pyramids
pyramids
- Corinna Rossi
Summary
Ancient Egyptian pyramids were funerary monuments. Besides the three world-famous pyramids at Giza, Egypt contains the remains of over eighty other large royal pyramids that were built during the Old and Middle Kingdoms, and of hundreds of smaller pyramids that adorned the New Kingdom tombs of private individuals; large groups of small royal pyramids were later built in Nubia, modern Sudan. Symbols of the connection between earth and sky, pyramids were built along the Nile for nearly three thousand years, displaying a range of shapes, dimensions, and construction techniques.
Our knowledge of these monuments is extensive yet uneven: a linear evolution of shape and layout appears to proceed alongside the periodic appearance of unique elements; the few extant mathematical sources from ancient Egypt provide information on how the slope of these monuments was measured and calculated, but not on how it was chosen; the precision of the orientation of the sides towards the four cardinal points indicates a stellar alignment, but the identification of the stars involved in the process is still doubtful; the archaeological evidence suggests that ramps where used in the construction, but their structure and shape can only be guessed. Therefore, the main challenge in the ongoing study of pyramids is that of combining various sources and reckoning with the simultaneous presence of recurring elements and unique circumstances.
Keywords
Subjects
- Science, Technology, and Medicine
- Near East
Ancient Egyptian pyramids are among the best-known and most important monuments built in Antiquity, and are ubiquitous and recognisable symbols of both ancient and modern Egypt. The iconic image of the three pyramids of Giza (Figure 1) has always attracted the greatest attention, but in fact Egypt is punctuated by the remains of over eighty other royal pyramids, including some as large as the Giza structures, as well as by hundreds of much smaller private pyramids.1
Figure 1. Classic view of the Giza pyramids (Old Kingdom, Fourth Dynasty, c. 2550–2470 bce).
Only some of them are well preserved: their state of preservation depends mainly on the materials and building techniques that were employed in their construction, which in turn were strictly related to the size of each monument. The element shared by all Egyptian pyramids, which were parts of funerary complexes that evolved over the centuries, is their symbolic function of connecting earth and sun, but there were a wide variety of ways in which this might be physically achieved.
Because of the complexity and variety of the available material evidence and textual sources, our understanding of ancient Egyptian pyramids is still rather uneven: although their chronological sequence and attribution, the evolution of the associated funerary cult, and the specific building techniques employed are generally understood, the reasons behind pyramids’ changes of shape and location, as well as some technical aspects of their construction, are still under discussion. The mathematics of pyramid building, for instance, is often either ignored or treated in an abstract way divorced from both the historical and the material context.
History and Distribution
Figure 3. The (partly reconstructed) pyramid of a New Kingdom tomb at Deir al-Medina (1550–1070 bce).
Figure 4. View of a group of the Nubian pyramids at Meroe, in modern Sudan (4th century bce to 4th century ce).
From the Third Dynasty of the Old Kingdom to the Thirteenth Dynasty of the Middle Kingdom (c. 2650–1650 bce), large pyramids were built by all Egyptian kings with sufficient wealth and time to do so.2 During these thousand years, pyramids were symbols of royal status as well as of power and wealth, and were only built for the burials of pharaohs and their most important queens. Each pyramid contained a funerary apartment and was accompanied by a small satellite pyramid and a pair of temples for the cult of the deceased: the funerary temple, attached to the pyramid, and the valley temple, located down at the foot of the plateau and connected to the other by a ceremonial ramp. These funerary complexes punctuate the plateau rising above the western edge of Egypt’s cultivated area from Abu Rawash in the north to al-Lisht in the south.
Set against the sunset, Old Kingdom pyramids literally shaped the landscape to the west of the ancient capital Memphis. On clear days, they could be seen from Heliopolis on the other side of the Nile (Figure 2), which was the seat of the solar cult centred on the so-called benben stone, an elongated fragment of meteoritic origin from which the pointed- and round-topped traditions of pyramids and stelae derived.3 During the Middle Kingdom, after the hiatus of the First Intermediate Period, the powerful Twelfth Dynasty kings resumed building pyramids for themselves and their queens. The centre of power was relocated slightly to the south, closer to the Fayum, where the same layout was replicated, with the royal funerary complexes rising along the edge of the plateau behind the new capital.
After the troubled Second Intermediate Period, the beginning of the New Kingdom (c. 1550 bce) saw a substantial shift in royal funerary arrangements. The rise of Thebes (modern Luxor) determined the expansion of the burial ground located on the west bank. There the three elements (pyramid, funerary apartment, and cult place) were physically separated: the temples were built at the edge of the cultivated area, whilst the funerary apartments were quarried into the sides of either the Valley of the Kings or the Valley of the Queens, tucked away under the imposing, pyramid-shaped Theban Mountain (Meret Seger-“she-who-loves-the-silence”). The pyramidal shape disappeared from the royal burials, and became instead a widespread element of private burials, albeit on a much smaller scale, during the next five centuries (Figure 3).
After a break of three hundred years from the last private pyramids (eight hundred from the last royal pyramids) a revival of pyramid building took place in Nubia (modern Sudan), where the kings of the Twenty-Fifth Dynasty had originated. During the next thousand years, a total of over 180 pyramids were built there; smaller than their earlier royal counterparts, but larger than the New Kingdom private monuments, they were the last pyramids to be built along the Nile (Figure 4).
Geometry
Figure 5. Schematic representation of a mastaba tomb.
Figure 6. Step pyramid and ancillary buildings of the funerary complex of Djoser at Saqqara (Old Kingdom, Third Dynasty, 2630–2611 bce).
Figure 7. Schematic vertical sections of the Bent Pyramid of Sneferu showing the second change of project (Old Kingdom, Fourth Dynasty, 2575–2551 bce).
Figure 8. The Bent Pyramid of Sneferu at Dahshur (Old Kingdom, Fourth Dynasty, 2575–2551 bce).
Figure 9. The Red Pyramid of Sneferu at Dahshur (Old Kingdom, Fourth Dynasty, 2575–2551 bce).
Figure 10. Schematic representation of the seked (measuring the slope of an inclined plane) of five palms (corresponding to c. 54°30′).
Figure 11. The Fifth Dynasty pyramids of Neferirkara and Niuserra at Abusir (Old Kingdom, 2465–2323 bce).
Figure 12. The (partly reconstructed) pyramid of the New Kingdom tomb of Tia and Tia at Saqqara (reign of Ramses II, 1290–1224 bce).
The long-lived smooth pyramidal shape of ancient Egyptian burials appeared slightly after, and probably was a re-elaboration of, an earlier type of monument, the so-called step pyramids. Consisting of a stepped sequence of progressively smaller volumes converging to the top, step pyramids in turn represent a re-elaboration of the earliest type of wealthy burial, which consisted of a subterranean portion capped by a parallelepipedal mass called a “mastaba” from the Arab name for the similarly-shaped bench located at the entrance of a traditional Egyptian house (Figure 5). The two major turning points in terms of shape, the transition from mastaba tomb to step pyramid and that from step pyramid to smooth pyramid, took place under the reigns, respectively, of Netjerikhet-Djoser (Third Dynasty, 2630–2611 bce) and Sneferu (Fourth Dynasty, 2575–2551 bce).
Mastaba tombs had been widely used by the earliest Egyptian kings and their most powerful officials; they were mainly built of mud bricks, whilst stone was reserved for some specific parts of the structure. Djoser built for himself a large mastaba at Saqqara, all of stone, and then made a groundbreaking decision: he had a sequence of progressively smaller mastaba-like structures built upon it, creating the first step pyramid. This large monument rose above a maze of subterranean passages and was surrounded by a series of ancillary structures (Figure 6).
From a technical point of view, three aspects of this funerary complex are especially interesting. First of all, it is the first Egyptian monument to be built entirely of stone, and as such it showed, for the first time, the innovative potential of this building material. Additionally, it retains (and thus reveals) many details of the older architecture that used lighter materials. The other buildings that line the funerary complex are representations in stone of pavilions that until then had been built with wood, reeds, and mats.4 In the space of a few generations, a new aesthetic of massive stone constructions took the stage: some older shapes originating from the use of lighter materials survived (the cavetto cornice, for instance) but the overall appearance of the buildings changed dramatically. The stone blocks that compose Djoser’s pyramid are relatively small, all similar in terms of shape and size, and are basically used as if they were mud bricks; a comparison with the monolithic pillars of the Sphinx Temple, built at Giza only a hundred years later, demonstrates the quick technical evolution that took place in such a short time.5
A later tradition identified Imhotep, a high-ranking officer under Djoser, as the architect behind the construction of the step pyramid of Saqqara. In ancient Egypt, however, the construction of major monuments was represented as the result of a collective effort captained by the king himself and entrusted to one of his most important officials. Generally, the names of the officials in charge of the construction of other royal funerary monuments were neither recorded nor celebrated, and the king took full credit for the enterprise.6
Of the subsequent kings who began constructing step pyramids for themselves, only Sneferu completed his project, at the southern site of Meidum. For an unknown reason, he then took the decision to begin construction of a smooth pyramid in the flat desert of Dahshur. The terrain struggled to sustain the pyramid’s weight, and the repeated appearance of cracks within the growing mass forced the builders first to surround the initial core with a flatter layer, and then to drastically reduce the amount of stone by reducing the slope halfway up the sides (Figure 7).7 The peculiar profile of this monument, in which the faces and edges abruptly change slope, is the reason for its modern name, the Bent Pyramid (Figure 8). In parallel with the construction of the Bent Pyramid, Sneferu built another pyramid, the Red Pyramid, nearby, using the flatter and safer slope employed in the upper part of the Bent Pyramid (Figure 9). Sneferu then went back to Meidum and turned his step pyramid into a smooth one with the addition of a stone casing; in doing so, he experimented with a slightly steeper slope.8 At the end of his reign, Sneferu thus left three completed pyramids, including both the last stepped pyramid and the first smooth pyramid. His son Khufu succeeded him on the throne, and the new king embarked upon the construction of what remains the largest pyramid ever built, using the steepest of the slopes that had been successfully adopted thus far, that of his father’s pyramid at Meidum (see Figure 1).
The earliest Egyptian mathematical sources which illustrate a method for calculating the slope of an inclined plane date to the Middle Kingdom (2040–1640 bce), but the available evidence suggests that the same method was also adopted earlier, during the Old Kingdom (from at least the Fourth Dynasty onwards).9 Whilst today we measure the amplitude of angles, ancient Egyptians expressed the slope as a linear measurement called seked, which may be defined as the horizontal displacement of the sloping face at the height of one royal cubit. (The cubit was the basic unit of measurement used in architecture and corresponded to the length of a forearm, or to six palms; the royal cubit included one additional palm.) The Egyptian method meant constructing a right-angled triangle in which one orthogonal side (or cathetus) was equal to one royal cubit (or seven palms) and the other to the seked; the latter was generally expressed using palms and fingers, the submultiples of the cubit (Figure 10). Another possible approach was to refer to right-angled triangles with specific geometrical properties. The seked of Sneferu’s first plan for the Bent Pyramid was probably four palms (corresponding to an angle of 60°); that of the lower part, after the addition of the additional layer, eventually corresponded to five palms (c. 54°30′). On the other hand, the dimensions of the upper part of the Bent Pyramid and of the contemporary Red Pyramid suggest the use of a right-angled triangle constructed on the Pythagorean triple 20-21-29 (which would correspond to an odd seked of seven palms and one finger).10
After sekeds of four and five palms both failed to produce a complete pyramid, for the casing of the Meidum pyramid Sneferu tried a slightly flatter seked of five palms and two fingers, corresponding to an angle of nearly 52°. This was the steepest slope achieved thus far and was selected by Khufu for his own monument. This proportion (a right-angled triangle in which the catheti correspond, respectively, to seven palms and five palms and a half) is sometimes referred to by modern scholars as the ratio 14:11; although geometrically equivalent, this modern expression has no parallel in/does not correspond to the mathematical language employed in the ancient Egyptian sources.
Khufu’s successor Khafra opted for a slightly smaller monument but a slightly steeper seked, five palms and one finger (just over 53°), which gave his pyramid a more slender and pointed appearance (cf. Figure 1).11 With the construction of the third pyramid, built for Menkaura (who adopted the same seked as Khufu), there was no space left on the Giza plateau for other major monuments or the associated cemeteries for royal family members and officials, and the succeeding kings moved south to Abusir and Saqqara.
Fifth Dynasty pyramids appear to have been designed using a seked of five palms and either one or two fingers, and thus are similar to the pyramids of Khufu and Khafra (Figure 11). By the time of the Sixth Dynasty, the possibility of expressing the slope of Khafra’s pyramid as a 3-4-5 triangle (based on the best-known Pythagorean triple) appears to have been fully understood, as all the kings’ pyramids were 150 cubits wide and 100 cubits tall.12
From the Fifth Dynasty onwards the slope of the pyramids of the queens was not aligned any longer to that chosen by their king. A favourite shape for these smaller pyramids both in this period and in the Middle Kingdom was the one in which base and height were the same (using a right-angled triangle in which one cathetus is half as long as the other, generating a slope based on an angle of 63°30′).
The ruined Middle Kingdom pyramids suggest that the kings of this period chose simple and linear sekeds of five and six palms (c. 54°30′ and slightly over 49°, respectively).13
The later New Kingdom private pyramids, being extremely small, could be built using extremely steep slopes (Figure 12). At Deir al-Medina sekeds of four, three, and two palms (60°, c. 67° and c. 74°, respectively) were used in the construction of the small mud brick pyramids surmounting the chapels located near or above the funerary shafts.14 The tendency throughout the Old, Middle, and New Kingdoms to adopt steeper slopes in the construction of smaller pyramids suggests that the underlying idea was that pyramidal shapes were expected to be as steep as possible.
Orientation
Figure 13. Schematic axonometric view of the pyramid of Khufu (Old Kingdom, Fourth Dynasty, 2551–2528 bce). The function of some internal passages and rooms is unclear; the narrow diagonal conduits departing from the two burial chambers are unique features that cannot be found in other pyramids.
Pyramids were oriented with their sides facing the four cardinal points. The precision which was achieved is likely to have been the result of a process of alignment based on the movement of stars, rather than on shadows cast by the sun.15 A new wave of studies on this subject was triggered by the appearance of computer programmes able to simulate the appearance of the sky in the distant past; at the time of the construction of the first pyramids, the north celestial pole was not marked by any star, and the position of the visible stars was different from what it is today.
The orientation of eight Old Kingdom pyramids built during the Fourth and Fifth Dynasties appears to progressively deviate from the true north, suggesting that the method used to obtain the orientation was based on observing the movements always of the same stars, and that the deviation corresponded to the effect of the precession of the equinoxes.16 The method is likely to have been based on the so-called simultaneous transit, that is, on detecting the moment at which two circumpolar stars aligned with one another perpendicularly to the horizon, thus identifying the direction of the north celestial pole. The precise identification of the stars that were actually used remains a matter of debate.17 The main difficulty that scholars must face in solving this riddle is the incompleteness of the available information: precise survey data are available only for a small number of pyramids and astronomical data must be combined with chronological information for the various kings who reigned during a particular period.
A stellar orientation has been postulated also for specific features of the pyramids, such as the slope of the diagonal shafts of the pyramid of Khufu (Figure 13), which apparently point in the direction of the culmination of specific stars, as well as for the enigmatic so-called trial passage located near the pyramid of Khufu.18 Such hypotheses are difficult to prove since they refer to unique architectural features and the lack of parallels prevents generalization.
The theory which suggests that the positions of the Giza pyramids reflected those of three stars in the constellation of Orion is not accepted by the scientific community. But a solar element in the layout of the area is suggested by the fact that on summer solstices the sun, seen from the area of Sphinx Temple, sets between the two pyramids of Khufu and Khafra; this seems to indicate a desire to redesign the landscape in the shape of the hieroglyph meaning “horizon,” which depicted the sun appearing between two rounded peaks.19 This large-scale design may have been yet another way to reinforce, this time at the level of the landscape, an idea that had already been clearly enunciated by Khufu when he named his own funerary monument the “Horizon of Khufu.”20 The idea is that of being reborn again after death, just as the sun rises again in the morning after setting in the west in the evening.
Materials and Construction Techniques
Figure 14. The entrance to the Fourth Dynasty pyramid of Khufu, protected by a large stone block.
Figure 15. The Twelfth Dynasty pyramid of Senusret II at Lahun (Middle Kingdom, Twelfth Dynasty, 1897–1878 bce), showing the internal structure made of stone walls and mud bricks, exposed after the removal of the stone casing.
Figure 16. The pyramidion of the Twelfth Dynasty pyramid of Amenemhat III at Dahshur (Egyptian Museum in Cairo, Middle Kingdom, Twelfth Dynasty, 1844–1797 bce).
The development of large pyramids is deeply intertwined with the exploitation of stone as a building material. Pyramids were mainly solid masses, within which relatively small voids were laboriously and carefully protected by thick stone padding and casing. Internal spaces started to appear when the size of stone blocks increased, and in fact are absent in the Third Dynasty step pyramids; in those early structures, the funerary apartments consisted of shafts and deep cuts in the underlying rock which would be progressively covered and then surmounted by the superstructure, a procedure already employed in the construction of mastaba tombs.
The ability to quarry, cut, and smooth blocks in order to achieve seamless joints allowed the builders to exploit in the best possible way the strength of stone and its resistance to pressure (Figure 14).21 At the same time, a building technique in which blocks were laid on sloping beds which converged towards the interior of the monument was abandoned in favour of a technique using horizontal layers that better distributed the pressure. Together, the two developments allowed the introduction of voids into the built-up part of the structure. In this respect, the ten-year-long open-air laboratory of the Bent Pyramid must have represented a crucial experience for an entire generation of builders and stonecutters.22
Old Kingdom pyramids were built entirely in stone. The original casing, which made the external faces truly smooth, has in nearly all cases been removed over the course of millennia, exposing the regular layers that were created just beneath it to support the pyramid’s outer skin. The innermost core of well-preserved pyramids is difficult to study, and it is possible that different technical solutions were adopted in different cases: some smooth pyramids, for instance, might contain stepped cores designed in a variety of ways.23
Middle Kingdom pyramids, by contrast, lie in a totally ruined state. The reason is the new building technique that was adopted by the kings of that period: a skeleton made of stone walls was filled in by either debris or mud bricks and then cased with fine limestone. This method was certainly more economical and probably quicker as well; however, once the external casing was removed by people looking for easily available stone, wind, rain, and gravity reduced the internal structures to shapeless mounds (Figure 15).
At Giza, most of the stone blocks used to build the pyramids were quarried nearby; in the area stretching from Abusir to Dahshur, however, a large quantity of building material must have been transported from other locations. This operation was probably facilitated by artificial canals that allowed the boats to reach the foot of the plateau even outside the period of the annual inundation.24 Temporary ramps were built to drag the stone blocks to the building site by smoothing and strengthening the desert surface with mud brick fills and wooden beams.25
Linear mud brick ramps were probably used to lift the stones up the first courses of the monument under construction, but the exact geometric and structural relationship that must have existed between the ramps and the growing pyramid once a certain height had been reached is unclear.26 Linear ramps would have rapidly become too long (or too steep), whilst spiral ramps encapsulating the entire structure would have hidden the monument and prevented the builders from checking the alignment of the corners. The most logical solution would have been a relatively narrow spiral ramp clinging to the pyramid’s faces; this would also explain why large pyramids were not too steep, never exceeding a seked of five palms (i.e., a slope of c. 54°30′).27 There is evidence that the casing blocks were placed in situ well dressed on all sides apart from the external face, which was left bulging and unfinished until the very end of the work.28 This irregular surface could be exploited to attach the ramp to the pyramid.
The uppermost stone to be placed on top of the pyramid was the so-called pyramidion, a large block shaped like a miniature version of the pyramid itself (Figure 16); for this reason, pyramidia can sometimes be used to derive geometrical information about the pyramids on which they were placed.29 Once the pyramidion was placed on the top, all the auxiliary structures built to reach the apex were progressively dismantled, whilst the surface of the pyramid was being smoothed from top to bottom.
The construction of the much smaller private pyramids that adorned the New Kingdom private tombs, reaching heights of just a few metres, was a completely different matter: the majority of them were entirely built of mud bricks, and simple scaffoldings or short linear ramps were sufficient to achieve the desired result. It is possible that similar solutions were adopted in the construction of the later Nubian pyramids (770 bce–350 ce).
Directions of Research
From a technical point of view, several aspects of pyramid building are still unclear. The main difficulty is in finding a balance between general practices and specific cases—that is, in looking for the general rules that must have existed, and yet remaining flexible enough to identify unique or rare deviations from those rules.
Several Old Kingdom pyramids show signs that plans were changed during their construction, and exhibit architectural elements that were never replicated; on the other hand, the evolution of such aspects of pyramid-building as the construction of the roofs of the burial chambers can be clearly reconstructed.30 The criteria for choosing the slope for each pyramid remain unclear: large pyramids are flatter than their smaller counterparts, but within these two large categories it is unclear how the choice to adopt a particular geometrical model was made; the reason might be sought in a desire to refer directly to a previous king, or it might depend on factors that are unknown to us.31 Investigation into the planning and building methods of ancient Egyptian pyramids is likely to produce further important results by combining information from different fields and parallel areas of research, encompassing fieldwork as well as the analysis of historical and textual sources.
Bibliography
- Arnold, Dieter. Building in Egypt: Pharaonic Stone Masonry. Oxford: Oxford University Press, 1991.
- Badawy, Alexandre. Le dessin architectural chez les anciens Egyptiens. Cairo, Egypt: Imprimerie Nationale, 1948.
- Belmonte, Juan Antonio. “On the Orientation of Old Kingdom Egyptian Pyramids.” Journal for the History of Astronomy 32, Archaeoastronomy supplement 26 (2001): S1–S20.
- Clarke, Somers, and Reginald Engelbach. Ancient Egyptian Construction and Architecture. Oxford: Oxford University Press, 1930.
- Haack, Steven C. “The Astronomical Orientation of the Egyptian Pyramids.” Journal for the History of Astronomy 15, no. 7 (1984): S119–S125.
- Imhausen, Annette. Mathematics in Ancient Egypt: A Contextual History. Princeton, NJ: Princeton University Press, 2016.
- Kemp, Barry J. Ancient Egypt: Anatomy of a Civilization. London: Routledge, 2018.
- Lauer, Jean-Paul. La pyramide à degrés: L’architecture. Cairo, Egypt: Service des Antiquités de l’Égypte, 1936–1939.
- Lehner, Mark. The Complete Pyramids. London: Thames & Hudson, 1997.
- Lightbody, David. “Moving Heaven and Earth for Khufu: Were the Trial Passages at Giza Components of a Rudimentary Stellar Observatory?” Journal of Ancient Egyptian Architecture 4 (2020): 29–53.
- Magli, Giulio. Architecture, Astronomy and Sacred Landscape in Ancient Egypt. Cambridge, UK: Cambridge University Press, 2013.
- Rossi, Corinna. Architecture and Mathematics in Ancient Egypt. Cambridge, UK: Cambridge University Press, 2004.
- Rossi, Corinna. “Egyptian Architecture and Mathematics.” In Handbook of the Mathematics of the Arts and Sciences. Edited by Bharath Sriraman, 1–12. Cham, Switzerland: Springer, 2018.
- Spence, Kate. “Ancient Egyptian Chronology and the Astronomical Orientation of Pyramids.” Nature 408 (2000): 320–324.
- Stadelmann, Reiner. Die Ägyptischen Pyramiden: Vom Ziegelbau zum Weltwunder. Mainz, Germany: Philipp von Zabern, 1985.
Notes
1. Reiner Stadelmann, Die Ägyptischen Pyramiden: Vom Ziegelbau zum Weltwunder (Mainz: Philipp von Zabern, 1985); Mark Lehner, The Complete Pyramids (London: Thames & Hudson, 1997).
2. Lehner, Complete Pyramids.
3. Giulio Magli, Architecture, Astronomy and Sacred Landscape in Ancient Egypt (Cambridge, UK: Cambridge University Press, 2013), 105–115; and Barry J. Kemp, Ancient Egypt: Anatomy of a Civilization (London: Routledge, 2018), 142.
4. Jean-Paul Lauer, La pyramide à degrés: L’architecture (Cairo: Service des Antiquités de l’Égypte, 1936–1939); and Alexandre Badawy, Le dessin architectural chez les anciens Egyptiens (Cairo, Egypt: Imprimerie Nationale, 1948).
5. Dieter Arnold, Building in Egypt: Pharaonic Stone Masonry (Oxford: Oxford University Press, 1991), 120–122.
6. Corinna Rossi, “Egyptian Architecture and Mathematics,” in Handbook of the Mathematics of the Arts and Sciences, ed. Bharath Sriraman (Cham, Switzerland: Springer, 2018), 3–6.
7. Stadelmann, Ägyptischen Pyramiden, 80–97.
8. In the long run, these experiments proved too much for this pyramid: the outer part eventually collapsed and left the monument in the shape of a tower rising from pile of rubble.
9. Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History (Princeton, NJ: Princeton University Press, 2016), 112–116; The slopes of all the Old Kingdom pyramids can easily be expressed in terms of seked and/or Pythagorean triples, apart that of the Fifth Dynasty king Unas: see appendix to Corinna Rossi, Architecture and Mathematics in Ancient Egypt (Cambridge, UK: Cambridge University Press, 2004). It must be borne in mind that the majority of pyramids were measured only once decades ago, and that in some cases no detailed records or description exists of how these measurements were taken.
10. Pythagorean triples bear this name because, ex post, they represent clear examples of the Theorem of Pythagoras; however, there is no need to suppose that the practical use of the former in constructing right-angled triangles implied a codified knowledge of the latter.
11. Rossi, Architecture and Mathematics, 221–226.
12. Rossi, Architecture and Mathematics, 219.
13. For a list of pyramids and their slope, see appendix to Rossi, Architecture and Mathematics.
14. Sekeds of 2 and 3 palms (respectively c. 74° and c. 67°) could also be achieved by means of the Pythagorean triples 7-24-25 and 5-12-13; Rossi, Architecture and Mathematics, 219–221, 231–235.
15. Steven C. Haack, “The Astronomical Orientation of the Egyptian Pyramids,” Journal for the History of Astronomy 15, no. 7 (1984): S119–S125; Magli, Architecture, Astronomy and Sacred Landscape, 89–93.
16. Kate Spence, “Ancient Egyptian Chronology and the Astronomical Orientation of Pyramids,” Nature 408 (2000): 320–324.
17. Juan Antonio Belmonte, “On the Orientation of Old Kingdom Egyptian Pyramids,” Journal for the History of Astronomy 32, Archaeoastronomy supplement 26 (2001): S1–S20.
18. Magli, Architecture, Astronomy and Sacred Landscape, 78–83; David Lightbody, “Moving Heaven and Earth for Khufu: Were the Trial Passages at Giza Components of a Rudimentary Stellar Observatory?,” Journal of Ancient Egyptian Architecture 4 (2020): 29–53.
19. Magli, Architecture, Astronomy and Sacred Landscape, 93–101.
20. Stadelmann, Ägyptischen Pyramiden, 106–107.
21. Somers Clarke and Reginald Engelbach, Ancient Egyptian Construction and Architecture (Oxford: Oxford University Press, 1930), 96–116.
22. Lehner, Complete Pyramids, 104.
23. Arnold, Building in Egypt, 160–161.
24. Lehner, Complete Pyramids, 82–83.
25. Arnold, Building in Egypt, 79–98.
26. Arnold, Building in Egypt, 98–101.
27. Rossi, “Egyptian Architecture and Mathematics,” 9–10.
28. Arnold, Building in Egypt, 132–141.
29. Rossi, Architecture and Mathematics, 205–211.
30. Arnold, Building in Egypt, 182–201.
31. Rossi, Architecture and Mathematics, 236–237.
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