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Keywords:

  • Egypt;
  • Old Kingdom;
  • mathematics;
  • physics;
  • pyramids;
  • ramps;
  • construction;
  • workforce;
  • time

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

Archaeological aspects of the various building methods are reviewed, paying special attention to the construction method described by Müller-Römer (2011) and the reason behind the stepped shape of Menkaure's pyramid. The main purpose is to present an alternative description of the building process according to this method, based on the physical principles of force and power. It is shown that this approach leads to results that are generally applicable and permits adjustment of the transportation capacity of the proposed system by adjusting the slope of the ramps.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

On the basis of an extensive review of the various building methods proposed in the literature, Müller-Römer (2011), with special reference to the stepped shape of the core of Menkaure's pyramid, concludes that none of these are fully acceptable from an archaeological point of view. As an alternative, he proposes a construction by means of ramps built on the steps of the core, parallel to its sides, as shown in Figure 1. This implies that the construction must be done in two phases. After completion of the core, the ramps have to be demolished and, during the second phase, the backing stones and the casing put in place by means of working platforms and newly built ramps. Finally, smoothing of the casing is done from the top down, while the ramps and the working platforms are being removed.

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Figure 1. The zig-zag ramp concept (from Müller-Römer 2011, 371; with permission from the author).

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Müller-Römer determines his volumes geometrically, rather than by deriving a general mathematical equation, and thus of necessity anew for each phase of the construction and each pyramid considered. Furthermore, his estimates of the project duration are based on postulated values of the time needed to lift a block one step higher, rather than on the principle of power.

The purpose of this paper is twofold: first, to reconsider briefly the pros and cons of the various building methods and, second, to offer an alternative calculation method that is more generally applicable and will provide additional information, such as relationships between the volumes and the height, and between the duration of the project and the number of workers required. The paper does not pretend to offer precise answers but, rather, to establish verifiable relationships. Two different descriptions will be used: a discrete method with the correct number of ramps at each step; and a continuous method which is more generally applicable, but which must be adjusted to ensure that it matches the total time of the discrete method.

The paper only describes the actual building process. For a more complete picture, other activities, such as quarrying and transporting the building material from the quarry to the building site, would also have to be taken into account, as well as the need to balance supply and demand of the building material (De Haan 2009, 2010).

Details of the calculations are presented in the appendices. In Appendix 1, various relationships are derived for the transportation of a single block, that are valid both for the discrete and the continuous calculation method, namely between:

  • the ramp volume and its dimensions;
  • the number of hauliers required per block and the maximum force per man;
  • the time to lift a block one step higher and the maximum power per man; and
  • the number of hauliers and the additional number of workers needed to relieve them regularly.

On the basis of the results of Appendix 1, equations are derived in Appendices 2 and 3 for the volumes of the core and the ramps, for the building rate and time, as well as for the effort (the number of man-hours) and the number of workers required. In Appendix 2, this is done for the discrete case and in Appendix 3 for the continuous case. Appendix 4 describes a procedure for averaging the distances that the blocks have to cover on the horizontal top surface of the truncated pyramid. The meaning of the symbols used is explained in Appendix 5. Basic data used are listed in Table 1 and the results are shown in Tables 2 and 3 and Figures 2-9. The outer part of the pyramid, constructed during the second phase after completion of the core, will be denoted as the ‘shell’.

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Figure 2. Results versus time in years (bold, continuous method; thin lines, discrete method). Khufu's pyramid: volume and effort, tan α = 0.5.

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Figure 3. Results versus time in years (bold, continuous method; thin lines, discrete method). Khufu's pyramid: number of workers, tan α = 0.5.

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Figure 4. Results versus time in years (bold, continuous method; thin lines, discrete method). Khufu's pyramid: volume and effort, tan α = 0.331.

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Figure 5. Results versus time in years (bold, continuous method; thin lines, discrete method). Khufu's pyramid: number of workers, tan α = 0.331.

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Figure 6. Results versus time in years (bold, continuous method; thin lines, discrete method). Menkaure's pyramid: volume and effort, tan α = 0.5.

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Figure 7. Results versus time in years (bold, continuous method; thin lines, discrete method). Menkaure's pyramid: number of workers, tan α = 0.5.

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Figure 8. Results versus time in years (bold, continuous method; thin lines, discrete method). Menkaure's pyramid: volume and effort, tan α = 0.3.

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Figure 9. Results versus time in years (bold, continuous method; thin lines, discrete method). Menkaure's pyramid: number of workers, tan α = 0.3.

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Table 1. Basic data
  1. a

    1 N (newton) = 0.102 kg force.

  2. b

    100 000 J h−1 = 27.78 W.

 Common constants   
Gravitational acceleration, g 9.81m s−2
Density, ρ 2500kg m−3
Maximum force per man, Fm 200Na
Maximum power per man, Pm 200000J per man-hourb
Weight per man, G 600N
Friction factor, μ 0.25 
Block size, V1 1.2m3
Time schedule, Fy 3000hours per year
Averaging factor, horizontal distance, λ 0.271 
Ramp dimensions:   
Length of horizontal sections of ramp, d 5m
Length of horizontal section of ramp, c 1.5m
Slope, tan β 1 
 Constants depending on size   
 KhufuMenkaure 
Height of pyramid, H146.565.8m
Base of pyramid230104m
Step height, Δh118.4m
Table 2. Main results for Khufu's pyramid
Volume of core, Vc, equation (3.1.3)1 895 173 m3
Volume of shell840 807 m3
Volume of pyramid2 583 283 m3
Slope of the ramp, tan α0.50.331 
Length of ramp, a, equation (1.1.1)44.555.7m
Ramp factor of core, Frc0.8750.815 
Ramp factor of shell, Frs0.7500.750 
Maximum height of core113105m
Maximum height of shell119112m
Maximum fraction of core completed98.897.7%
Maximum fraction of shell completed99.398.7%
Volume of single ramp, Vr, equation (1.1.2)1 3601 715m3
Number of ramps228160 
Volume of ramps as a fraction of the pyramid volume12.010.6%
Number of hauliers per block, Na, on staircase DE, equation (1.2.3)3226men
Number of auxiliary workers per block, Nb, on ramp BC, equation (1.3.5)94men
Number of hauliers per block on top surface, Nh, equation (1.3.7)3131men
Distance between hauliers on staircase DE, with three rows, equation (1.3.1)1.51.8m
Distance between auxiliary workers with one row, equation (1.3.3)2.910.0m
Step time for zero delay, Δτ, equation (1.2.9)5.37.4min
Total time for zero delay, equation (3.2.3)6.611.7years
Total time for 5 min delay, equation (3.2.3)12.819.7years
Total effort for maximum height, zm, equation (3.3.6)19 64816 938man-years
Total effort for completed pyramid, z = 1, equation (3.3.7) 17 746man-years
Average workforce corresponding with the maximum height1 535860men
Average workforce corresponding with the completed pyramid 901men
Table 3. Main results for Menkaure's pyramid
Volume of core, Vc, equation (3.1.3)140 267 m3
Volume of shell139 587 m3
Volume of pyramid237 051 m3
Slope of the ramp, tan α0.50.3 
Length of ramp, a, equation (1.1.1)36.747.9m
Ramp factor of core, Frc0.7650.600 
Ramp factor of shell, Frs0.6250.730 
Maximum height of core35.626.4m
Maximum height of shell44.437.9m
Maximum fraction of core completed90.378.5%
Maximum fraction of shell completed96.692.4%
Volume of single ramp, Vr, equation (1.1.2)650857m3
Number of ramps6444 
Volume ramps as a fraction of the pyramid volume17.615.9%
Number of hauliers per block, Na, on staircase DE, equation (1.2.3)3225men
Number of auxiliary workers per block, Nb, on ramp BC, equation (1.3.5)93men
Number of hauliers per block on top surface, Nh, equation (1.3.7)3131men
Distance between hauliers on staircase DE, with three rows, equation (1.3.1)1.11.4m
Distance between auxiliary workers with one row, equation (1.3.3)2.210.5m
Step time for zero delay, Δτ, equation (1.2.9)4.26.4min
Total time for zero delay, equation (3.2.3)1.01.9years
Total time for 5 min delay, equation (3.2.3)2.33.3years
Total effort for maximum height zm, equation (3.3.6)945443man-years
Total effort for completed pyramid, z = 1, equation (3.3.7) 937man-years
Average workforce corresponding with the maximum height411134men
Average workforce corresponding with the completed pyramid 284men

Archaeological Background

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

A variety of building methods is described by Müller-Römer (2011, 261–348). In a similar, but less detailed, review by De Haan (2010, 59–78), the emphasis is more on the quantitative aspects of these methods. These methods can be roughly subdivided in three groups: (1) methods without ramps; (2) straight or linear ramps; and (3) ramps built parallel to the sides of the pyramid (‘parallel’ ramps).

Methods without ramps

These methods have the obvious advantage that only the material for the construction of the pyramid has to be lifted. Naturally, without a ramp, the building material must be lifted somehow along the sides of the pyramid. A method proposed and proved experimentally is levering (Hodges 1989). He proposed that the building blocks are lifted along the side walls of the pyramid by means of staircases. In other concepts, the blocks are lifted by means of cranes mounted on the steps of the pyramid (Croon 1925) or pulled up along a lubricated wooden plank, assisted by levering (Isler 2001, 259–61). Illig and Löhner (1994) suggest the use of a pulley fixed at the top to reverse the direction of the pull, so that the hauliers can benefit from their own weight. Riedl's concept (1980) relies on the use of winches—which, however, were not known to the Egyptians prior to the 12th Dynasty (Arnold 1991, 71).

Linear ramps

There is ample evidence that linear ramps were used for construction purposes in ancient Egypt. Various ramp concepts are reviewed by Arnold (1991, 79–101). The oldest construction ramps, made of pebbles, are found at Sakkara and belong to Sekhemkhet's complex. There are four ramps around the pyramid of Sinki, 12 m long, with a gradient of 21–26%, which seems acceptable for the transportation of the small blocks used. The height would have been 6 m, if finished. Presumably the later use of brick (rather than rubble) ramps is related to the use of large building stones. Two large ramps at Meidum were heightened several times. There are also short ramps to the roofs of mastabas, made of bricks and pebbles that were needed for the construction of the mastaba and for the installation of a sarcophagus. A ramp is also pictured in the tomb of Rekhmire. Remains of a ramp can still be observed at the temple of Karnak.

Probings suggest that the whole south edge of the Giza plateau is a gigantic rubbish infilling, consisting of ramp debris and remains of building sites and additional work camps (Lehner 1997, 132; Kemp 2006, 191). At Giza, there is also a ramp with stone walls near Khufu's pyramid, connecting the quarries west of the Sphinx to the plateau. Nevertheless, as pointed out by Arnold (1991, 98), there is no indication as to which type of ramp may have been used for the construction of Khufu's pyramid. That is, of course, not surprising, as the ramps would have been demolished after completion of the pyramid. Linear ramps have been proposed by several authors (Borchardt 1928; Arnold 1991, 98; Lehner 1997, 129–32; Stadelmann 2003, 128). In some cases, multiple ramps were proposed, or a combination of a linear ramp with a parallel ramp wrapping round the pyramid. The linear ramp is illustrated in Figure 10, but there is no reason why the ramp must be oriented perpendicular to the pyramid.

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Figure 10. The linear ramp.

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Müller-Römer (2011, 238–9) discards the possibility of a linear ramp on the grounds that its construction would take a disproportional amount of effort, that it would lead to interruptions in transportation, and that it would allow insufficient transportation capacity and that therefore four ramps would be required to achieve the necessary high initial building rate. Moreover, he points out that the linear ramp does not do justice to the stepped shape of the core of Menkaure's pyramid. By contrast, De Haan (2010, 228, 37 and 44–6) showed that a single 82 m high ramp would require a relatively small fraction of the total effort in man-years, but would suffice for the completion of more than 90% of Khufu's pyramid. The top 10% of the pyramid would have to be completed by means of another method; for instance, levering or the zig-zag ramp system proposed by Müller-Römer. This combination would provide sufficient transportation capacity for completion of the pyramid in 20 years. The ramp would only cover less than 20% of the pyramid surface, so that the ramp would not seriously hamper its alignment. Admittedly, the proponents of the linear ramp are left with the question of why the core of Menkaure's pyramid was apparently constructed with steps that are 4–5 m wide. On the other hand, one cannot perhaps exclude the possibility that the steps of the core structure were filled in concurrently with the construction of the pyramid by means of a linear ramp.

In this connection, one specific aspect of Menkaure's pyramid should not be overlooked, namely that in contrast to the other pyramids, a large part—that is, the first 16 courses—of its casing consists of granite. This corresponds with a height of 16 m according to Müller-Römer (2011, 192, fig. 5.1.2.6.4). For a thickness dc of 1 m and a slope γ of the pyramid, this corresponds to a volume of

  • display math

In addition, the limestone for the pyramid had to be quarried, with a volume of

  • display math

Based on experiments reported in the literature, De Haan (2010, 13) estimated a quarrying efficiency for granite of 0.00024 m3 per man-hour = 2624 × 0.00024 = 0.63 m3 per man-year, and for limestone 0.03 m3 per man-hour = 2624 × 0.03 = 78.7 m3 per man-year. Thus it would take 7433/0.63 = 11 800 man-years to produce the granite and 234 000/78.7 = 3014 man-years to produce the limestone. This compares with a mere 937 man-years to build the actual pyramid (Table 3). In other words, the effort to quarry the material for the casing far exceeded the combined effort required to build the pyramid and provide the necessary limestone. Thus the supply of the casing material was in fact a bottleneck in the building process, and one can well imagine that Menkaure wanted to have the bulk of his pyramid—the stepped core—completed first and the casing installed later. This problem did not occur for the other pyramids and thus did not interfere with the simultaneous construction of the core and the ‘shell’. In fact, Arnold (1991, 160, 161) believes that, in general, the 4th Dynasty pyramids, such as those of Khufu and Khafre, consist of horizontal layers, presumably constructed in a single phase. It would seem that the presence of a granite, rather than a limestone casing, also casts some doubt on Müller-Römer's supposition that the casing was smoothed at the end, from the top down, bearing in mind how difficult it is to work granite. It seems more logical that the blocks were finished on the ground, so as to ensure a more or less smooth surface, sloping at the correct angle (Arnold 1991, 169).

Parallel ramps

These ramps are supposed to be constructed parallel to the sides of the pyramid. We distinguish two types of parallel ramps: (1) ascending ramps that wrap around the pyramid in the way shown in Figure 11, denoted as spiral ramps; and (2) ramps constructed on all four sides on the (horizontal) steps of the pyramid, as shown in Figure 1 and thus creating a zig-zag path to the top, which we shall denote as ‘zig-zag’ ramps.

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Figure 11. The spiral ramp.

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Spiral ramps

Figure 11 shows how the slope of the side walls would become steeper with increasing height, and eventually vertical for a ramp with a moderate top width of 10 m and a slope of 10%. As pointed out by Hodges (1989, 12–14), this entails a serious stability problem. Other disadvantages are that the ramp obscures the faces of the pyramid, thus making its alignment next to impossible, and that it would be difficult to drag the sledges around the sharp corners of the ramp. De Haan (2010, 46) has drawn attention to the limited transportation capacity of the spiral ramp. To summarize, the spiral ramp does not seem to be suitable as a construction method.

Zig-zag ramps

This concept was introduced by Hölscher (1912) and also considered by Graefe (2003). Both Hölscher and Graefe consider ramps occupying the entire width of the pyramid. In Graefe's case the ramp is a staircase, with transportation by levering. This method calls for a two-phase construction, but requires a small total ramp volume compared with the linear ramp. Having dismissed the linear ramp, Müller-Römer (2011, 259–60) concludes that this is the only building concept that finds support in the stepped shape of the core of Menkaure's pyramid. His concept, depicted in Figure 1, differs from that of the previous authors by the use of smaller ramps, several of which will fit on a step of the core.

Conclusion

To summarize, it seems that only the spiral ramp and the method making use of winches must be dismissed. The separate construction of the stepped core of Menkaure's pyramid indeed suggests the use of a zig-zag ramp system, but might also find an explanation in the fact that the supply of the granite for the casing was lagging behind the construction of the actual pyramid. Thus other building methods cannot be excluded. This applies in particular for the linear ramp, in view of the fact that remains of this type of ramp can still be observed in various places.

Transportation of a Single Block

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

The transportation method for a single block, as proposed by Müller-Römer, is shown schematically in Figure 12. The building block, mounted on a sledge, is supposed to be hauled up the ramp from B to D by hauliers moving down a staircase from D to E. The circles near C and D represent some type of fixed (non-rotating) pulleys, which serve to change the direction of the rope. The dimensions shown, together with the width of the ramp, determine its volume according to equation (1.1.2). All steps have the same height Δh and width b, and comprise several layers of building blocks. The ramps on the topmost step reached are assumed to be increased gradually in height, together with the building, until the step is completed. We shall assume that this aspect can be ignored, as all the lower ramps are completed and thus have a constant height.

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Figure 12. The transportation of a single block.

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The number of hauliers per block follows from the condition that their combined force must be sufficient to overcome the counteracting force F1 exerted by the block. This force consists of two terms: the component of the block weight along the ramp surface, W sin α, and the friction term, which equals the normal force (perpendicular to the ramp surface) times the friction factor: μW cos α. This friction factor depends on the properties of the two surfaces that are in contact. Based on various literature data (Hodgman 1961, 2222; Cotterell and Kamminga 1990, 28, 221), a value of 0.25 appears to be realistic.

In Figure 12, the circle on slope DE represents a single haulier, who exerts his maximum force Fm and adds his own weight component G sin β. As this is a staircase, friction can be ignored. To put the block in motion, these forces, multiplied by the total number of hauliers Na, must balance the above force F1. This force equilibrium is expressed by equation (1.2.2) and enables us to find the minimum number of hauliers according to equation (1.2.3).

There is a limit, not only to the force that a man can exert, but also to the power that he can exert, in other words, the effort he can put in per hour or per day. Since power equals force × speed, we can derive the speed of the block by dividing the power exerted by the hauliers by the force F1 exerted on the block (1.2.4). Equations (1.2.5) and (1.2.6) then give the vertical component of this velocity and the velocity on the horizontal sections of the ramp.

To determine the time it takes to lift a block from one step to the next, denoted as the step time, Δτ, we divide these velocities into the respective distances covered on the ramp (1.2.9). The time determined in this way is an ideal value. Actually, additional time is needed; for instance, to load and unload and manoeuvre the sledges, return the empty sledges and install the fixed pulleys. We therefore have to introduce a delay time, Δτd, which increases the step time to Δτc = Δτ + Δτd. By dividing the step time into the step height, we find an effective vertical displacement speed vveff, which can be used directly to determine the time taken to lift a block to a certain height.

After lifting a block to the top of the truncated core, it must still be moved horizontally to its final position. The force, the number of men per block and the block speed are then found by setting both α and β to zero in the relevant equations, resulting in equations (1.2.7) and (1.2.8).

To ensure a constant block speed, the number of hauliers must be constant. This implies that for every haulier arriving at the bottom of ramp DE, a colleague must be ready to replace him at the top. We shall assume that in order to keep this ‘Jacob's ladder’ in motion, those leaving the staircase climb up slope BC, as indicated in Figure 12. In the ‘Traffic on the ramps’ section of Appendix 1, relationships are derived for the number and the speed of these auxiliary workers and the distance between them.

Volumes and the Maximum Number of Ramps

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

In the ‘Dimensions’ section of Appendix 2, the total volumes of the core and the ramps corresponding with a certain height of the core have been derived by adding the volumes for layers of a finite thickness Δh according to equations (2.1.5) and (2.1.6). Summing these volumes over all the layers completed then gives the total volume (2.1.7). In Appendix 3, these layers are assumed to be infinitely thin. This enables us to describe the volume of the core as a continuous function of the height according to equation (3.1.3), so that the total volume is obtained simply by substituting the height in this equation.

For the discrete method, the number of ramps is obtained by determining how many ramps of length a will fit on the steps, a number that will of course decrease with increasing height. With the continuous calculation procedure, the width of the core varies gradually, rather than abruptly. Consequently, by dividing the width by the length of the ramp, the result obtained is not an integer and therefore will lead to an overestimation of the number of ramps that will fit on the step. To compensate for this, we must introduce a correction factor or ‘ramp factor’, Fr, in equation (3.1.5). This factor, which is different for the core and the shell, is adjusted so as to match the total time for the discrete method. The total number of ramps is then found by determining how many times the rectangular vertical cross-sectional area a Δh, effectively occupied by a ramp, will fit into the vertical cross-sectional area of the core, as illustrated by Figure 13. This ratio must be multiplied by four to take into account the fact that ramps are built on all four sides of the core, and corrected with the ramp factor. Through multiplying this by the volume of a single ramp, Va, according to (1.1.2), we find the total ramp volume (3.1.6).

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Figure 13. The determination of the total ramp volume.

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The maximum height that can be reached using this method is determined by the length a of the ramps and is given by equation (2.1.4). For reasons discussed below, two different slopes of ramp BC have been considered: tan α = 0.5, as used by Müller-Römer, and reduced slopes—tan α = 0.331 for Khufu's pyramid and tan α = 0.3 for Menkaure's pyramid—that lead to larger ramps and, as a result, reduce their number and increase the total time. A larger ramp also reduces the maximum height that can be reached. The corresponding heights for the two pyramids are given in Tables 2 and 3, both for the core and for the shell, together with the corresponding fractions completed at these heights. It appears that for Khufu's pyramid, 98% or more can be completed; whereas for Menkaure's pyramid, in all cases but one, more than 90% can be completed. Also shown are the sizes of the ramps for the various cases considered.

Figures 2, 4, 6 and 8 show that by matching the total time for the continuous method to that for the discrete method, excellent agreement is obtained between the volumes determined using the two methods, at least for a minimum number of steps.

The Building Rate

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

The number of blocks that can be lifted simultaneously from one step to the next equals the number of ramps on that step. Thus the maximum transportation capacity at a certain level equals the number of ramps at that level times the block volume, divided by the corrected step time, Δτc, as expressed by equation (2.2.1). However, as there can be no accumulation on the pyramid, the number of blocks lifted from a given step cannot exceed the lifting capacity of the next step. As the number of ramps and thus the lifting capacity decreases with increasing height due to the decreasing width of the core, this implies that, effectively, the building rate equals the transportation capacity of the topmost step, as this step contains the fewest ramps. This means that equation (2.2.1) represents the building rate only if the step number i refers to the topmost level. Hence at a lower level, with a larger number of ramps, parts of the ramps are necessarily inactive. Eventually, only four active ramps are left on every step, one on each face of the core. In other words, the building rate is proportional to the top width of the truncated pyramid and thus decreases linearly from the initial value to a minimum value corresponding with the maximum height q(zm), obtained by substituting z = zm in equation (3.2.1).

Time

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

By dividing the volume of a slice by the prevailing building rate, we find the time needed to install it, represented by equation (2.2.2). Summation or, for the continuous case, integration then results in the total time corresponding to a certain height, given by equations (2.2.3) and (3.2.2). Substitution of the maximum height zm = hm/H in equation (3.2.2) then gives the total time according to equation (3.2.3).

As shown in Tables 2 and 3, for the case of tan α = 0.5 and zero delay time, step times of 5.3 and 4.2 minutes are obtained, corresponding to total times of 6.6 and 1.0 years, respectively, for the two pyramids. The addition of an arbitrary delay time of 5 minutes, as assumed by Müller-Römer, still leads to unrealistic total times of no more than 12.8 years, rather than about 20 years for Khufu's pyramid and 2.3 years for Menkaure's pyramid. As indicated in Table 1, these results are based on a conservative value of 200 kJ per man-hour = 55.6 W per man. The reason why Müller-Römer arrives at larger total times is that implicitly he uses a much lower power per man. For Khufu's pyramid, he assumes (Müller-Römer 2011, 364, 398–9) that a block weighing 3000 kg is lifted 11 m by 40 men in 7 minutes, exclusive of any delays. This corresponds to a power per man of

  • display math

As pointed out by De Haan (2010, 17–21), there is ample evidence that a normal healthy man can produce 70–100 W. Apparently, with a maximum number of ramps installed, the zig-zag system has an overcapacity. Of course, reducing the power per man is not an option, as that would imply a (further) underutilization of the available workforce. Increasing the step time by increasing the friction factor does not help either. Quadrupling this factor from 0.25 to 1.0 increases the total time for Khufu's pyramid by no more than a year. The only two ways to arrive at a realistic total time is to increase the delay time Δτd and/or decrease the number of ramps. According to equations (1.1.1), (1.2.9), (1.3.5) and (3.2.3), a convenient and effective way to reduce the number of ramps and at the same time increase the step time Δτ is to reduce the slope α of ramp BC and thereby increase its length. For the reduced slopes, the ramp factors must be adjusted again to ensure that the total times still match those corresponding to the discrete method. The results of Tables 2 and 3 show that in this way, realistic total times can be achieved.

Naturally, one can hardly expect the ancient builders to have adjusted their building schedule in such a systematic way. It is more likely that it was a process of trial and error, with the number and the slope of the ramps being adjusted as the project advanced.

Effort

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

The effort is defined as the number of man-hours or man-years required. Thus the effort required to lift a block to a certain level equals the number of men per block times the height, divided by the effective vertical speed, as expressed by equations (2.3.1) and (3.3.1). The second term in these equations represents the effort needed to move the blocks along the horizontal top surface to their final positions in the pyramid. The ratio f of the two terms is defined by equation (2.3.2). This must be multiplied by the number of blocks ΔVic/V1, to obtain the effort to install the entire slice according to equation (2.3.3). Similarly, the effort for the construction of the ramps on this step is given by equation (2.3.4). The addition of equations (2.3.3) and (2.3.4) and summation over all the steps then gives the total effort needed to construct both the core and the ramps according to equation (2.3.6). For the continuous description, this result is obtained by integration of equations (3.3.2) and (3.3.4), which leads to equation (3.3.6). The factor f is quite small, 0.02 at a maximum. In fact, the effort for horizontal transportation is significant only at the beginning, when relatively large distances along the horizontal surface have to be covered.

According to equation (3.3.3), the effort is proportional to both the height and the volume. This is the reason why the total effort for the construction of Khufu's pyramid is more than 20 times that for Menkaure's pyramid for tan α = 0.5, and even as much as 40 times for the reduced slope of ramp BC. The reason why this factor is so much higher in the latter case is that the maximum height, and thus also the maximum volume, is more strongly reduced for Menkaure's pyramid (Table 3). The effort and time needed to smooth the casing and to demolish the ramps have been neglected. Also indicated in Tables 2 and 3 is the effort corresponding to the completed core, according to equation (3.3.7). The top part, which exceeds the maximum height that can be achieved with the zig-zag method, would have to be completed by using a different building method; for instance, levering. The effort for the various cases has been plotted against time in Figures 2, 4, 6 and 8, which show again good agreement between the discrete and the continuous case.

The Workforce

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

Once we know both the effort and thus the number of man-hours and the time required for a certain height, the corresponding number of workers—the workforce—is found as the incremental effort according to equations (2.3.5) and (3.3.8) divided by the incremental time according to equations (2.2.2) and (3.3.9), respectively. This results in equations (2.3.7) and (3.3.10). We have neglected the relatively small number of workers who may have assisted with the transportation by means of levers and lubrication. The results are plotted in Figures 3, 5, 7 and 9, which show a good agreement between the continuous and the discrete approaches for Khufu's pyramid, especially in Figure 3, but somewhat less in Figures 5, 7 and 9, in that order, due to the decreasing number of steps. In fact, these curves are the time derivatives of the curves representing the effort in Figures 2, 4, 6 and 8. By dividing the total effort by the total time, we find an average (corrected) workforce of 860 and 134 men for the two pyramids. The reason for the low number for Menkaure's pyramid is that the total effort of 443 man-years does not include the effort devoted to the labour-intensive construction of the top 21.5%, which takes another 937 minus; 443 = 494 man-years. Completing the pyramid in 3.3 years thus would actually require 937/3.3 = 284 men. Of course, to these numbers we have to add those involved in other activities, such as quarrying.

In equation (3.3.10), the second term, which represents the workers on the horizontal surface, is small due to the factor f. This means that, except at the very beginning (z << 1), the workforce is mainly determined by the number of workers on the ramps, and thus by the number of steps and the number of ramps on the topmost step according to equations (2.3.8) and (3.3.11). The number of workers is small at the beginning due to the small number of ramps, and because relatively few workers suffice to move the blocks along the horizontal surface. It decreases towards the end because the number of active ramps decreases as a result of the decreasing top width of the truncated pyramid. This is the reason why in Figures 3, 5, 7 and 9 the workforce shows a maximum. In reality, the strong variation of the workforce both during the two construction phases and due to the two phases may be alleviated to some extent; for instance, by exchanging builders with quarrymen. By determining the ratio (3.3.12) of the two terms in equation (3.3.10), we find that initially all of the workers are active on the horizontal surface and towards the end the majority are active on the ramps.

Also shown in Tables 2 and 3 are the numbers of hauliers per block. According to equation (1.3.7), reducing the slope of ramp BC reduces the number of auxiliary workers, Nb, far more than that of the hauliers Na on ramp DE. As, moreover, BC is longer than DE, the distance between them is larger, in spite of the fact that there are three rows on DE and only one on BC.

Our equations also enable us to evaluate the consequences of using other building methods. As an illustration, consider the use of winches, rather than hauliers, to lift the blocks, as suggested by Müller-Römer in his earlier publications (2008a,b). For the same ramp dimensions and the same maximum power per man, it follows from equations (3.2.3) and (1.2.9) that for a zero delay time, the total time is inversely proportional to the number of workers actively involved in the transportation of a block. Hence eight winch operators, rather than our 32 hauliers per block, would increase the step time to 4 × 5.3 = 21.2 minutes, and the total time without delay to 4 × 6.6 = 26.4 years, for Khufu's pyramid. A 5 minute delay per step would further increase this to (26.2/21.2) × 26.4 = 32.6 years. In other words, here we have the opposite problem: this method is too slow rather than too fast. In this case, two remedies offer themselves: either more men at the winches or a larger power per man.

Main Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

It appears that the supply of the granite for the casing must have been the bottleneck in the building process for Menkaure's pyramid. Thus the stepped shape of the core is not necessarily related to the zig-zag building method, but may as well be due to the need to complete the core first, rather than wait for the supply of the granite.

Although not the only possible building method, the zig-zag ramp system proves to be an effective method. For realistic values of the power per man and with a maximum number of ramps installed, the system even appears to have an overcapacity, resulting in unrealistically short total building times. It has been shown that using our calculation method, this problem can be solved conveniently by reducing the slope of the ramps. This also reduces the number of men per block and increases the distance between them on the ramps.

The total effort in man-years is vastly different for the two pyramids, because the effort is proportional both to the height and the volume. The reason why the curves representing the number of workers show a maximum is that at the beginning the available space is small due to the small number of ramps, and that towards the end the number of active ramps decreases as a result of the decreasing top width of the truncated pyramid. It has been shown that initially all the workers are active on the horizontal surface and towards the end the majority are active on the ramps.

The maximum height that can be reached and thus the maximum volume that can be built using this method is determined by the length of the ramps. It appears that for Khufu's pyramid, 98% or more of the core or the shell can be completed; whereas for Menkaure's pyramid, in all cases but one more than 90% can be completed.

In contrast to the effort, the time and the building rate, the workforce is practically independent of the power per man and the delay time. The reason is that the workforce is almost exclusively determined by the number of ramps.

To summarize, it has been shown that an analysis based on the physical principles of power and force enables us to obtain a wealth of additional information from the available data, as is evident from Tables 2 and 3 and Figures 2-9. It appears that the building process can be adequately described by a continuous calculation procedure, provided that this is properly calibrated by means of a discrete calculation. It has been shown that an acceptable fit between these two calculation methods can be obtained by equating the total times.

The advantage of the continuous description is that it shows explicitly how the time, the building rate and the workforce depend on the dimensions of the core and the ramps, and on the power per man, and thus how and why certain results are obtained. Without this method, it would have been impossible to adjust the transportation capacity in a consistent way by reducing the slope of the ramp. The most obvious advantage of this approach is that the equations derived are generally applicable. All one has to do to obtain similar results for another pyramid or another building method is to substitute the relevant data. In other words, there is no need to start the calculation all over again.

Although we shall never know in any detail how the ancient Egyptians went about building their pyramids, we are at least able to show that they must have been able to create even the Great Pyramid without having recourse to a gigantic workforce for the actual building. In other words, it must have been mainly a matter of organization, rather than of numbers.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation
  • Arnold, D., 1991, Building in Egypt, Pharaonic stone masonry, Oxford University Press, Oxford.
  • Borchardt, L., 1928, Die Entstehung der Pyramide an der Baugeschichte der Pyramide bei Meidum nachgewiesen, Springer-Verlag, Berlin.
  • Cotterell, B., and Kamminga, J., 1990, Mechanics of pre-industrial technology, Cambridge University Press, Cambridge, UK.
  • Croon, L., 1925, Lastentransport beim Bau der Pyramiden, Doctoral thesis, Technische Hochschule Hannover.
  • De Haan, H. J., 2009, Building the Great Pyramid by levering—a mathematical model, Palarch's Journal of Archaeology of Egypt, 6(2), 122; available at http//www.palarch.nl/category/egypt/
  • De Haan, H. J., 2010, The large Egyptian pyramids—modelling a major engineering project, BAR International Series 2057, Archaeopress, Oxford.
  • Graefe, E., 2003, Über die Determinanten des Pyramidenbaus, bzw Wie habe die Alten Ägypter die Pyramiden erbaut? In Alter Orient und Altes Testament (ed. G. J. Selz ), 274, 113152, Münster.
  • Hodges, P., 1989, How the pyramids were built, ed. J. Keable, Aris & Phillips, Warminster, UK.
  • Hodgman, C. D. (ed.), 1961, Handbook of chemistry and physics, 43rd edn, Chemical Rubber Publishing Company, Cleveland, OH.
  • Hölscher, U., 1912, Das Grabdenkmal des Königs Chephren, Veröffentlichungen der Ernst von Sieglin Expedition in Ägypten 1, Leipzig.
  • Illig, H., and Löhner, P., 1994, Der Bau der Cheops Pyramide, Mantis Verlag, Gräfelfing.
  • Isler, M., 2001, Sticks, stones and shadows, University of Oklahoma Press, Norman, OK.
  • Kemp, B. J., 2006, Ancient Egypt—anatomy of a civilization, Routledge, London.
  • Lehner, M., 1997, The complete pyramids, Thames and Hudson, London.
  • Müller-Römer, F., 2008a, A new consideration of the construction methods of the ancient Egyptian pyramids, Journal of the Egyptian Research Centre in Egypt, 44, 113140.
  • Müller-Römer, F., 2008b, Die Technik des Pyramidenbaus im Alten Ägypten, Münchener Studien zur alten Welt, München.
  • Müller-Römer, F., 2011, Der Bau der Pyramiden im Alten Ägypten, Herbert Utz Verlag, München.
  • Riedl, O. M., 1980, Der Pyramidebau und seine Transportprobleme, Wien.
  • Stadelmann, R., 2003, The pyramids of the Fourth Dynasty, in The treasures of the pyramids (ed. Z. Hawass ), 2831, Edizioni White Star, Vercelli, Italy.

Appendix 1: Transportation of a Single Block

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

The equations in this appendix apply both to the discrete and to the continuous case. For notation, see Appendix 5.

Dimensions

According to Figure 12, the length of the completed ramp, including the horizontal sections c and d at both ends, is as follows:

  • display math(1.1.1)

With a width b, the ramp volume thus becomes

  • display math(1.1.2)

Number of hauliers and block speed

With a block weight W = ρgV1, the minimum force needed to move a block of volume V1 and density ρ along a ramp sloping at an angle α with a friction factor μ is given by:

  • display math(1.2.1)

The blocks are supposed to be hauled up the ramp from B to C with slope α by Na hauliers moving down the other side of the ramp with slope β, who use their own weight G as counterweight (Fig. 12). The corresponding normal component G cos α does not play a role, because on the staircase the hauliers are not supposed to experience friction. Their combined force must be sufficient to set the block in motion:

  • display math(1.2.2)

We shall assume that the hauliers exert their maximum force Fm. Their minimum number is therefore as follows:

  • display math(1.2.3)

The maximum block speed equals the maximum power divided by the force exerted, which equals F1. If Pm is the maximum power per man, the maximum speed is as follows:

  • display math(1.2.4)

The vertical component of the velocity va is as follows:

  • display math(1.2.5)

The velocity along the horizontal sections of the ramps, with α = 0 and with the same number of hauliers, is according to equation (1.2.4):

  • display math(1.2.6)

For transportation along the horizontal top surface, the force, the number of hauliers per block and the speed are found by setting α = β = 0 in equations (1.2.1), (1.2.3) and (1.2.4). This results in

  • display math(1.2.7)

and

  • display math(1.2.8)

The step time equals the sum of the time needed for the displacement along the horizontal sections and that along the inclined section of the ramps. According to Figure 12:

  • display math

Combination with equations (1.2.5) and (1.2.6) then leads to the following:

  • display math(1.2.9)

To take into account delays due to loading and unloading, putting in place the fixed pulleys, manoeuvring the sledges and returning the empty sledges, we introduce a delay time Δτd and a corrected step time Δτc:

  • display math(1.2.10)

The time needed for the construction of the ramp will be taken into account in the building time, represented by equations (2.2.3) and (3.2.2), and the number of workers is adjusted accordingly in equations (2.3.7) and (3.3.9). The effective vertical speed is obtained by dividing the corrected step time into the step height Δh:

  • display math(1.2.11)

As we shall see, on average, the displacement along the horizontal top surface plays a minor role and for the sake of simplicity any delays will therefore be neglected.

Traffic on the ramps

To ensure a constant block speed, the number of hauliers must be constant. This implies that for every haulier arriving at the bottom, a colleague must be ready to replace him at the top. We shall assume that in order to keep this ‘Jacob's ladder’ of workers in motion, those leaving the ramp climb up ramp BC, assuming that there is enough space for them to walk from E to B in Figure 12. We shall neglect the time needed to cover this distance. To prevent congestion, the number of workers climbing up—denoted as auxiliary workers—per unit time must equal the number moving down the other slope per unit time. These numbers equal the speed of the workers divided by the distance between them, times the number of rows. If the Na hauliers are arranged in ra rows along a ramp of length La (DE in Fig. 12), the longitudinal distance between them is as follows:

  • display math(1.3.1)

If they move at a speed va, the downward ‘traffic’ is given by

  • display math(1.3.2)

In other words, the number of rows disappears from the equation. Similarly, the distance between the auxiliary workers on ramp BC is as follows:

  • display math(1.3.3)

The upward ‘traffic’ on this ramp is given by the following equation:

  • display math(1.3.4)

The upward traffic must be equal to the downward traffic, so that

  • display math(1.3.5)

As the climbers only have to carry their own weight, their (maximum) speed up the ramp is as follows:

  • display math(1.3.6)

The number of climbers now follows from equation (1.3.5), combined with equations (1.2.3), (1.2.4) and (1.3.6):

  • display math(1.3.7)

For the sake of simplicity, we shall ignore the additional workers who assist with the transport by means of lubrication and levering, both for the transport on the ramps and on the top surface of the truncated pyramid.

Appendix 2: Construction of the Entire Core—Discrete Description

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

Dimensions

We introduce the relative height:

  • display math(2.1.1)

Here, H is the total height of the core or shell and hi is the height halfway between the top and bottom of slice i, so that

  • display math(2.1.2)

As illustrated in Figure 13, because of the similarity between the core and its top part, the width of the core at height hi as a fraction of the width of the base Lc follows from:

  • display math(2.1.3)

Thus the maximum relative height zm is reached when the width of the core equals the length of the ramps: Li = a, so that according to equation (2.1.3):

  • display math(2.1.4)

According to equation (2.1.3), the volume of slice i is given by the following equation:

  • display math(2.1.5)

In this way, the slice volume is related to the step height and the overall dimensions of the core or shell. The combined volume of the ramps at a step that can accommodate ri ramps is as follows:

  • display math(2.1.6)

where Va follows from equation (1.1.2). The total volume of the core and the ramps for a number of layers i is then obtained by summation:

  • display math(2.1.7)

The topmost layer then corresponds to the maximum height according to equation (2.1.4).

Building rate and time

The building rate equals the number of blocks lifted from one step to the next per unit time times the block volume. If ri is the number of ramps that can be accommodated at level i, the maximum lifting capacity at that level becomes:

  • display math(2.2.1)

This would be equal to the rate if all ramps at that level were in operation. However, as there can be no accumulation of building material on the pyramid, the rate must be the same at all levels and thus equal to the rate at the level containing the minimum number of ramps; in other words, the topmost level. Consequently, only if we interpret i as the topmost level does equation (2.2.1) represent the actual building rate. Hence at a lower level k, with a larger number of ramps rk, rk − ri ramps are inactive. The value of Δτc follows from equation (1.2.10). As, according to equation (1.2.9), Δτ is proportional to the weight W and thus to the volume V1 of the block, the building rate—and, in fact, the whole building process—would be independent of the block size for Δτd = 0.

By dividing the combined volumes of a slice and the corresponding ramps at that level by the prevailing building rate, we find the time needed to install these volumes:

  • display math(2.2.2)

Summation over all the steps then gives the total time to reach a height hi:

  • display math(2.2.3)

Effort and workforce

The number of man-hours or effort required to install a block of volume V1 at height h1 is proportional to the number of men and the time:

  • display math(2.3.1)

The first term represents the effort needed to lift the block to level z, and the second term the effort to move the block along the top surface of the truncated pyramid, so as to place it in its final position. The ratio of these two terms then is given by the following factor:

  • display math(2.3.2)

The factor λ is introduced to take into account the fact that the blocks have to cover different distances along the top surface to their final locations (De Haan 2009, 16; De Haan 2010, 90). Its value is obtained by subdividing the top surface into, for example, 20 × 20 squares and determining the average distance to the centres of these squares, as illustrated in Figure  A1 and explained in Appendix 4. The value of vveff follows from equation (1.2.11). The first term in equation (2.3.1) corresponds to the vertical displacement, and the second term to the displacement along the top surface of the core. Thus the factor f represents the ratio of the average effort needed to move a block along the top surface to its final position at height hi to the effort needed to lift it to that height.

The effort needed to install the entire slice i is found by replacing V1 by the slice volume ΔVic:

  • display math(2.3.3)
figure

Figure A1. Averaging of the horizontal distances.

Download figure to PowerPoint

For the construction of the ramps, any horizontal displacement can be neglected, so that:

  • display math(2.3.4)

and hence the total effort to install the slice, including the ramps, becomes:

  • display math(2.3.5)

By summation over all the slices, we find the total effort for the construction of the core and the ramps:

  • display math(2.3.6)

We can now determine the number of workers needed to install a slice by dividing the incremental effort in man-hours according to equation (2.3.5) by the incremental number of hours required according to equation (2.2.2). Moreover, taking into account equations (1.2.11) and (2.2.1), we find, for the total workforce corresponding with height hi:

  • display math(2.3.7)

By setting f = 0, we eliminate the term corresponding to the displacement on the top surface and thus find the number of workers on the ramps. According to equation (2.1.2), this reduces equation (2.3.7) to the following:

  • display math(2.3.8)

As mentioned before, ri is the number of ramps at the topmost step. The number of workers on the ramps according to equation (2.3.8) thus corresponds to the situation in which the same number of workers, (Na + Nb)ri, are active on all steps except the topmost one. At that step, only half the number of workers is needed, because the slice under construction needs to be lifted on average only over half the step height. As the levels below the topmost one contain more than ri ramps, it follows that the additional ramps are necessarily inactive, as was pointed out above. Equation (2.3.8) also explains the irregular ups and downs of the graphs representing the workforce, as shown in Figures 3, 5, 7 and 9, for if the number of ramps decreases more steeply than the height increases, the workforce will decrease.

The equations for the shell are similar, except that the horizontal dimensions must be adapted. Where applicable, the base of the core must be replaced by

  • display math

where Ls is the base of the core plus that of the shell. The base area then is given by:

  • display math

Appendix 3: Construction of the Entire Core—Continuous Description

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

Dimensions

The thickness of the slices is now reduced until they are infinitely thin. This enables us to describe the various quantities as continuous functions of the height by using the mathematical tools known as differentiation and integration. The volume of a truncated core (or shell) of height h equals the difference between the completed core Vc1 with height H and the top part Vctop of similar shape and height H minus; h:

  • display math

We again introduce the relative height z:

  • display math(3.1.1)

Similar to equation (2.1.3), the width of the core at height h then follows from:

  • display math(3.1.2)

so that

  • display math

Consequently,

  • display math(3.1.3)

Differentiation of equation (3.1.3) gives:

  • display math(3.1.4)

The number of ramps at this level is determined by dividing L by the overall length a of the ramp and multiplying this ratio by 4, to take into account the fact that the ramps are constructed on all four sides of the core. As this number must be an integer, this leads to an overestimation of the number of ramps. To compensate for this, we must introduce a correction factor, or ‘ramp factor’, Fr, which may be different for the core and the shell. If r0 is the number of ramps for z = 0, we have, with equation (3.1.2):

  • display math(3.1.5)

The ramp factor is adjusted so as to match the total time for the discrete method. The number of ramps built on one side of the core equals the vertical cross-sectional area of the core for a given height, divided by the rectangular area a Δh effectively occupied by a ramp, as illustrated in Figure 13. The equation for the trapezium-shaped vertical cross-sectional area of the core as a fraction of the total area is similar to that for the volume, except that the areas are proportional to the square of the height, rather than to the third power:

  • display math

so that

  • display math

By dividing this by the effective cross-sectional area of the ramp, we find the number of ramps on one side of the pyramid. This result must therefore be multiplied by 4, and by the ramp volume according to equation (1.1.2) and corrected with the ramp factor Fr, to obtain the total ramp volume:

  • display math(3.1.6)

Here, Vr1 = FrbHLc (1 − c/a) represents the total ramp volume for z = 1.

Differentiation of equation (3.1.6) gives:

  • display math(3.1.7)

Building rate and time

Similar to equation (2.2.1) for the discrete case, the building rate is given by:

  • display math(3.2.1)

To determine the time taken to reach height h, we make use of equations (3.1.4), (3.1.7) and (3.2.1):

  • display math(3.2.2)

By means of the following equation, any relationships between the various quantities of interest and the height can be converted into relationships with time. The total time, corresponding to the maximum height, becomes:

  • display math(3.2.3)

Effort and workforce

Equation (2.3.2) for the factor f is also valid for the continuous case. Similar to equation (2.3.1), the effort needed to install one block of volume V1 is given by the following equation:

  • display math

The effort required per unit volume is then as follows:

  • display math(3.3.1)

By reducing V1 and ΔM1 to small increments dVc and dMc, this takes the form of a differential equation:

  • display math(3.3.2)

The total effort is found by integration in combination with equation (3.1.4):

  • display math(3.3.3)

For the construction of the ramp, the horizontal transport can be neglected, so that the term with the factor f in equation (3.3.2) disappears:

  • display math(3.3.4)

Substitution of equation (3.1.7) and integration yields:

  • display math(3.3.5)

Combination of equations (3.3.3) and (3.3.5) gives the total effort, both for the construction of the core and the ramps:

  • display math(3.3.6)

The effort corresponding to the maximum height is obtained by substituting z = zm in equation (3.3.6). By substituting z = 1, we find the effort corresponding with the completed core, provided that the various parameters still apply for zm < z < 1:

  • display math(3.3.7)

According to equation (3.3.3), the effort is proportional to both the height and the volume of the pyramid and thus strongly dependent on its dimensions. The second term in equations (3.3.3) and (3.3.5) corresponds to the transportation along the top surface of the core or the shell. Differentiation of equations (3.3.6) and (3.2.2) gives, respectively:

  • display math(3.3.8)

and

  • display math(3.3.9)

By combining equations (3.3.8) and (3.3.9), we find the number of workers at this height:

  • display math(3.3.10)

N(z) is seen to be independent of vveff and thus of the power per man and the retardation Δτd. The reason is that both the effort and the time are inversely proportional to vveff. By setting f = 0 in equation (3.3.10), we find the number of workers on the ramps:

  • display math(3.3.11)

As is to be expected, this number is proportional to the number of levels built, the number of ramps at the level reached and the number of men per ramp. The curve representing this equation is symmetrical with respect to z = 0.5. The number of those working on the top surface of the pyramid corresponds with the term containing f. The ratio then becomes:

  • display math(3.3.12)

By substituting z = 0 and z = zm in this equation, we confirm that initially all the workers are active on the horizontal surface. It follows that towards the end, the majority are working on the ramps. This is still true for the rather extreme case of Menkaure's pyramid, with tan α = 0.3, where according to Table 3, zm = 26.4/65.8 = 0.40. As f < 0.2, this leads to the following:

  • display math

Appendix 4: Calculation of the Average Horizontal Travelling Distance

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation

This distance can be determined numerically by subdividing the horizontal area into n × n square elements (Fig. 4). Let us assume that the blocks arrive on one side of the top surface of the pyramid at a distance x = mL/n from a corner; then the distance that a block must travel from that point to the centre of an arbitrary element i,j is given by:

  • display math(4.1)

The average distance is obtained by adding the distances corresponding to all the elements i,j and dividing by their number n2. Expressing the average distance as a fraction of the side L, we obtain:

  • display math(4.2)

In our case, the blocks are supposed to be delivered at the midpoints of all four sides. This means that the surface can be subdivided into eight identical rectangular triangles, each containing n2/8 elements, with the blocks arriving at the right angle: m = n/2. The diagonals connecting the corners of the top surface are now no-flow boundaries. To ensure that the elements on the diagonal are counted only once, we have to halve the corresponding distances, for which j = i:

  • display math(4.3)

Application of this equation for n = 20 results in λ = 0.271. Doubling n changes these numbers by less than 0.15%.

Actually, the blocks arrive simultaneously at different points along the sides of the top surface, rather than all at the midpoint. This means that our calculation method leads to an overestimation of the average distance covered.

Appendix 5: Notation

  1. Top of page
  2. Abstract
  3. Introduction
  4. Archaeological Background
  5. Transportation of a Single Block
  6. Volumes and the Maximum Number of Ramps
  7. The Building Rate
  8. Time
  9. Effort
  10. The Workforce
  11. Main Conclusions
  12. References
  13. Appendix 1: Transportation of a Single Block
  14. Appendix 2: Construction of the Entire Core—Discrete Description
  15. Appendix 3: Construction of the Entire Core—Continuous Description
  16. Appendix 4: Calculation of the Average Horizontal Travelling Distance
  17. Appendix 5: Notation
aLength of ramp (Fig. 12)m
AcHorizontal cross-sectional area of corem2
Ac1Base area of corem2
ActopCross-sectional area of top part of core not yet completedm2
bWidth of rampm
cLength defined in Figure 12m
dLength defined in Figure 12m
EEnergyJ (joule)
fRatio effort for horizontal to vertical displacement, defined by equation (2.3.2) 
FmMaximum force per manN (newton)
FrRamp factor, defined by equation (3.1.5) 
F1Force required to set the block in motionN
gAcceleration due to gravitym s−2
GWeight of workerN
hHeightm
HTotal height of core/shellm
LWidth of core/shell at height hm
LaDistance DE in Figure 12m
LbDistance BC in Figure 12m
MEffort in man-hours or man-years 
NTotal number of workers 
NaNumber of hauliers per block on the ramp 
NbNumber of auxiliary workers 
NhNumber of hauliers per block on the top surface 
PmMaximum power per manJ h−1a
qBuilding ratem3 h−1
q0Initial building ratem3 h−1
rNumber of ramps per step 
raNumber of rows of hauliers 
r0Initial number of ramps 
RTotal number of ramps on the core 
tTimeh, year
vaBlock speed along the rampm h−1
vhrBlock speed on horizontal sections of the rampm h−1
vhtBlock speed on the top surfacem h−1
vveffEffective vertical block speedm h−1
V1Block volumem3
VaVolume of single rampm3
VcTotal volume of corem3
VctopVolume of top part of core not yet completedm3
Vr1Total volume of ramps for z = 1m3
WBlock weightN
zRelative height = h/H 
zmMaximum relative height 
αAngle defined in Figure 12 
βAngle defined in Figure 12 
γSlope of side of pyramid 
λAveraging factor for displacement along the top surface 
μFriction factor 
θAngle defined in Figure 12 
ρRock densitykg m−3
ΔhStep heightm
ΔLaLongitudinal distance between hauliers on staircase DEm
ΔLbLongitudinal distance between auxiliary workers on ramp BCm
ΔzRelative step height = Δh/H 
ΔτStep timemin
ΔτdDelay timemin
ΔτcStep time corrected for the delay timemin

Subscripts

  1. a

    1 W (watt) = 1 J s–1 = 3600 J h–1; 1 J = 1 N m; 1 kg force = 9.81 N.

cCore
hHorizontal
iStep number
mMaximum
pPyramid
rRamp
sShell
tTotal
vVertical