Orientation Analysis of the Pyramids of Khufu, Khafre, and Menkaure at Giza, Study notes of Astronomy

Download Orientation Analysis of the Pyramids of Khufu, Khafre, and Menkaure at Giza and more Study notes Astronomy in PDF only on Docsity! The Orientations of the Giza pyramids and associated structures Erin Nell and Clive Ruggles School of Archaeology and Ancient History, University of Leicester, Leicester LE8 0PJ, United Kingdom This paper, submitted to Archaeoastronomy: The Journal of Astronomy in Culture on 30 March 2012, has been accepted for publication and is due to appear in vol. 25 (2012). Long abstract Ever since Flinders Petrie undertook a theodolite survey on the Giza plateau in 1881 and drew attention to the extraordinary degree of precision with which the three colossal pyramids are oriented upon the four cardinal directions, there have been a great many suggestions as to how this was achieved and why it was of importance. Surprisingly, perhaps—especially in view of the various interpretations that have been offered and the many other astronomical hypotheses and speculations, both scholarly and popular, that have been offered in the intervening 130 years—there have been remarkably few attempts to reaffirm or improve on the basic survey data concerning the primary orientations. Existing discussions typically depend upon measurements dating back at least 25 years and in part, either explicitly or implicitly, right back to Petrie’s own. Nor have these data generally been considered in the broader context of the orientations of the numerous other structures—in particular, the smaller pyramids and mastabas, as well as the valley and mortuary temples—in the vicinity. This paper presents the results of a week-long Total Station survey undertaken by the authors during December 2006 whose principal aim was to clarify the basic data concerning the orientation of each side of the three large pyramids and to determine, as accurately as possible, the orientations of as many as possible of the associated structures. The principal difference between this and all previous surveys is that it focuses upon measurements of sequences of points along multiple straight and relatively well preserved structural segments, with best-fit techniques being used to provide the best estimate of their orientation, as opposed to simple triangulation between directly identified or extrapolated corners. This approach also facilitates a richer interpretative approach taking account of possible variations in orientation targets and alignment techniques at different stages of construction, both of individual monuments and among the various monuments and monument complexes on the plateau. We proceed to a comparative analysis between our data and those that have been published in the past, and to offer some preliminary interpretations, focusing on the following key questions, which motivated the survey: 1. Are the azimuths measured by Petrie and quoted ever since, often to a precision of 0.1 arc minutes, actually a fair reflection of the intended orientation of the main Giza pyramids? 2. What do the basic data suggest about which of the cardinal directions was used to determine the orientation? 3. What does the broader context of structure orientations on the Giza plateau suggest about the techniques and motives of orientation? Our results do not directly challenge the idea that pairs of circumpolar stars were used to determine the primary orientation of the largest pyramids, but they do suggest that there is only a very slight difference in orientation (≈0.5ʹ′) between the north-south axes of Khufu’s and Khafre’s pyramids, that the sides of Khafre’s are more perfectly perpendicular than those of Khufu’s, and that the east-west axis is closer to true cardinality in both cases. The broader context of associated structures suggests that the east-west orientation in relation to sunrise or (in one case) sunset may have been a, or even the, key factor in many cases. Nell and Ruggles (2012)—Orientations of the Giza pyramids 2 1 Introduction Putative celestial alignments on the Giza plateau, embodied both in sightlines between the structures and in the orientations of individual structures, have been the subject of considerable debate over the preceding century, with a renewed burst of interest taking place during the past decade. This paper is primarily concerned with the orientations of individual structures rather than alignments between them (regarding which see Lehner 1983; Houdin 2006; Magli 2008, 2009a, 2009b). The largest of the Giza pyramids, often referred to as the Great Pyramid, was built in the mid-26th century BC for the fourth-dynasty pharaoh Khufu, known to the Greeks as ‘Cheops’ It consists of approximately 2,300,000 blocks of stone, averaging more than two tonnes in weight (Lehner 1997: 108). While the pyramid of his son Khafre (‘Khephren’ to the Greeks) is slightly smaller than Khufu’s and that of his grandson Menkaure (‘Mycerinus’) is only about a quarter the size (in terms of area), all three pyramids are imposing monuments. Khafre was not Khufu’s only son, and was not his immediate successor. Khafre’s elder brother Djedefre ruled for a short period after the death of Khufu and Djedefre decided to build his pyramid north of the Giza Plateau at Abu Roash. Djedefre’s pyramid is much smaller than the three Giza Plateau pyramids, had a steeper incline and is poorly preserved (Reisner 1942; Lehner 1997: 120–121); therefore its orientation will not be considered further in this paper. Ancient Egyptian pyramids were usually built with subsidiary structures and therefore had a number of associated architectural elements. The concept of a pyramid complex was not finalized until early dynasty 4, from which time each pyramid complex consisted of six such elements: a temenos (sacred enclosure) wall, the main pyramid, one or more satellite pyramids, a mortuary temple, a valley temple at some distance, and a causeway that connected the mortuary and valley temples. A pharaoh’s family members or favourite courtiers were frequently buried in mastabas (flat-roofed rectangular tombs) next to his complex. Each of the three large pyramids at Giza formed part of such a complex, with its causeway running down to a valley temple some 500m away to the east. Adjacent to Khafre’s valley temple was the Sphinx temple and the Sphinx itself. 1.1 Egyptian survey methods Within a broad pattern of orientation in relation to the River Nile, many Egyptian temples were aligned cardinally, especially in Old Kingdom times.1 Yet it has long been recognized that three Giza pyramids were aligned to the cardinal points with remarkable accuracy (Petrie 1883; Dorner 1981; Haack 1984). On the question of how this was achieved, evidence other than that of the alignments themselves is fragmentary. No engineering documents or architectural plans have been found that give technical explanations as to how the ancient Egyptians aligned any of their temples or pyramids. No ancient Egyptian compasses have ever been discovered, nor has any other type of sophisticated survey equipment (Romer 2007: 318). Some records do exist, however, that appear to provide clues about the process of surveying. Depictions of foundation ceremonies for sacred buildings appear on temple walls throughout ancient Egyptian history, 1 The recent work of Belmonte and his collaborators (Shaltout and Belmonte 2005; Belmonte and Shaltout 2006; Shaltout et al. 2007; Belmonte et al. 2008; Belmonte and Shaltout 2009) has revealed a much more elaborate picture in which the temples of the Nile valley and delta were oriented both topographically and astronomically, the astronomical alignments being of three types: cardinal or inter-cardinal (by sighting on northern stars and then rotating if necessary); solar/calendrical; and upon the two brightest stars, Sirius and Canopus. Nell and Ruggles (2012)—Orientations of the Giza pyramids 5 seen in foundation ceremony depictions (Fig. 2b), which bears a striking resemblance to a Roman groma. However, this idea is not supported either by literary evidence or by temple illustrations. A question left unanswered regarding the ‘simultaneous transit’ strategy for determining the north direction is on what grounds the ancient Egyptians made the choice of a particular pair of stars. For example, the choice of Kochab and Mizar would, as Spence has shown, have achieved the greatest prevision at the time of Khufu and Khafre, but it is inconceivable that these stars were chosen for this reason some 50 years earlier; instead, we would have to assume that the level of precision achieved by those pharaohs was fortuitous. If, on the other hand, the level of precision achieved at Giza was deliberate rather than fortuitous, then the surveyors must already have determined that a particular pair of stars, when aligned vertically in the sky, would provide a particularly good indication of true north. One conceivable strategy, paralleling the Edwards method using a single star, would be to observe the same pair of stars twice, first one above the other and then vice versa, extending the plumb-line down to the horizon to determine the azimuth in each case, and using the mean point between them. However, this not only runs into the same problem of having to make the two observations in different seasons, in order to get a dark sky in each case, but also (in the Spence case) of having to determine the vertical over an even wider arc down to the horizon; furthermore this is also a problem for one of the two observations in the Belmonte case—the one where both the stars are above the celestial pole. Belmonte (2012: 138) has more recently suggested that, since Phecda and Megrez were aligned upon the ‘pole star’ of the time—Thuban (α Dra)—they might well seemed natural choices for that reason. In that case, only their lower culmination would have offered the required precision. 1.2.2 Noon shadows Neugebauer (1980) and Isler (2001: 163) have independently suggested that the pyramids could have been aligned by solar methods using a gnomon and measuring cord. Neugebauer proposed that measurements of the tip of the shadow of a pyramidal block could have been taken throughout a day, resulting in a curve to the north of the block (see Fig. 2c). If this did not intersect the extensions of the eastern and western sides at equal distances from the corners (C and D in the figure), then an adjustment to the orientation would be needed. Factors such as the levelness of the ground are critical. 1.2.3 Sunrise in the east One of the authors (EN) has explored the possibility that the easterly direction could have been determined using sunrise observations. However, there are several problems with this, related to the precise determination of the astronomical equinox compounded with problems due to the altitude of the horizon (cf. Ruggles 1997). On the other hand, sun worship was becoming one of the most dominant aspects of Egyptian religion during dynasty 4. We shall discuss this further in what follows. 1.3 Nature and quality of the existing survey data Between 1880 and 1882, Petrie (1883) conducted an extraordinarily comprehensive, careful and detailed survey including the pyramids of Khufu, Khafre, and Menkaure, as well as two of the smaller Queens’ pyramids of Khufu. Remarkably few independent surveys have been undertaken since that time. Those of J.H. Cole in the 1920s (Cole 1925) and Josef Dorner in the 1970s (Dorner 1981) were restricted to the pyramids of Khufu and Khafre. Nell and Ruggles (2012)—Orientations of the Giza pyramids 6 Petrie and Cole surveyed using vernier theodolites, triangulating from different theodolite stations to obtain distance measurements (Petrie 1883: 10–21, 24; Cole 1925: 1–2)4. Dorner, working several decades later, did not obtain distance measurements independently but relied upon those of Cole: “Auf Streckenmessungen mußten wir von vornherein verzichten, weil hier kien entsprechendes Instrumentarium zur Hand war. Dies war aber auch nicht notwendig, weil wir von Cole dort die Abstände übernehmen konnten…” [At the outset we had to forgo distance measurements, as there were no suitable instruments available. However, they were unnecessary, as we could use the distance measurements already made by Cole…”] (Dorner 1981: 74). Thus the conclusions of authors such as Spence and Belmonte depend, ultimately, upon survey data that dates back at least to the 1920s.5 Recent years have brought great improvements in technology, most notably Electronic Distance Measurement (EDM) devices that are integral to modern Total Stations, and the Global Positioning System (GPS) network. In 1984 the Giza Plateau Mapping Project (GPMP) established its own triangulation grid over the entire plateau, and themselves undertook surveys of the Khufu and Khafre pyramids (Lehner and Goodman 1984; Lehner 1985a). A concern with interpretations based upon azimuths quoted from the early surveys is that they pay too little attention to the subtleties of context. This is not to imply any criticism of the early archaeologists—indeed, Petrie’s achievements in 1881 are truly remarkable—but relates simply to the limitations of the technology available to them. One of the huge advantages of EDM devices is that they enable one efficiently to take measurements of a series of points at frequent intervals along a straight segment, for example of the edge of a casing layer or stone course, and then calculate a best-fit straight line to provide the best estimate of the azimuth. On a pyramid side, there may be many such segments, well preserved, at different levels. The early surveyors, on the other hand, would have found it impracticable (using just a theodolite and a tape or chain) to do more than measure the end points of such segments (for example, the corners of a pyramid to obtain a best estimate of the azimuth of the sides), triangulating them in from different survey stations in order to obtain a single best estimate of the azimuth of the line joining them. As Petrie (1883: 37) was well aware, different casing and masonry levels may provide different insights into the intended orientation as well as the process of how this was adhered to during construction. Among the obvious questions are: how accurately did the foundations, the casing stones and then successive courses of stones in the pyramid itself accurately preserve the same orientation? Indeed, were the courses of stones in the pyramid straight and precisely perpendicular, and did successive courses retain the same orientation? If not, and if the precise alignment was so important, which structure in fact defined it? Or was the alignment constantly re-determined and possibly by differing means? With the aim of clarifying the basic orientation data, the current authors undertook an intensive Total Station survey over a six-day period in December 2006. We collected orientation survey data from the pyramids of Khufu, Khafre, and Menkaure, five of the eight queens’ and satellite pyramids associated with those pharaohs, the causeways of Khafre and Menkaure, Khafre’s valley temple and the Sphinx 4 Cole (1925: 1–2) recounts that “Eight brass bolts were cemented into the rock round the base, one near each of the four corners, and, as these were not intervisible, four more were placed, one at about the middle of each side, in such a position that each point was visible from the adjacent points on each side of it. … The positions of these points were determined by means of precise traverse observations.” 5 Spence, for example, in the commentary to her own table of orientations (Spence 2000: 320), quotes Dorner for the orientations of the W and E sides of Khufu and Khafre, and concludes her paper with the intention of undertaking more accurate survey work herself; but this was something she was never able to bring to fruition (Spence, priv. comm., 2006). Nell and Ruggles (2012)—Orientations of the Giza pyramids 7 temple, and a sample of monuments within the western, eastern, and southern cemeteries (mastaba fields) surrounding Khufu’s pyramid. Alignment information for Menkaure’s valley temple was also incorporated by courtesy of the Giza Plateau Mapping Project.6 In this paper we present the principal results of this survey and provide a comparative analysis between our data and those that have been published in the past. We proceed to review the existing interpretations in the light of these new data and conclude by presenting some tentative conclusions of our own. 2 The 2006 survey: methods and procedures 2.1 Survey methods The authors set out to measure as many as possible of the orientations of monumental structures on the Giza plateau during an intensive six-day period starting on 16 December 2006. The work was undertaken using a Leica TCR705 Total Station loaned from the School of Archaeology and Ancient History at the University of Leicester. The survey was conducted under the Supreme Council of Antiquities umbrella permission granted to the Giza Plateau Mapping Project (GPMP), a division of Ancient Egypt Research Associates (AERA). Our broad strategy was to identify a set of survey stations over the plateau that would not only enable us to establish a robust triangulation network but also to optimize the accuracy of our sightings of reference points on structures around the site. A set of 11 survey stations was duly identified. The direction of true north was determined independently at each survey station by taking one or more sets of sun-azimuth readings (Ruggles 1999: 168–169) and a triangulation grid was established by direct measurements between intervisible stations together with sightings upon locatable control points on the GPMP triangulation grid set up in 1984 (Goodman and Lehner 2007; Goodman 2007). The latter was itself aligned to true north by astronomical means—Polaris observations (Goodman and Lehner 2007: 56; Goodman 2007: 98)—and we were able to verify that our own grid and the GPMP grid agree on the direction of true (astronomical) north to within our own margins of error (± 0.3ʹ′), thus eliminating the possibility that any significant systematic errors affected our entire survey. The EDM reflector was taken to prearranged locations along the base of each pyramid or monument and the position of each point determined as accurately as possible from at least one of the survey stations. For the large pyramids we aimed to obtain up to 30 readings from each of the four sides (at distance intervals of approximately 3m to 10m), depending upon the availability of original casing or foundation bases, in order to obtain best estimates of structure orientations and deviations from straightness. For the remaining monuments, depending on their size, at least 3 readings (and preferably many more) were taken from as many sides as possible given their condition of preservation and the time and logistical constraints. Numerous repeated measurements and consistency checks were made in order to reduce random errors and eliminate gross errors. As a result, distance measurements are generally considered accurate to 0.01m and plate bearing measurements to 0.3ʹ′. Best-fit straight lines through sets of points were determined using the least-squares method with perpendicular offsets (see http://mathworld.wolfram.com/LeastSquaresFittingPerpendicularOffsets.html). The accuracy of the deduced orientation of an alignment of measured points depends upon several factors, in particular: • their distance from the survey station; • the number of points from which a best fit can be determined; and 6 These data, and also those relating to the Khafre workmen’s village and the complex of Khentkawes (see below) have been supplied by GIS specialist Farrah Brown LaPan of the Giza Plateau Mapping Project. Nell and Ruggles (2012)—Orientations of the Giza pyramids 10 Satellite pyramid This small pyramid is poorly preserved and its foundation almost completely destroyed. We were only able to collect reliable survey data from one side. Causeway The central part of the causeway is sufficiently well preserved for a good estimate to be made of the intended orientation. The southern edge is less well defined than the northern edge. Valley temple and Sphinx temple Khafre’s valley temple (Fig. 4) is located at the eastern of the causeway and was split into lower and upper sections. The Sphinx temple (the Sphinx itself is to its west) abuts the northern side of the valley temple; the two temples share a similar alignment and give the impression of being one building split into two parts by a passageway. The interior plan of the Sphinx temple is unique in that it includes both an eastern and western sanctuary along the central axis of the temple. 2.3.3 Menkaure’s pyramid and its associated structures Like Khufu’s pyramid, Menkaure’s has three smaller queens’ pyramids, but they are on its south side. The causeway runs east. Queens’ pyramids These three pyramids, to which we refer as MQ1–MQ3 running from west to east, are badly deteriorated and we were only able to collect data from all four sides in the case of MQ1. Causeway The outer edge of the pavement on the northern side of the western half of the causeway is sufficiently well preserved for a good estimate to be made of the intended orientation. 2.3.4 Summary In summary, then, the structures targeted for our survey were as follows: • The pyramid of Khufu (Great Pyramid): each of the four sides. • The central one of the three pyramids of Khufu’s queens. • Khufu’s mortuary temple. • Khufu’s western cemetery (Western Mastaba Field): a sample of early mastabas immediately to the east of the Tomb of Hemiunu. • Khufu’s eastern cemetery (Eastern Mastaba Field), a selection of the early mastabas closest to Khufu’s pyramid. • Southern cemetery (mastabas believed to have been mostly constructed by courtiers of Khafre and Menkaure, not Khufu): the westernmost three in the row of nine mastabas. • The pyramid of Khafre: each of the four sides. • Khafre’s satellite pyramid. • Khafre’s causeway. • The Sphinx temple. • Khafre’s valley temple. Nell and Ruggles (2012)—Orientations of the Giza pyramids 11 • The pyramid of Menkaure: each of the four sides. • The three pyramids of Mankaure’s queens. We refer to these as MQ1–MQ3, running from west to east. • Menkaure’s causeway. 3 The 2006 survey: basic results In this section we present the basic survey results for each of the oriented structures just listed. For the main pyramids we treat each of the four sides independently, but even in the case of a single side there are various structures that might be taken as providing an indication of the intended orientation. The main possibilities are: • The casing foundation: a series of stone blocks, extending outward around the pyramids, which supported the casing stones. Readings were taken on the outer edge, specified as being along the base or top of this edge where the distinction was deemed significant. Petrie refers to the casing foundation as the ‘casing in situ’. • The outer foundation: a lower level of stone blocks supporting the casing foundation, which extends yet further out from the pyramid. Readings were taken on the outer edge. Petrie refers to the outer foundation as the ‘pavement’. • The stone courses of the pyramid, numbered from the 1st upwards. Readings were taken on the outer edge (top), or along the bottom of a stone course, as circumstances dictated. In one case at Menkaure’s pyramid, the best indication of the intended orientation was deemed to be a straight line etched along the stones as part of the construction process. In each case, we sought to identify straight segments of the edge of the casing foundation, outer foundation or stone courses that were sufficiently undamaged to yield a reasonable estimate of the original orientation. Points along the segment were then measured at intervals of between 3m and 10m. Khafre’s pyramid is surrounded by alignments of postholes, but we did not survey these because this work had already been undertaken by Lehner (1983; 1985a; 1986); similar postholes surrounding Khufu’s pyramid are in such a poor state of preservation that they seem to be randomly scattered. The results are summarised in the six tables that follow (Tables 1–6). The following notes apply to all of these tables. 1. Without loss of generality, azimuths are expressed in one of the two possible directions—eastwards for east-west alignments and northwards for north-south alignments. 2. Azimuths and deviations from cardinality (DFCs) are only quoted to a precision of 0.1´ where they are based upon a best-fit of some 9 or more points. Otherwise they are quoted to a precision of 1´. Where the azimuths is based upon only 3 points, the entry is given in light type to indicate that it may be optimistic to expect even this level of accuracy. 3. Italicised rows indicate segments that are subsequently considered together. 4. Bolded rows indicate data considered to be of particular interest or importance. Nell and Ruggles (2012)—Orientations of the Giza pyramids 12 3.1 The Great Pyramid of Khufu The basic data for the Great Pyramid are shown in Table 1a. Included in these data are the surviving three of four brass corner markers put in place by earlier surveyors. Table 1a: The Great Pyramid of Khufu: basic data Side Indication Part No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes North Outer foundation 7 89° 54´ –0° 06´ a North Extended casing below entrance, front 9 89° 56.4´ –0° 03.6´ North Brass corner markers 2 89° 56.4´ –0° 03.6´ c North Casing foundation, base Western 6 89° 47´ –0° 13´ b North Reconstructed casing foundation Western –Central 5 89° 56´ –0° 04´ East Casing foundation Northern 3 0° 02´ +0° 02´ East Casing foundation Central 10 359° 58.1´ –0° 01.9´ East Casing foundation Southern 4 359° 52´ –0° 08´ East Casing foundation 17 359° 56.6´ –0° 03.4´ South Casing foundation Eastern 10 89° 58.4´ –0° 01.6´ South Casing foundation Western –Central 11 90° 00.5´ +0° 00.5´ South Casing foundation 21 89° 59.5´ –0° 00.5´ West Outer foundation Southern 5 359° 53´ –0° 07´ West Casing foundation, base Southern 8 359° 54´ –0° 06´ West Casing foundation, base Northern 7 359° 58´ –0° 02´ West Casing foundation, base 15 359° 56.3´ –0° 03.7´ West Casing foundation, top Central 7 359° 54.6´ –0° 05.4´ West Brass corner markers 2 359° 52.7´ –0° 07.3´ c Notes a The identification of this as the outer foundation is not certain. b The identification of this as the casing foundation base is questionable. c In the case of the brass corner markers, four independent readings were taken of each pairwise alignment, consistent to within 0.2´ in each case, so the quoted precision is considered justified. Data from the west and south sides indicate that the edge of the casing foundation base on each of these sides was slightly convex. On the west side, the difference in azimuth at the two ends is about 4´, and the data from the south side are consistent with this figure. The data from the north side suggest a greater degree of convexity, but see (2) below. This is in contrast to the well-documented concavity of the stone Nell and Ruggles (2012)—Orientations of the Giza pyramids 15 Table 2a (ctd) Structure Side Indication No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes Mastaba EC-3b (G 7320) South Casing foundation, base 3 90° 17´ +0° 17´ Mastaba EC-3c (G 7330) West Casing foundation, base 3 359° 55´ –0° 05´ LS Mastaba EC-3d (G 7340) West Casing foundation, base 3 0° 41´ +0° 41´ LS Mastaba EC-3c/3d (G 7330/7340) West Casing foundation, base 6 0° 18´ +0° 18´ LS Mastaba EC-3e (G 7350) North North wall 3 87° 19´ –2° 41´ Mastaba EC-4b (G 7420) South Casing foundation, base 3 90° 6´ +0° 6´ Mastaba EC-4b (G 7420) West Casing foundation, base 3 0° 30´ +0° 30´ LS Mastaba EC-4c (G 7430) North Casing foundation, base 3 90° 24´ +0° 24´ Mastaba EC-4c/4d (G 7430/7440) West Casing foundation, base 4 0° 57´ +0° 57´ LS Mastaba EC-4e (G 7450) North Casing foundation, base 3 88° 29´ –1° 31´ Mastaba SC-1 North Casing foundation, base 4 90° 43´ +0° 43´ Mastaba SC-1 East Casing foundation, base 5 359° 50´ –0° 50´ LS Mastaba SC-1 South Casing foundation, base 4 87° 43´ –2° 17´ Mastaba SC-1 West Casing foundation, base 5 359° 43´ –0° 17´ LS Mastaba SC-3 North Casing foundation, base 3 89° 57´ –0° 03´ Mastaba SC-3 East Casing foundation, base 4 359° 08´ –0° 52´ LS Mastaba SC-3 South Casing foundation, base 4 88° 11´ –1° 49´ Mastaba SC-3 West Casing foundation, base 5 0° 02´ +0° 02´ LS Mastaba SC-4 North Casing foundation, base 4 90° 21´ +0° 21´ Mastaba SC-4 East Casing foundation, base 7 359° 48´ –0° 12´ LS Mastaba SC-4 South Casing foundation, base 4 90° 07´ +0° 07´ Mastaba SC-4 West Casing foundation, base 5 358° 56´ –1° 04´ LS Mastabas WC–2b/3b (G 4250/4350) North Casing foundation, base 3 88° 14´ –1° 46´ Mastaba WC-2e (G 4220) West Casing foundation, base 3 356° 52´ –3° 08´ LS Mastaba WC-3a (G 4360) North Casing foundation, base 3 87° 01´ –2° 59´ Mastaba WC-3a (G 4360) West Casing foundation, base 3 357° 32´ –2° 28´ LS Mastaba WC-4a (G 4460) North Casing foundation, base 3 86° 49´ –3° 11´ Mastaba WC–4a (G 4460) West Casing foundation, base 3 358° 31´ –1° 29´ LS Mastaba WC-4b (G 4450) West Casing foundation, base 3 357° 44´ –2° 16´ LS Notes LS Longer side (of mastabas) Nell and Ruggles (2012)—Orientations of the Giza pyramids 16 These data reflect the poor state of preservation of the structures involved, and in many cases we were only able to obtain lines of 3 points on any given side, with consequent uncertainties in the azimuth values obtained. In the case of the western cemetery, particularly, we obtain values for the DFC varying from –1.5° to –3.2° on different mastaba sides, with no reduction in this range if we restrict ourselves to the longer (west and east) sides. Yet, given that our data set comprises seven independent values obtained from six different tombs, it is safe to conclude that the DFC, averaging out at around –2.4°, was not only significantly but also deliberately different from that of the Great Pyramid itself. If we exclude the remaining azimuth estimates that are based on only 3 values, then a more coherent picture begins to emerge in relation to the remaining structures in the Khufu complex. The data are summarised in Table 2b. Values from the shorter (north and south) sides of mastabas in the southern cemetery are included where lines of more than 3 points have been measured, but are placed in brackets to reflect the fact that they may be of secondary significance. Table 2b: Great Pyramid of Khufu and associated structures: summary data Structure Approx. Deviation from cardinality (DFC) foundation date * North side East side South side West side Mean (all sides) Mean (E & W sides) Great pyramid 2587 BC –0° 04ʹ′ –0° 03ʹ′ –0° 01ʹ′ –0° 04ʹ′ –0° 03ʹ′ –0° 03ʹ′ Khufu mortuary temple 2587 BC 0.0° Queen’s pyramid KQ2 2585 BC –0.3° EC row 1 (7100) 2583 BC –0.5° to –0.3° EC row 2 (7200) 2583 BC –0.5° EC row 3 (7300) 2583 BC +0.3° EC row 4 (7400) 2583 BC +1.0° SC–1 2535 BC (+0.7°) –0.8° (–2.3°) –0.3° –0.7° –0.5° SC–3 2535 BC –0.9° (–1.8°) 0.0° –0.5° SC–4 2535 BC (+0.4°) –0.2° (+0.1°) –1.1° –0.2° –0.7° Notes * See note 7. The dates quoted here (purely for reference and not critical to the arguments presented) are arrived at by taking Shaw’s (2000: 480) most likely date for Khufu’s accession, 2589 BC, and then tentatively assuming that the main pyramid, mortuary temple and western cemetery were started a couple of years into the reign while the queens’ pyramids and eastern cemetery were started a little later. Some SC tombs were built during Khafre’s reign (2558 – 2532 BC according to Shaw) while others were built during Menkaure’s reign (2532 – 2503 BC), so we have adopted an approximate starting date for these of 2535 BC. The data from the western part of the southern cemetery indicate that the mastabas here were oriented consistently, with a mean DFC of around –0.5°, although the eastern and western sides of individual mastabas were off-parallel by up to c. 1.0°. From the remaining data we can tentatively conclude the following: 1. The orientation of the mortuary temple reflected that of the Great Pyramid itself. 2. The orientation of KQ2 was significantly different, with a DFC of some –0.3°. Nell and Ruggles (2012)—Orientations of the Giza pyramids 17 3. The orientations of the westernmost two rows of mastabas in the eastern cemetery, closest to the Khufu’s pyramid and his queens’ pyramids, follow a similar orientation, with DFCs between –0.3° and –0.5°. 4. The orientations of the next two rows of mastabas, in contrast, deviated from cardinality in the opposite sense, with the DFC of row 4 (+1.0°) being significantly greater than that of row 3 (+0.3°). On the basis of these data we can draw two general conclusions. First, none of the burial structures of Khufu’s sons and advisors achieved the degree of perfection to which his own pyramid was cardinally oriented. The orientation of KQ2 suggests that the same may have been true of those of his three wives. Second, there appears to have been a systematic variation from west to east, with the mastabas of the western cemetery having DFCs between around –3.0° and –1.5°, those of the southern cemetery around –0.5°, and those of the eastern cemetery varying from around –0.5° on the western side to around +1.0° further to the east. 3.3 The Pyramid of Khafre The basic data for Khafre’s Pyramid are shown in Table 3a. Table 3a: The pyramid of Khafre: basic data Side Indication Part No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes North Outer foundation Western 3 89° 01´ –0° 59´ T1 North Casing foundation 13 89° 56.2´ –0° 03.8´ T1 North 1st course, outer edge Western 7 90° 01´ +0° 01´ T1 North 1st course, outer edge Central 6 89° 33´ –0° 27´ T1 North 1st course, outer edge Eastern 3 89° 41´ –0° 19´ T1 North 1st course, outer edge 16 89° 58.7´ –0° 01.3´ T1 East Casing foundation, base 19 359° 56.0´ –0° 04.0´ T4 East 1st course, outer edge Northern– Central 14 359° 55.9´ –0° 04.1´ T4 East 1st course, outer edge Central– Southern 4 359° 53´ –0° 07´ T4 East 1st course, outer edge Southern 6 0° 19´ +0° 19´ T4 East 1st course, outer edge 24 359° 58.8´ –0° 01.2´ T4 South Outer foundation Western 11 89° 57.2´ –0° 02.8´ T4 South Casing foundation, base 11 89° 54.2´ –0° 05.8´ T4 South Casing foundation, base Western 6 89° 56´ –0° 04´ T5 South 1st course, outer edge Western 6 89° 35´ –0° 25´ T4 South 2nd course, bottom Eastern 5 90° 40´ +0° 40´ T4 South 2nd course, bottom Central– Eastern 3 90° 08´ +0° 08´ T4 South 2nd course, bottom Central 5 90° 07´ +0° 07´ T4 Nell and Ruggles (2012)—Orientations of the Giza pyramids 20 Table 4a (ctd) Structure Side Indication Part No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes Sphinx temple North Enclosure rock wall 6 89° 31´ –0° 29´ Sphinx temple North Main wall 8 90° 22´ +0° 22´ Sphinx temple East Main wall 9 359° 58.8´ –0° 01.2´ Sphinx temple South Passage wall Western 5 94° 33´ +4° 33´ Sphinx temple South SE corner chamber, south wall 4 92° 44´ +2° 44´ Sphinx temple South SE corner chamber, north wall 4 92° 42´ +2° 42´ Sphinx temple West SW corner chamber, west wall, outer face 4 1° 41´ +1° 41´ Sphinx temple West SW corner chamber, west wall, inner face 3 2° 13´ +2° 13´ Sphinx temple West Inner wall, outer face Southern 3 0° 04´ +0° 04´ Valley temple East Foundation wall, base 8 89° 48´ –0° 12´ Valley temple East Main wall 12 89° 37.0´ –0° 23.0´ Valley temple South Outer wall, outer face 10 90° 06.2´ +0° 06.2´ Valley temple South Main wall 7 90° 32´ +0° 32´ Valley temple West Foundation wall 8 0° 44´ +0° 44´ Valley temple West Outer wall, outer face 3 359° 31´ –0° 29´ Valley temple West Main wall 6 0° 33´ +0° 33´ To judge by the only 3-point alignment that we could identify, on the north side, the satellite pyramid of Khafre is oriented slightly clockwise at +0° 04´. Two additional pairs of points on short straight segments of the west wall yielded a deviation from cardinality of –0° 12´ (towards the southern end) and +0° 02´ (northern end) respectively, but such readings yield no error checks and can only be taken as generally affirming the +0° 04´ figure. We measured a line of 8 points on the northern edge of Khafre’s causeway, which yield our best estimate of its orientation (azimuth 103° 26´/283° 26´). The discrepancy between this and the figure obtained from 4 points on the southern edge (103° 33´/283° 33´) appears to be due to the poorer state of the latter. A 7´ discrepancy over the entire length of the causeway (495m) would mean a difference between the width at both ends of 1.01m, but existing plans (e.g. Maragioglio and Rinaldi 1966, pl. 5, fig. 1 & pl. 14, fig. 1) provide no indication that the two sides of the Khafre causeway are anything but parallel. Our measurements yield a mean width of 8.29m over the section surveyed. The south and west walls of the valley temple display a consistent DFC close to +0° 32ʹ′, but that of the east wall is almost a degree different, at –0° 23ʹ′. This implies that the building is somewhat narrower at its northern end, but also that the mean north-south axis is very close to the true meridian, with a DFC of just +0° 05ʹ′. Nell and Ruggles (2012)—Orientations of the Giza pyramids 21 The most reliable sets of data relating to the Sphinx temple were obtained from its east and north walls, yielding DFCs of –0° 01ʹ′ and +0° 22ʹ′ respectively. As it was not possible to obtain readings from the west wall, the east wall orientation—which is extremely close to true north-south—gives our best estimate of the axial orientation. A small set of readings from the southernmost segment of the western inner wall yielded a DFC value of +0° 04ʹ′, suggesting that the two long interior walls were also carefully aligned cardinally. A limited sample of readings from other internal structures gave much more disparate DFC values, between +1.5° and +3°. It is evident that the passageway separating the two temples is skewed away from true east-west by several degrees, in the same sense as the Khafre causeway. Measurements of the southern wall of the Sphinx temple yielded a DFC of +4° 33ʹ′. The data are summarised in Table 4b. Table 4b: Khafre’s pyramid and associated structures: summary data Structure Approx. Deviation from cardinality (DFC) foundation date * North side East side South side West side Mean (all sides) Mean (E & W sides) Main pyramid 2556 BC –0° 04ʹ′ –0° 04ʹ′ –0° 06ʹ′ –0° 04ʹ′ –0° 05ʹ′ –0° 04ʹ′ Satellite pyramid 2556 BC +0° 04ʹ′ Valley temple 2556 BC –0° 23ʹ′ +0° 32ʹ′ +0° 33ʹ′ +0° 05ʹ′ Sphinx temple 2556 BC +0° 22ʹ′ –0° 01ʹ′ (+4° 33ʹ′) Causeway 2556 BC +13° 26´ N/A +13° 33´ N/A +13° 29´ N/A Notes * See note 7. The dates quoted here (purely for reference and not critical to the arguments presented) are arrived at by taking Shaw’s (2000: 480) most likely date for Khafre’s accession, 2558 BC, and then tentatively assuming that all the structures were started a couple of years into the reign. 3.5 The pyramid of Menkaure Surveying this pyramid proved a challenge because its foundations were covered with rubble; and we were only able to survey the alignment of courses of stones on the pyramid itself. Largely intact courses of casing stones are only found on the north and east sides. A few casing stones remained on the south side so we surveyed a combination of casing stones and coarse core blocks on the second and seventh courses respectively. On the west side no casing stones remain so we surveyed the extant raw core blocks on the ninth and eleventh courses. Table 5a: The pyramid of Menkaure: basic data Side Indication Part No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes North 4th course, outer edge Western 5 90° 22´ +0° 22´ North 6th course, outer edge Central 5 90° 20´ +0° 20´ North 7th course, outer edge Central 6 90° 18´ +0° 18´ North 7th course, etched line 20 90° 18.5´ +0° 18.5´ North 9th course, outer edge 22 90° 21.5´ +0° 21.5´ Nell and Ruggles (2012)—Orientations of the Giza pyramids 22 Table 5a (ctd) Side Indication Part No. of points Best-fit azimuth Deviation from cardinality (DFC) Notes East 3rd course, outer edge Northern 11 0° 06.5´ +0° 06.5´ East 6th course, outer edge 21 0° 16.5´ +0° 16.5´ South 2nd course, outer edge Western 3 89° 48.5´ –0° 11.5´ South 7th course, outer edge 26 90° 31.8´ +0° 31.8´ c West 9th course, outer edge 23 0° 29.5´ +0° 29.5´ c West 11th course, outer edge 21 0° 19.7´ +0° 19.7´ c Notes c Measurements taken from coarse blocks only, in the absence of casing stones An issue to address at the outset is the concavity of the faces. Maragioglio and Rinaldi (1967: 36–37) have stated that, at least in the upper part of the pyramid, the extant casing is perfectly flat while the internal packing blocks formed a concavity in the center of each face: “The lower part of the filling of the steps, up to the height of the granite casing, in general follows an inclined surface like that of the outer faces of the pyramid. The thickness of the casing in this area, except for the bonding blocks, was thus fairly constant. The upper part, on the other hand, corresponding to the limestone casing, has on all of the faces—and particularly to the south—a marked concavity which seems intentional. The concavity appears greater in the lower part of this section and diminishes as it goes upwards. The thickness of the white limestone casing was thus greater towards the centre of the faces than towards the corners, as in Cheops” (Maragioglio and Rinaldi 1967: 36). To see whether our data are consistent with Maragioglio and Rinaldi’s statement we have taken each of the instances where we have readings from the entire length of a course and calculated the best-fit azimuth for each half as well as the whole. The data are presented in Table 5b. Table 5b: The Pyramid of Menkaure: concavity Mean deviation from cardinality (DFC) Wall Course Type Whole length ‘CCW’ half ‘CW’ half ‘CCW’ half – ‘CW’ half North 7th Casing +0° 18´ +0° 42´ –0° 08´ +0° 50´ North 9th Casing +0° 22´ +0° 57´ –0° 04´ +1° 01´ East 6th Casing +0° 17´ +0° 37´ +0° 16´ +0° 21´ South 7th Internal +0° 32´ +0° 34´ –0° 02´ +0° 36´ West 9th Internal +0° 30´ +1° 00´ +0° 23´ +0° 37´ West 11th Internal +0° 20´ +0° 13´ +0° 13´ 0° 00´ Notes • ‘CCW half’ means the half of the wall in the counter-clockwise sense, i.e. the western half of the north wall, northern half of the east wall, etc., while ‘CW half’ is the half in the clockwise direction, i.e. the eastern half of the north wall, southern half of the east wall, etc. Nell and Ruggles (2012)—Orientations of the Giza pyramids 25 Menkaure’s pyramid itself. The DFC value obtained from the west side of MQ1, at least, is consistent with this. A small segment of well-preserved pavement edge on the northern side of the Menkaure causeway yielded an azimuth of 89° 10´ (DFC –0° 50´). Thus Menkaure’s causeway, although approximately cardinal (unlike the causeways of Khufu and Khafre), deviates from cardinality by a significant amount and in the opposite sense to the pyramids of Menkaure and his queens. These data (excluding lines of only 3 points) are summarised in Table 6b. Table 6b: Menkaure’s Pyramid and associated structures: summary data Structure Approx. Deviation from cardinality (DFC) foundation date * North side East side South side West side Mean (all sides) Mean (E & W sides) Main pyramid 2530 BC +0° 20ʹ′ +0° 12ʹ′ +0° 32ʹ′ +0° 25ʹ′ +0° 22ʹ′ +0° 18ʹ′ Satellite pyramid MQ1 2530 BC –1° 01ʹ′ +1° 42ʹ′ +0° 31ʹ′ +1° 11ʹ′ Satellite pyramid MQ2 2530 BC +0° 27ʹ′ +0° 12ʹ′ Satellite pyramid MQ3 2530 BC +0° 39ʹ′ Causeway 2530 BC –0° 50ʹ′ N/A N/A N/A Notes * See note 7. The dates quoted here (purely for reference and not critical to the arguments presented) are arrived at by taking Shaw’s (2000: 480) most likely date for Menkaure’s accession, 2532 BC, and then tentatively assuming that all the structures were started a couple of years into the reign. In addition, alignment data for Menkaure’s valley temple—usually buried by sand but cleared during excavation—have been provided courtesy of the GPMP. They give the north-south alignment as 359.8°/179.8° and the east-west alignment as 90.2°/270.2°. 4 Comparative analysis In this section we compare our 2006 survey results with those published by other authors. The surveys conducted by Cole, Dorner, and Lehner and Goodman were restricted to the pyramids of Khufu and Khafre, so full comparative analyses are only possible in the case of these two pyramids. 4.1 Pre-Giza pyramids: the Third and Early Fourth Dynasties As a preliminary, we present and review alignment data collected by other authors from pre-Giza pyramids in dynasties 3 and 4. Included in this review are the ‘Bent Pyramid’ and ‘Red Pyramid’ (the south and north pyramids, respectively) at Dashur, both built by Snefru early in dynasty 4 and located approximately 40 km south of Giza (Reisner 1942: 1-3). We also include the Meidum pyramid, about 100km south of Giza, whose construction was also completed in Snefru’s reign although it has been speculated that it was begun by Huni (Snefru’s father) (Stadelmann 1980; Malek 2000: 93). Of the other ten pyramids two, belonging to Djoser and Sekhemkhet respectively, are located at Saqqara (between Dashur and Giza); and one at Zawiyet el-Aryan was (possibly) built by the pharaoh Khaba (Lehner 1997: 95). The other seven are small step pyramids located between the Fayoum and Elephantine Island (the southern border of upper Egypt) (ibid.). The function of these provincial step pyramids is unknown because they do not have burial chambers or (except in Seila’s case) associated chapels (Lehner 1997: 96). Nell and Ruggles (2012)—Orientations of the Giza pyramids 26 The published data are summarized in Table 7. We have retained the level of precision quoted in the original publications, although in the case of the three values quoted to the nearest arc second it is clear from the level of discrepancy between these and independent measurements of the same structures that they are unlikely to be accurate to better than a few arc minutes. In the case of El Kula and Seila, there is a gross discrepancy between the two independent readings. Table 7: Pre-Giza pyramid alignments Location Pharaoh Approx dates of reign* Deviation from cardinality Source Notes Saqqara Djoser 2667–2648 +3° Lauer 1960: 99; Romer 2007: 279 Saqqara Sekhemkhet 2648–2640 –11° Lauer 1960: 99; Romer 2007: 279 Zawiyet el-Aryan (Layer pyramid) ?? ?2640–2613 –9 to –8° Lehner 1996: 510; Romer 2007: 279 a Elephantine ?Huni 2637–2613 –17° Lehner 1997: 96; Belmonte & Shaltout 2009: 312, appendix II Edfu (El Ghoneimiya) ?? ?2640–2613 +3.5° Belmonte & Shaltout 2009: 312, appendix II a, b El Kula ?? ?2640–2613 –43.75° Belmonte & Shaltout 2009: 312, appendix II a, c Ombos/Naqada ?? ?2640–2613 +24.5° Belmonte & Shaltout 2009: 312, appendix II a, d Sinki ?? ?2640–2613 –41.25° Belmonte & Shaltout 2009: 312, appendix II a, e Zawiyet el-Meitin (Layer pyramid) ?? ?2640–2613 –19° Belmonte & Shaltout 2009: 312, appendix II a, f Seila Snefru 2613–2589 0° Belmonte & Shaltout 2006: 179–181 g Meidum Snefru 2613–2589 –0° 18ʹ′±2ʹ′ Belmonte 2001: S2 Dashur, south (‘Bent Pyramid’) Snefru 2613–2589 –0° 12ʹ′±2ʹ′ Belmonte 2001: S2 –0° 12ʹ′±2ʹ′ Belmonte 2001: S2 Dashur, north (‘Red Pyramid’) Snefru 2613–2589 –0° 5´ 0ʺ″ Isler 2001: 158. Measured by Joseph Dorner –0° 9ʹ′±2ʹ′ Belmonte 2001: S2 Notes * See note 7. The dates quoted here (purely for reference and not critical to the arguments presented) are those of the pharaoh’s reign taken from Shaw (2000: 480). a It is uncertain for which pharaoh this pyramid was built. The date range given covers the reigns of Khaba (2640–2637 BC) and Huni (2637–2613 BC), again according to Shaw (2000: 480), but this assignation is far from certain. b Lehner (1997: 96) lists the DFC as ~ 0°, which is broadly consistent with Belmonte and Shaltout’s more accurate value. Nell and Ruggles (2012)—Orientations of the Giza pyramids 27 c Lehner (1997: 96) lists the DFC as 0° but this seems to refer to the diagonal, which is close to cardinal. d Lehner (1997: 96) lists the DFC as ~ –12°, which is inconsistent with Belmonte and Shaltout’s value. e Lehner (1997: 96) lists the DFC as ~ 0° but this seems to refer to the diagonal, which is close to cardinal. f Lehner (1997: 96) lists the DFC as ~ –20°, which is consistent with Belmonte and Shaltout’s more accurate value. g Lehner (1997: 96) lists the DFC as –12°, which is inconsistent with Belmonte and Shaltout’s value. Based on these data, it is clear that the practice of orientating royal pyramids cardinally with remarkable precision was well established during Snefru’s reign. It is also possible that the orientations of Meidum, The Bent Pyramid at Dashur, and the Red Pyramid at Dashur represent successive refinements getting ever closer to true cardinality, while always deviating counterclockwise, as argued by Spence (2000). The data from Seila are less accurate, and this pyramid is perfectly cardinal within the larger measurement errors. In contrast, the data from the earlier pyramids, and particularly from the small step pyramids, really show no obvious trends at all. 4.2 The Great Pyramid of Khufu Four noteworthy surveys of the Great Pyramid have been conducted prior to the one reported in this paper: those of Petrie (1883), Cole (1925: 1–9), Dorner (1981) and Lehner and Goodman (1984; Lehner 1985a). As already noted, Dorner’s survey combined his own theodolite measurements with distance measurements taken from Cole. The results of all five surveys are summarized in Table 8. For clarity, earlier survey data, originally expressed in degrees, minutes and seconds, have been converted to degrees, minutes and decimals of minutes. Table 8: The Great Pyramid of Khufu: comparative data Side Petrie (1883: 39) Cole (1925: 6)11 Dorner (1981: 77) Lehner and Goodman (1984) Nell & Ruggles Mean deviation from cardinality (DFC) North –0° 03.3ʹ′ –0° 02.5ʹ′ –0° 02.5ʹ′ –0° 03.1ʹ′ –0° 03.6ʹ′ –0° 03.0ʹ′ East –0° 04.0ʹ′ –0° 05.5ʹ′ –0° 03.4ʹ′ –0° 03.7ʹ′ –0° 03.4ʹ′ –0° 04.0ʹ′ South –0° 03.7ʹ′ –0° 02.0ʹ′ –0° 02.5ʹ′ +0° 00.5ʹ′ –0° 00.5ʹ′ –0° 01.7ʹ′ West –0° 03.9ʹ′ –0° 02.5ʹ′ –0° 02.8ʹ′ –0° 07.1ʹ′ –0° 03.7ʹ′ –0° 04.0ʹ′ Mean deviation from cardinality –0° 03.7ʹ′ –0° 03.1ʹ′ –0° 02.8ʹ′ –0° 03.4ʹ′ –0° 02.8ʹ′ –0° 03.2ʹ′ These values are broadly consistent, but with some obvious discrepancies such as Lehner and Goodman’s value for the west side and Cole’s for the east side, and the fact that Lehner and Goodman’s and our values for the south side are somewhat higher than those of the three earlier surveyors. The obvious possibility is that some if not all of these discrepancies are due to different authors measuring different things. 11 “East” and “south” have been mistakenly transposed in the comparative table at the foot of page 7. Nell and Ruggles (2012)—Orientations of the Giza pyramids 30 Table 10: Menkaure’s pyramid: comparative data Side Petrie (1883: 111) Course North +0° 16.8ʹ′ 4th (W end); 1st (E end) East +0° 12.4ʹ′ Casing foundation base (N end); “foot of rough casing” (S end) South +0° 13.0ʹ′ 3rd (E end); 1st (W end) West — Mean deviation from cardinality +0° 14.1ʹ′ Dorner did not carry out his own survey of Menkaure’s pyramid, merely quoting Petrie’s results (Dorner 1981: 82). While we did obtain a DFC of +0° 07´ from the 3rd course on the east side, our DFC values obtained from higher stone courses are between +0° 17´ and +0° 32´. Petrie’s data are not directly comparable with ours (Tables 5a–5c), but the two sets of data perhaps complete a picture in which the higher levels were ‘twisted round’ with respect to the foundation, a possibility that Petrie himself seems to have ruled out.13 5 Discussion 5.1 Patterns of orientation In order to examine some of the broader questions relating to the motives for orienting the various structures on the Giza plateau and the techniques used, it is useful to summarise the overall orientation trends that emerge from all the data discussed so far. Table 11 represents an attempt to summarize these trends. Note that columns 3 and 4 of this table refer to the N-S and E-W directions, the N-S direction being that indicated by the alignments of the east and west sides or walls of the structures concerned, and the E-W direction being that indicated by the north and south sides or walls. Table 11: Overall patterns of orientation: summary data Approx Mean deviation from cardinality (DFC) date* Structure N-S direction E-W direction Notes Khufu complex 2587 BC Main pyramid –0° 04.0ʹ′ –0° 02.4ʹ′ A 2587 BC Mortuary temple 0.0° 2585 BC Queen’s pyramid KQ2 –0.3° 2585 BC Western cemetery –2.3° (–2.6°) b 2535 BC Southern cemetery –0.6° (–0.6°) cd 2583 BC Eastern cemetery –0.5° to +1.0° 13 “It must be remembered that if any different base level should be supposed to have been intended, it will make no difference in the above azimuths, nor in the differences between the sides.” (Petrie 1883: 212). Nell and Ruggles (2012)—Orientations of the Giza pyramids 31 Table 11 (ctd) Approx Mean deviation from cardinality (DFC) date* Structure N-S direction E-W direction Notes Khafre complex 2556 BC Main pyramid –0° 05.3ʹ′ –0° 05.0ʹ′ E 2556 BC Satellite pyramid +0.1° f 2556 BC Causeway N/A +13.3° 2556 BC Valley temple +0.1° +0.5° 2556 BC Sphinx temple ..0.0° +0.4° g Menkaure complex 2530 BC Main pyramid +0° 18ʹ′ +0° 26ʹ′ 2530 BC Queen’s pyramid MQ1 +1.2° –1.0° 2530 BC Queen’s pyramid MQ2 +0.2° +0.5° 2530 BC Queen’s pyramid MQ3 +0.6° 2530 BC Causeway N/A –0.8° 2530 BC Valley temple –0.2° +0.2° h Notes * See note 7 and Tables 2b, 4b, 6b. • Bold face indicates a figure derived as the mean of measurements on opposite sides; otherwise the figure is from one face only. The alignments of shorter sides of mastabas in the Khufu complex, less likely to be significant (or precisely defined or accurately measured) in themselves, are shown in brackets. Readings considered less reliable (reasons given in the notes) are italicized. A The figures obtained from our data alone are –0° 03.6ʹ′ and –0° 02.1ʹ′ respectively. b Based solely on averages of measurements of straight segments with only three surveyed points (see Table 1a). c Measurements are from the mastabas in the western half of the cemetery only. d The five measurements from shorter sides (Table 2b) yield a mean DFC in the east-west direction of –0.6°, but the wide scatter of these readings suggests that they merely reflect a broad practice of constructing the shorter walls to be broadly perpendicular to the longer ones, with no great precision. E The figures obtained from our data alone are –0° 04.1ʹ′ and –0° 04.8ʹ′ respectively. f Based on the measurement of a single straight segment with only three surveyed points (see Table 4a). g Data from the skewed south (passage) wall (see Table 4b) have been excluded. h GPMP data Our own data from the three main pyramids, and the comparative analysis with earlier surveys, reveals a level of complexity in determining the intended orientations that is not always acknowledged in discussions of the factors that may have influenced those intended orientations. Certainly, azimuths quoted to second-of-arc precision are completely unjustified, and decimals of a minute of arc may be misleading. Even minute-of-arc precision may be unjustified in the case of Menkaure’s pyramid, given that relevant data can only be obtained at the present time from stone courses, not casing foundations. That said, our data support the widely quoted assertion that the mean orientation of Khafre’s pyramid deviates slightly more from true cardinality than does Khufu’s. Nonetheless, according to our own data, the difference between the north-south orientation of the two pyramids is only 0.5ʹ′. Khufu’s pyramid is some 1.5ʹ′ to 2ʹ′ off perpendicular, much more so than Khafre’s, and the north and south sides are Nell and Ruggles (2012)—Orientations of the Giza pyramids 32 significantly non-parallel. Yet it is the east-west direction that is closer to true cardinality, and our data concur with Lehner and Goodman’s—though not to the earlier surveys—in singling out the south wall as the one closest of all to true cardinality, deviating by no more than half a minute (see Table 8). As we have already noted, the orientations of the Khufu and Khafre pyramids appear to follow a precedent for anti-clockwise orientation already evident during the earlier part of dynasty 4, and likely refined over time so as to be ever closer to true north. Menkaure broke with this tradition. Not only was his pyramid oriented clockwise, but it also deviated significantly more from true cardinality. While this could be because Menkaure’s surveyors were less precise than those of Khufu and Khafre, the fact that his pyramid was constructed no less finely than the others (despite being significantly smaller with its facing never completed) suggests otherwise. What about the ‘bigger picture’ provided by all the other structures on the plateau? Starting with Khufu’s complex, there was a marked difference in orientation between his pyramids—his own and those of his queens—(DFC between –0.3° and 0.0°) and the burial structures of his sons and advisors in the surrounding cemeteries (DFC [WC] ≈ –2.3°; DFC [SC] ≈ –0.6°; DFC [EC] between –0.5° and +1.0°). While this could be due to poorer techniques being applied to ensure the cardinal orientation of the smaller tombs, two factors at least suggest otherwise. The first is a systematic increase in the DFC (orientation change in the clockwise sense) from west to east, the value being c. –3.0° in the case of the westernmost mastabas and c. +1.0° in the case of the easternmost ones. If a similar method was used to align all of these structures, this might give an insight into that method. The second factor is that the successive rows in the eastern cemetery—while generally increasing in DFC from west to east (the DFCs for the first four rows being –0.4°, –0.5°, +0.3° and +1.0°—see Table 2b)—switch from orientation more anticlockwise than Khufu’s pyramids to orientation more clockwise. While it is tempting to suggest that the complete avoidance of the –0.3° to 0.0° range could mean that a deliberate attempt was made to avoid replicating the precise orientation of the larger pyramids, the evidence is too thin to give serious support to this speculation. The available data from Khafre and Menkaure complexes offer a less complete picture. Khafre’s Valley Temple and the Sphinx Temple are both aligned very closely to the north-south meridian, deviating, if anything, clockwise rather than anticlockwise. As far as can be ascertained, Menkaure’s satellite pyramids seem to follow the clockwise orientation pattern of his own pyramid. The alignment of the causeways of Khufu and Khafre, although not that of Menkaure, differ greatly from all other monuments on the Giza Plateau, and may offer some useful insights as a result. Khafre’s causeway, as already discussed, is oriented with a DFC of between +13° and 14°, while that of Khufu is around –14° to –15° (Lauer 1947: 246; Goyon 1969; Lehner 1985b: 118–119). The approximate symmetry with respect to due east-west is noteworthy although not necessarily an end in itself for the builders. The skewing of Khafre’s causeway is more likely to have arisen as a consequence of other considerations, for example ensuring that the axis of the Sphinx temple, the south side of the Sphinx, and the south side of Khafre’s pyramid were in alignment (Lehner 1997: 129–130). 5.2 The question of the primary direction: planning and construction at Giza Lehner (1997: 107) argues that construction on the Giza Plateau was carefully planned, so that it would fulfil the needs of more than one pharaoh, even before Khufu’s pyramid began to be constructed. This idea is supported by the fact that the location of Khufu’s pyramid was as far north as was possible on the Mokkatam limestone ridge, thereby leaving room for the construction of the other two pyramids. It is also supported by various evident alignments among the monuments themselves: for example, the eastern sides of Khafre’s and Menkaure’s Mortuary Temples are aligned respectively with the western sides of Khufu’s and Khafre’s pyramids. Jeffreys (1998) and Magli (2009a) extend this ‘master plan’ concept, suggesting that a diagonal line running through the south-eastern corners of all three main pyramids (the ‘Giza Nell and Ruggles (2012)—Orientations of the Giza pyramids 35 he has observed (pers. comm., 2005), the Sphinx itself is surrounded by a giant cartouche. Khafre’s Valley Temple seems to have had a connection with the day-night cycle, its upper level being associated with daytime and the sun, while the lower level is thought to be associated with the Duat (underworld): twenty- four statues placed there may have represented the 24 hours of the day and night (Ricke 1970; but see Lehner 1997: 126). In view of these likely connections with the sun, solar hypotheses need to be taken into serious account in discussing possible methods of determining structural orientations at Giza alongside, rather than necessarily as an alternative to, sightings upon stars. 5.4 Equinoctial alignment strategies One method of identifying east is to locate the position of the rising or setting sun at the equinox. If we are going to postulate that various structures on the Giza plateau were aligned in this way then we must clarify (a) which ‘equinox’ we are referring to; (b) how the ancient Egyptians could have set out such alignments and what precision is likely to have been achieved; and (c) how likely is it that the equinox was significant to the ancient Egyptians? Regarding the first question, there are at least four different ‘practical’ definitions of the equinox (Ruggles 1997): 1) The spatial midway point of the sun along the horizon between the winter and summer solstices, either in the spring or in the autumn. 2) The midpoint in time between two successive solstices. 3) The day on which the sun rises and sets at exactly opposite points on a level horizon. 4) The 24-hour period in which ‘night’ and ‘day’ (defined as the time between sunset and sunrise and vice versa) are equal in length. Each of these methods will produce somewhat different results owing to varying horizon altitudes and refraction, and (in the case of method 2) the inconstant rate at which the earth travels around the sun. For example, method 2 will typically produce declinations in the range +0.1° to +0.8° (Ruggles 1999: 54–55); which for an eastern horizon altitude of 0° corresponds to the azimuth range 88.7° to 89.5°; each increase of 1° in the altitude will increase the horizon range by some 0.7°, so that for altitude 3° the range will be 90.7° to 91.5°. Method 1 would have involved the construction, say, of a north-south wall at a point where the horizon altitude was the same in the direction of the summer solstice and the winter solstice sunrise. The surveyors would have needed to mark the direction of both winter and summer solstice sunrise on this wall, and then divide the distance between the two into equal parts. While this is not inconceivable, there is no evidence for the existence of such a wall. Method 3 would depend upon the use of gnomons and cords but, critically, would depend upon finding a location where the altitude of both the eastern and western horizons is close to 0°. As Table 12 shows, high horizons to the west seem to rule out the use of this method on the Giza plateau. Method 4 presupposes the use of accurate timing devices. Star clocks date back at least to the 9th dynasty (Claggett 1995: 52) and, according to Parker (1974: 53–55), diagrams of stars inside coffin lids indicate that a system of 12 night hours was in use by at least ca. 2150 BC. Wells (1993; 1996: 36–37) argues that a previous system, recorded in the Book of Gates (a New Kingdom funerary text), used a 24-hour day. Be this as it may, as early as dynasty 5 there were ‘watchers of the hour’ priests (Imyw-wnwt) who kept accurate time at night by sighting and recording stellar decans, so it is not inconceivable that this was done during dynasty 4. Method 2 might seem the most straightforward, since the ‘watchers of the hour’ could have determined when the equinox was near by stellar observations. Each of the 36 decans was represented by a Nell and Ruggles (2012)—Orientations of the Giza pyramids 36 constellation that rose for 10 days—an Egyptian decade—before a new one appeared, with 5 intercalary days completing the 365-day year. Thus the equinox would fall a quarter of a year, about 9 decades and 1–2 days, after the solstice. A major issue with this method would have been to keep track of the date of the solstice in the first place, since it would have moved forward one day in every four years. A full analysis of the efficacy of the four methods, given the numerous cultural and technical factors that are relevant, is beyond the scope of this paper. However, we can conclude broadly, on technical grounds, that the best precision in identifying true east-west that could be achieved by any of these methods is about 0.5° (cf. Belmonte 2001: S2). As a ‘rule of thumb’ we would look for a range of declinations from about 0.0° to +1.0’ if method 2 were used, but about –0.5° to +0.5° in other cases. However, no records have been recovered to indicate that the ancient Egyptians were cognizant of equinoxes or that they held any importance for them. 5.5 Possible astronomical orientation methods used at Giza As we have seen, in the two cases where we have direct evidence—in the form of ‘survey’ postholes—of survey baselines and offsets being set up, the evidence suggests that the primary baseline was oriented north-south. The question we explore in this section is whether the orientations and relative situations of the various structures on the plateau can, in themselves, furnish any clues as to the astronomical method used to determine the orientation of the baselines. The possibilities we shall consider are: • north-south baselines set up using observations of circumpolar stars in the north; • north-south baselines set up using observations of shadows at local noon; and • east-west baselines set up using observations of the rising (or setting) equinoctial sun. We are also mindful that, in all cases, the perpendicular lines could have been set out using geometrical methods and/or set out or corrected using independent astronomical measurements. Belmonte (2001: S1–3) has reviewed the various astronomical methods proposed in the literature to determine a cardinal orientation and concluded that only observations of a pair of circumpolar stars that simultaneously cross the meridian (as proposed by Spence 2000) could explain the extraordinary precision with which this was achieved in the case of the Khufu and Khafre pyramids. Yet Belmonte’s critique of Spence’s choice of stars highlights the fact that there is no clear-cut choice for us as to the candidate pairs of stars that best fit the orientation and dating evidence, let alone the contextual evidence regarding the possible instruments that could have been used to make the observations. This raises the concern that the mere existence of a pair of stars that fit the constraints does not necessarily provide strong proof that this was actually the method used. On the other hand, the fact that there is only a tiny (ca. 1.5ʹ′) difference in DFC between the Khufu and Khafre pyramids argues strongly that such a pair of stars was chosen, and that the same pair was used to orient both pyramids. Furthermore, if the builders’ margins of error could have been no better than ca. 2ʹ′, as Belmonte (2001: S1) has argued, then we cannot rule out from the orientation data the possibility that the two pyramids were, in fact, aligned almost simultaneously. Spence (2000) introduced the idea that the substantial difference in DFC between Menkaure’s pyramid and the other two (some 27ʹ′ between Menkaure and Khafre, and in the opposite sense) could be due to precession. Another possibility is that a different method was used to determine Menkaure’s orientation. The fact that its clockwise DFC is repeated in the burial monuments of Menkaure’s immediate successors, though not with any systematic increase in the DFC (Shepseskaf +0.5°; Userkaf +0.25°; Sahura +1.5°; Neferirkara +0.25°; Belmonte 2012: 184) suggests that a distinct orientation practice was used for several pharaohs from Menkaure onwards, but that it did not involve the consistent use of the same sighting stars. Then there are all the other structures on the plateau. How should their more disparate orientations be explained? In the light of the conclusions of the previous sections, we should explore the possibility that observations of sunrise or sunset could have been important. In Table 12 we present the principal data concerning east-west alignments of structures on the plateau, including the elevation of the structure Nell and Ruggles (2012)—Orientations of the Giza pyramids 37 concerned, the altitude of the horizon, and the declination. Azimuth values are quoted to the nearest 0.1°, while altitude values (measured by EN using a hand-held compass-clinometer) are quoted to the nearest 0.25°.wherever readings were checked several times, 0.5° otherwise. A precision of 0.1° is therefore justified for the declinations, which were calculated using CR’s GETDEC program (Ruggles 1999: 169; still available from www.cliveruggles.net), which takes account of celestial refraction. A mean latitude value of 29.975 was assumed. Elevations, estimated from the published elevations of control points on the GPMP grid (Goodman 2007) and Google Earth, are included to give a sense of the relative heights of the different locations on the plateau:.16 Table 12: East-west alignments: summary data Alignment to east Alignment to west Approx date* Structure Elev (m) Az (°) Alt (°) Dec (°) Notes Az (°) Alt (°) Dec (°) Notes Khufu complex 2587 BC Main pyramid 59m 90.0 +0.25 –0.1 270.0 +5.0 +2.4 2587 BC Mortuary temple 61m 90.0 +0.25 –0.1 a 270.0 +5.0 +2.4 a 2585 BC Queen’s pyramid KQ2 56m 89.7 +0.25 +0.1 a 269.7 +5.0 +2.2 a 2585 BC Western cemetery 69m 87.7 >+0.25 >+1.8 a, b, c 267.7 +4.5 +0.2 a 2535 BC Southern cemetery 61m 89.4 +0.25 +0.4 269.4 +6.0 +2.4 2583 BC EC row 1 (7100) 57m 89.6 +0.25 +0.2 269.6 +6.0 +2.6 2583 BC EC row 2 (7200) 55m 89.5 +0.25 +0.3 269.5 +6.0 +2.5 2583 BC EC row 3 (7300) 53m 90.3 +0.25 –0.4 270.3 +6.0 +3.2 2583 BC EC row 4 (7400) 51m 91.0 +0.25 –1.0 271.0 +6.0 +3.8 Khafre complex 2556 BC Main pyramid 69m 89.9 +0.25 –0.1 b 269.9 +15.0 +7.3 d 2556 BC Satellite pyramid 65m 90.1 +0.25 –0.2 b 270.1 +10.0 +5.0 d 2556 BC Causeway (top) 60m 103.5 +0.25 –11.8 b — — — 2556 BC Causeway (bottom) 22m — — — 283.3 +8.0 +15.4 2556 BC Valley temple 20m 90.5 +0.25 –0.6 e 270.5 +8.0 +4.3 2556 BC Sphinx temple 24m 90.4 +0.25 –0.5 e 270.4 +8.0 +4.3 Menkaure complex 2530 BC Main pyramid 72m 90.4 +0.25 –0.5 270.4 +6.5 +3.5 2530 BC Queen’s pyramid MQ2 67m 90.5 +0.25 –0.6 270.5 +6.5 +3.6 2530 BC Causeway (top) 68m 89.2 +0.25 +0.6 b — — — 2530 BC Causeway (bottom) 20m — — — 269.2 +6.0 +2.2 2530 BC Valley temple 20m 90.2 +0.25 –0.3 270.2 +6.0 +3.1 16 The earliest reference to the relative elevations of the three main pyramids is on a plan given by Vyse (1840: opp. 148). In all copies seen by the authors this information is illegible, but Maragioglio and Rinaldi (1967: 32–35) quote Vyse as saying that the difference in level between the bases of the pyramids of Cheops and Mycerinus is 12.68m and between Chephren and Mycerinus 2.57m. The values quoted in Table 12 are broadly consistent with this. Nell and Ruggles (2012)—Orientations of the Giza pyramids 40 well have been oriented simply by offsetting from the southern side of Khafre’s pyramid, the valley temple must have been independently aligned. It is actually closer to true cardinality than the main pyramid. As we have noted, sunrise observations are too imprecise to explain the extraordinary level of precision achieved in the case of Khufu and Khafre’s pyramids, but it is conceivable that the orientations of three Menkaure structures were all fixed using the rising sun. With the easterly declinations for Menkaure’s pyramids and valley temple lying between –0.6° and –0.3°, it is just possible that equinoctial sunrise could have been used consistently for all three structures, albeit with good fortune in achieving such consistent alignments. It should also be noted that Menkaure’s queens’ pyramids could not have been aligned using northern stars once the main core of Menkaure’s pyramid had been built (and Khufu’s pyramid would also have been in the way). However, it is also possible that the main pyramid and valley temple were aligned using (the same) northern stars, while the queens’ pyramids were aligned by offsetting. But in view of our conclusions at Khufu’s and Khafre’s complexes, perhaps the most plausible explanation is that the orientation of Menkaure’s pyramid was fixed using circumpolar stars, while the valley temple was aligned using sunrise, matching the day when the sun rose in line with the easterly orientation of the main pyramid. It is tempting to suggest that the 0.8° anticlockwise deviation of Menkaure’s causeway from true east-west might be due to its orientation having been determined by equinoctial alignment. However, a lower horizon altitude in the east would fit such an argument better: an altitude of +0.5° would yield a declination of +0.7°, altitude 0.0° would yield declination +0.4°, and altitude –0.25° would yield declination +0.2°. 5.5.4 Khafre’s workmen’s village One other group of structures is associated with Khafre’s pyramid complex: the village of the workmen who built the pyramid (Lehner 2002). This is situated to the SSE of the Sphinx, beyond the temenos wall of the ancient necropolis (‘Wall of the Crow’) and is currently being excavated by the Giza Plateau Mapping Project. The core of the village comprises sets of long narrow galleries running roughly north- south, not quite perpendicular to three east-west streets. Alignment data provided by the GPMP show that the galleries were aligned with azimuths between 356.6°/176.6° and 357.6°/177.6° while the streets were aligned with azimuths between 87.9°/267.9° and 88.9°/268.9°. The elevation of the workmen’s village is similar to that of the Sphinx temple and Khafre’s valley temple, but it is laid out on a different orientation, skewed anticlockwise from cardinality by an average of 1.6° (streets) and 2.9° (galleries). We cannot entirely rule out the possibility that the orientation was determined by observations of the rising sun at the equinox (az = 87.9°, alt = +0.25° => dec = +1.7°; az = 88.9°, alt = +0.25° => dec = +0.8°) although the altitude of the western horizon is too great for the alignment to have been determined by observations of the setting equinoctial sun (az = 267.9°, alt = +8.0° => dec = +2.1°; az = 268.9°, alt = +8.0° => dec = +3.0°). The question of what determined the alignment of the village remains open. 5.5.5 The complex of Khentkawes For completeness, we conclude this section with mention of the temple complex of Khentkawes, unique in being a female ruler during the transition between dynasties 4 and 5. Her tomb (built c. 2494 BC) is located just to the north of Menkaure’s valley temple, with its causeway on the eastern side. The base of Nell and Ruggles (2012)—Orientations of the Giza pyramids 41 the tomb was carved out of the natural rock, and the area around it was quarried out and used for other monuments, in the same way as the area around the Sphinx had been quarried out and used to construct the Sphinx temple and Khafre’s valley temple. Khentkawes’ complex is unique in that it included a town (Lehner 1997: 138). Khentkawes’ tomb and entrance corridor have an orientation pattern markedly different from that of Menkaure’s valley temple, causeway, and pyramid. Alignment data provided by the GPMP show that the ‘east-west’ alignment of the tomb, temple and town were between 82.9°/262.9° and 83.1°/263.1°. This is too far from cardinality for the orientation to have been determined by observations of the rising sun at the equinox: the altitude of the eastern horizon here, again +0.25°, yields a declination between +5.8° and +6.0°. To the west, the deviation overcompensates for the horizon altitude (+8.0°): to obtain a declination in the range –0.5° to +1.0° we would need an azimuth in the range 264.8° to 266.6°. Once again, the question of what determined the alignment remains open. 6 Conclusions The pyramids of Giza have been the focus of academic, religious and spiritual attention for centuries. However, practically all of the focus has been on the pyramids of Khufu and Khafre, with scant attention paid to Menkaure’s pyramid, and little or none to the orientations of the surrounding structures. It has been widely accepted that the three Giza pyramids were aligned to northern circumpolar stars, and that the slight variations between the orientations of these pyramids was due either to precession or to measurement errors. The main aim of this paper has been to re-examine and augment the basic data relating to the orientations of the pyramids of Giza Plateau and their associated monuments. These basic data not only underlie all theories concerning how the extraordinarily precise cardinal orientation of the main pyramids was achieved, but also to enrich the contextual picture concerning the methods used when locating and orienting satellite pyramids, valley temples, other tombs, and even potentially the associated workmen’s villages. As part of our Total Station survey in 2006, we were able to collect extensive new survey data pertaining to the main pyramids, including Menkaure’s, which has not been re-examined since the survey carried out by Petrie in the late 1890s. Given that the definitive baselines used by the original surveyors are not identifiable, our data emphasize the importance of considering such factors as • variations between different levels such as casing foundations and stone courses, • irregularities and overall concavity and convexity in individual sides and walls, and • departures from strict parallelism in opposite sides/walls and perpendicularity in adjacent ones before producing estimates of the actual orientations and how they were achieved at different stages, let alone a single estimate of the ‘intended orientation’. These data, together with the evident variability in the results obtained by the few archaeologists from Petrie onwards who have carried out their own surveys, emphasize the false precision inherent in orientations quoted to the nearest arc second and the caution that is needed if basing detailed astronomical hypotheses on such data. Our data suggest, for example, that there is only a very slight difference in orientation (≈0.5ʹ′) between the north-south axes of Khufu’s and Khafre’s pyramids, that the sides of Khafre’s are more perfectly perpendicular than those of Khufu’s, and that the east-west axis is closer to true cardinality in both cases. Discussions in recent years as to how the ancient Egyptians achieved such extraordinary precision have tended to assume that the astronomical reference was the northern stars, and Spence (2000) and Belmonte (2001) have shown in particular how this could have been achieved using the simultaneous transit of two Nell and Ruggles (2012)—Orientations of the Giza pyramids 42 circumpolar stars. Our results do not directly challenge these ideas but do suggest some possible adjustments, e.g. in Spence’s correlation between orientation and date (2000: fig. 1). However, noting a range of evidence that suggests a strong link between the Giza monuments—and particularly the Sphinx and its temple and the Khafre Valley Temple—to solar cults, we felt it important seriously to consider the possibility that at least some of the temples ands tombs were oriented using observations of sunrise or sunset rather than circumpolar or other stars. A key question was whether observations of the equinox, despite being less precise than many postulated stellar observation strategies, could have played a major role in orienting some of the subsidiary monuments. Given the lack of any evidence that the equinox concept held any importance at all for the ancient Egyptians, we would need strong alignment evidence for this possibility to be taken seriously. Our data do not provide this. On the other hand, our evidence has raised the possibility that while the main pyramids, and very possibly also the valley temples and the Sphinx temple, were aligned upon stellar targets to the north, many of the more minor associated structures, and particularly the many mastabas surrounding the Great Pyramid, were aligned using sunrise or (where this was not possible) sunset, using the already-established eastern axis of the main pyramid to identify the day when the sun was rising directly to the east. In so far as our own data and tentative interpretations have lead to a consistent general picture, it is this. The observation of circumpolar stars in the north remains the most viable way of explaining the precision of cardinal orientation achieved by the pyramids of Khufu and Khafre, and the closeness of their two orientations. The stellar explanation remains the most viable at Menkaure’s pyramid too, but with a significant change in the DFC due to a change in practice (now favouring a clockwise deviation from cardinality), a change in the stars used, or precession over a gap of up to 60 years—or some combination of these factors. In each complex, the satellite pyramids and valley temple (where we have the data) follow the orientation of the main pyramids to within c. 0.3°. It seems most likely that the satellite pyramids were simply aligned to the relevant side of the respective main pyramid, while the orientations of the valley temples were determined independently using similar techniques to the main pyramids. While the orientations of the mastabas surrounding Khufu’s pyramid may simply have been determined by offset, with errors accumulating at further distances from the Great Pyramid, a serious possibility is that those in the eastern cemetery were aligned primarily to the east, by reference to sunrise on the day when the sun was seen to rise in line with the northern or southern side of the Great Pyramid. In the western cemetery, where the eastern horizon was obscured by the Great Pyramid, sunset was used instead. This hypothesis is not without its difficulties: the most serious of these, perhaps, is that fact that the principal (longer) axes of the mastabas are all north-south, suggesting that this was the more important axis. That the orientations of Khafre’s valley temple and the Sphinx temple may have been fixed in a similar way remains an intriguing possibility given the possible solar associations of these temples. A complex picture involving alignment by reference to both the northern stars and sunrise to the east may seem less satisfying than one simply involving observations of northern stars, but it may fit better with the contextual evidence. For example, it is clear from the fragmentary texts and inscriptions that the ancient Egyptians had the necessary tools to align structures to the sun—gnomons, cords, merkhets, beys, plumb bobs, and stakes—and that they were also expert at land measurement, as they needed to re-stake out land annually following the inundation of the Nile. Furthermore, there is textual evidence to suggest that both stellar and sun/shadow observations could form part of the process of orienting a sacred Egyptian building. 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Žába, Zbyněk 1953 L’Orientation Astronomique dans l’Ancienne Égypte, et la Précession de l’Axe du Monde. Prague: Académie Tchécoslovaque des Sciences. Nell and Ruggles (2012)—Orientations of the Giza pyramids 50 FIG. 1(d) use of the bay with a gnomon (after Isler 2001: 172). Nell and Ruggles (2012)—Orientations of the Giza pyramids 51 FIG. 2. Suggested methods for determining cardinal orientations: FIG. 2(a) A Roman stele from Pompeii depicting a Roman groma (after Miranda et al. 2009: fig. 3 left) FIG. 2(b) Seshat’s crown for comparison, as shown in the Temple of Horus at Edfu (photograph by Erin Nell) Nell and Ruggles (2012)—Orientations of the Giza pyramids 52 FIG. 2(c) Neugebauer’s solar orientation suggestion (see text) (after Neugebauer 1980: 1). FIG. 3. The eleven survey stations together with an indication of the principal structures (excluding satellite pyramids and mastabas) surveyed from each of them. Underlying plan after Romer (2007: 15).