belief that the Egyptians were not acquainted with the wheel at the time is the information from the depictions of colossal statues transported by being dragged on top of sleds, as described above. Our new hypothesis for rolling the stones upward applies to the prevailing number of regularlyshaped stone blocks (with a weight of some twothree tons). This method of transportation is of course not applicable for very large blocks of the type used for topping the King’s and the Queen’s chambers, the pyramidion capstone of the pyramid, and other irregular shapes such as statues.^{25} Such components must have been moved into place using the dragging method, but they were relatively few in number.
To accomplish the enormous amount of work, the Egyptians must have chosen the optimal size for the typical block carefully in order to obtain the minimum total quantity of work for quarrying, transporting and arranging the blocks. A simple calculation shows that the amount of quarrying work decreases significantly with the increase of the block dimensions. For a block of 1 × 1 × 1 m the ratio between the surface to be cut and the volume is 6 (6 m^{2} per 1 m^{3}); for 2 x 2 x 2 m block the ratio is 3 and for 3 × 3 × 3 m block the ratio is 2, i.e. the quarry of a 27 m^{3} single block requires three times less work than the work for 27 single 1 m^{3} blocks. Once the Egyptians were able to haul heavy blocks they should have utilized the largest possible size to make the total work effort more efficient. Hauling heavier loads will require more people in the team, but the same number per ton. The amount of total transportation work is not related to the weight of the typical block. Having smaller numbers of heavier blocks would permit the increase of the transportation speed and the reduction of the duration of construction. However, instead of choosing 8, 10, or 20 m^{3} for the typical block, the ancient builders used blocks with an average volume slightly above 1 m^{3}. This required millions of square meters to be cut additionally. What was their reason for choosing that option? The Egyptians must have possessed a different technology allowing them to transport 1.5 m^{3} blocks far more efficiently than, say 10 m^{3} blocks. The most likely explanation is that this method was to “roll” the relatively small blocks upward. These blocks were as large as they could still be rolled, and large enough to minimize labor at quarrying. The limitations of block size and weight might be caused by the restricted strength capacity of the rolling device connections and the lower bearing pressure of the ramp material. Here we may add that the process of quarrying blocks, judging by blocks that were not completely cut away and remain in the quarries, is entirely compatible with our theory. Before being detached, a block was essentially completely shaped except for a soontobe discarded stump that carried it. At this point it would have been very easy to affix the wooden parts around the formed edges of the block, allowing it to be rolled once completely quarried.
This new hypothesis is based on three issues left unexplained by the existing theories for the construction of the Great Pyramid. These issues are the incredible speed of transportation necessary to complete the pyramid within the time framing, the intriguing inscription “this side up” found on some of the blocks, and the typical block size chosen by the builders. The concept developed above provides a viable answer to these questions. The Great Pyramid was most probably built using rolling devices for the transportation of the bulk of the blocks up the ramps. The rolling devices were removable and reusable parts of solid wheels fitted around the blocks, transforming each block into an axis. Once the block reached its intended place, the device was disassembled and brought down to be reused. The “dragging” method was utilized for the few blocks weighing over 5 or 6 tons, or with irregular shapes. It is also possible that some lifting devices were used to move and adjust part of the cladding exterior blocks, even from one level to another, but such methods could account for only a minor portion of the construction work. We feel that the theory advanced here explains better than any other the mysteries of the construction of the Great Pyramid.
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