Multi-star target model for astronomical orientation of the old kingdom Egyptian pyramids

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Multi-star target model for astronomical orientation of the old kingdom Egyptian pyramids

A. Puchkov

PhD student

Department of World History

Oles Honchar Dnipro National University Dnipro,



During the Pyramid Age, the ancient Egyptians erected some of the most iconic monuments in the world, but their method of alignment and the exact dates of construction remain in dispute. This paper presents new archaeoas- tronomical evidence that both explains the ostensibly erratic orientation of the Old Kingdom pyramids and offers a novel solution to the dating problem. An analysis of the alignment of pyramids built during the 3^rd to 6^th Dynasties reveals that they were not oriented to true north, as expected by one of the prevailing current models, but to prominent stars in the northern circumpolar region. A distinct pattern emerges when the time-dependent position of these stars is compared with the orientation of a series of pyramids whose align- ments are known. The pattern explains all the available azimuth data of the pyramids from Djoser to Unas and predicts older dates of construction for these structures with an accuracy of no more than five years, up or down. In conclusion, the age of the Old Kingdom is approximately two centuries older than conventionally estimated, according to traditional textual reconstruc- tions of Egyptian chronology. These results are consistent with previous ra- diocarbon data obtained from samples collected at known Old Kingdom sites thereby aligning archaeological physical with archaeoastronomical evidence. The Egyptian chronology serves as a standard reference to establish chrono- logies in the entire ancient Near East of the 3^rd millennium BCE. Therefore, the revised chronology based on the findings presented here warrants a fresh look at the historical timelines of other ancient civilizations contemporary with Ancient Egypt. old kingdom pyramid astronomical orientation

Keywords: Old Kingdom pyramids, astronomical orientation, Meskhetiu, imperishable stars, Egyptian chronology, archaeoastronomy, radiocarbon dating



О. В. Пучков

МоДЕЛЬ МУЛЬтИЗІрКоВоЇ ЦІЛІ ДЛя АстроНоМІЧНоЇ орІЄНтАЦІЇ ЄГИПЕтсЬКИХ ПІрАМІД ДАВНЬоГо ЦАрстВА

В епоху пірамід єгиптяни спорудили деякі з найбільш знакових па- м'ятників у світі, але їх метод вирівнювання та точні дати будівництва залишаються предметом суперечок. Ця стаття представляє нові архео- астрономічні докази, які пояснюють як нібито нерегулярну орієнтацію пірамід Давнього царства, так і пропонують нове вирішення проблеми датування. Аналіз вирівнювання пірамід, побудованих під час 3-6 ди- настій, показує, що вони були орієнтовані не на північ, як передбачається однією з панівних сучасних моделей, а на видатні зірки в північному циркумполярному регіоні. Чітка закономірність виявляється, коли за- лежне від часу положення цих зірок порівнюється з орієнтацією ряду пірамід, вирівнювання яких відомо. Шаблон пояснює всі наявні азиму- тальні дані пірамід від Джосера до Унаса та передбачає більш давні дати будівництва цих споруд із похибкою, що не перевищує п'ять років. Під- сумовуючи, вік Давнього царства приблизно на два століття старший, ніж традиційно вважається згідно з наявними текстовими реконструк- ціями єгипетської хронології. Ці результати є сумісними з радіовугле- цевими даними, отриманими зі зразків, зібраних у відомих спорудах Давнього царства, таким чином узгоджуючи археологічні дані з архео- астрономічними доказами. Єгипетська хронологія служить стандартом для встановлення регіональних хронологій на всьому стародавньому Близькому Сході III тис. до н. е., отже, переглянута хронологія, заснова- на на представлених тут висновках, вимагає нового погляду на історичні шкали інших цивілізацій, сучасних Стародавньому Єгипту.

Ключові слова: піраміди Давнього царства, астрономічна орієнта- ція, Месхетіу, нетлінні зірки, єгипетська хронологія, археоастрономія



Introduction

Several of the Old Kingdom Egyptian pyramids are oriented to the cardinal points with striking accuracy. This is exemplified by the two big pyramids of Giza - those attributed to Khufu and Khafre, the foundations of which deviate from the meridian line by no more than 3? and 5?, respectively. These values are near those the human eye can resolve. What method of orientation to the cardinal directions did the pyramid builders use to achieve such impressive results? Over the past century and a half, researchers have proposed various hypotheti- cal methods¹ to explain this feat. These methods largely fall into two groups:

1) “True north” methods, the accuracy of which depends only on the quality of the observations:

? Observing the elongations of the orbit of a circumpolar star [Petrie 1883, 211-212; Edwards 1947, 209-211]²;

? Observing the meridian transit of a circumpolar star [Romieu 1902, 135-142];

? Observing the shortest shadow produced by a gnomon [Zin- ner 1931, 1-32];

? Observing the rising and setting position of the sun [Gallo 1998, 77-90].

2) “Precession-susceptible” methods, the results of which contain an additional error because of the precession of the Earth's axis:

? Observing the rising position of a star [Haack 1984, 119- 125];

? Observing the vertical alignment of a pair of circumpolar stars [Polбk 1952, 219-220; Spence 2000, 320-324; Bel- monte 2001, S11-S15];

? Observing the horizontal alignment of a pair of circumpolar stars [Rawlins, Pickering 2001, 699].

The first group of methods was favored by scholars until 1984, when S. Haack [1984, 119-125] discovered that the azimuths of pyramids from the 4^th and 5^th Dynasties tended to vary with time; “precession-susceptible” methods have been proposed to explain this trend. However, no proposal from either category could explain the entire set of available azimuth data of all those pyramids for which accurate measurements exist. The fundamental weakness of the methods from the “true north” group is that the expected ran- dom orientation errors would not form a distinctly systematic trend3 (Fig. 1). The methods from the “precession-susceptible” group suf- fer from exceptions to the rule they attempt to establish, unable to account for data collected from the pyramids of Djedefre, Unas and those from the 3^rd Dynasty. Despite significant differences otherwise, both groups of methods have in common the assumption that the goal of the ancient Egyptians was to orient the monuments to true north⁴. Any deviation from due north, consequently, must be the re- sult of ancient surveying errors (first group), or ignoring precession (second group).

Fig. 1. Moreover, the azimuths of two trend-forming post-Khufu pyramids change sign to confirm the trend. ⁴ “All plausible methods of stellar orientation involve establishing the di- rection of true north, either through bisecting observed positions of a cir- cumpolar or near-circumpolar star, or through alignments directly to cir- cumpolar stars. […] Establishing north must thus be considered the goal of the act of celestial alignment, regardless of the method used […]” [Spence 2010, 173].

Fig. 1. Average azimuths of the 4^th - 5^th Dynasty pyramids. Data from Table 2 (see below).

However, some pyramids' orientations are incompatible with the predictions made within the framework of this key assumption that ancient surveyors were fixated on true north:

1) The pyramid of Teti is rotated by almost 10° relative to the cardinal directions, while the 30-year older pyramid of Unas merely one kilometer to the south in the necropolis of Saqqara follows the meridian line almost exactly.

2) The pyramid of the 3^rd Dynasty king Djoser deviates circa 3° from true north. Thus, it represents the first large Egyptian struc- ture known to have been oriented to the cardinal points. Howe- ver, an explanation is lacking as to why subsequent rulers like Sekhemkhet and Khaba built pyramids that significantly devia- te from cardinality by circa 11° and 8-9°, respectively⁵.

3) Of the 4^th Dynasty pyramids, that of Djedefre deviates the most from cardinality, some 10 times more than the pyramids of Khufu and Khafre (47? vs. 3? and 5?), built before and after, re- spectively (Fig. 1).

In the first two of the listed cases the deviations in the azimuths from due north are so great that attributing them to observational er- rors seems incorrect. An alternative explanation - that only some pyramids were oriented to true north while others were aimed at dif- ferent targets - necessitates a new look at the subject of ancient survey methods to ask if prior models made false assumptions. Therefore, a reexamination of the expanded data set including new azimuth data of Old Kingdom pyramids is needed.

Analyzing the data

To date, all existing 4^th Dynasty pyramids have been carefully mea- sured. The three Giza pyramids have been scrutinized the most. The situation is less clear concerning monuments of the 5^th Dynasty: azi- muth data exist for three of these seven pyramids, while the data for two (Sahure and Neferirkare) are not accurate enough due to the poor state of their bases. For the three pyramids of the 3^rd Dynasty, the data have been rounded to whole degrees, as they have never been mea- sured to higher accuracy. Data for the pyramids from the 6^th Dynasty are missing. Table 1 shows a compilation of the known azimuths of thirteen Old Kingdom pyramids.

Pyramid	Azimuth (N side), arcmin.	Azimuth (E side), arcmin.	Azimuth (S side), arcmin.	Azimuth (W side), arcmin.	Azimuth (sides), arcmin.	Azimuth (passage), arcmin.
Djoser6					+180 ± 30
Sekhemkhet7					-660 ± 30
Khaba8					-510 ± 30

Table 7]. All the listed sources except of the second give the azimuth value of -11° (-660?). The second source gives -11.5°. ⁸ See: [Lehner 1996, 510; Romer 2007, 279; Nell, Ruggles 2014, 329, Table 7]. All the listed sources give the range of values from -9° to -8° (-510? on average).

Meidum9	-35.4 ± 1.0	-20.6 ± 1.0	-23.6 ± 1.0	-18.1 ± 1.0		-21.6 ± 1.0
Bent10	-07.5 ± 0.3	-17.3 ± 0.3	-04.2 ± 0.3	-11.8 ± 0.3		-01.0 ± 0.3
Red11	-	-08.7 ± 0.3	-	-		+02.9 ± 0.3
Khufu12	-03.6 ± 0.3	-03.4 ± 0.3	-00.5 ± 0.3	-03.7 ± 0.3		-03.7 ± 0.3
Djedefre13	-51.7 ± 1.0	-43.9 ± 1.0	-48.4 ± 1.0	-50.8 ± 1.0		? (-20…- 30)
Khafre14	-03.8 ± 0.3	-04.0 ± 0.3	-05.8 ± 0.3	-04.2 ± 0.3		-05.6 ± 0.3
Menkaure15	+16.8 ± 1.0	+12.4 ± 1.0	+13.0 ± 1.0	-		+13.3 ± 1.0
Sahure16					-23 ± 10
Neferirkare17					+30 ± 10
Unas18	+17.8 ± 0.3	+17.1 ± 0.3	+17.5 ± 0.3	+17.4 ± 0.3

Table 1. Azimuth data on the sides and descending passages of the measured Old Kingdom pyramids. Error margins refer to the accuracy of measure- ments (±0.3? is allowed for recent measurements taken with a meridian- seeking theodolite ^(10,11); ±1.0? - for measurements with a less accurate the- odolite ^(9,13,15); ±10? - for pyramids the orientation of which was calculated from figures in excavation reports ^(16,17); ±30? - for pyramids the orientation of which was reported rounded to degrees ^(6,7,8)).

All plausible¹⁹ orientation methods from the above list imply the use of circumpolar stars to determine the N-S axis of the structure under construction. Therefore, either (a) the East side, or (b) the West side, or (c) the axis of the descending passage would have been initial- ly oriented during an astronomically themed ritual, and, afterwards, all other sides relative to it using geometry. We have no information about which option (a), (b) or (c) took place. Since there are more data from the bases than on the descending passages (13 vs. 6 rows in Ta- ble 1), the average azimuth of the East and West sides (or the average azimuth of all sides in the absence of specific values) will hereafter be used in all computations²⁰.

Pyramid	Dyn.	Avg. azimuth, arcmin.	Accession date, BCE	Start of construction date, BCE
Djoser	3rd	+180 ± 30	2678 [+11]	2677
Sekhemkhet	3rd	-660 ± 30	2659 [+11]	2658
Khaba	3rd	-510 ± 30	2651 [+11]	2650
Meidum	4th	-19.3 ± 1.0	2624 [+11]	2623
Bent	4th	-14.6 ± 0.3	-	2613
Red	4th	-08.7 ± 1.0	-	2604
Khufu	4th	-03.6 ± 0.3	2589	2588
Djedefre	4th	-47.3 ± 1.0	2566	2565

claim that there is no trend in the descending passages' data, since the Bent and Red pyramids, the passages of which are misaligned with the bases (Table 1), deviate from it. The other accurately measured pyramids (Khufu, Khafre, Menkaure, Meidum) have descending passages co-aligned with their bases, hence the misalignment in two Sneferu's pyramids could not be the goal of the builders, and may indicate either their mistake (Bent and Red pyramids were built successively), or a survey error (both values refer to Dorner's sur- veys). The assumption that the orientation of the passages of these two pyra- mids reflect the plan of the builders, while the bases are mistakenly rotated by almost the same value (-9? and -12? respectively), is unlikely, since in this case the sides' data would “accidentally” confirm the trend given by the other pyramids.

Khafre	4th	-04.1 ± 0.3	2558	2557
Menkaure	4th	+14.1 ± 1.0	2532	2531
Sahure	5th	-23 ± 10	2487	2486
Neferirkare	5th	+30 ± 10	2475	2474
Unas	5th	+17.3 ± 0.3	2375	2374

Table 2. Average azimuths of the E-W sides (or the average azimuth of all sides) of Old Kingdom pyramids and corresponding dates. The accession dates are from Shaw's [2000, 482] chronology²¹ except for the length of Sneferu's reign, for which the middle estimate of 35 years²² is used (differ- ences from Shaw's dates are in square brackets). The dates of construction of Sneferu's pyramids are calculated using Stadelmann's proportion²³. The start of construction date is defined as the year following the date of acces- sion.

Upon initial inspection (Fig. 2), the data belonging to the 3^rd Dy- nasty pyramids appear non-contributory because of the large error margins and could be excluded from the analysis. Nevertheless, these very data are most interesting, as they fail to confirm the expected pattern due to their great azimuthal deviations.

Fig. 2. Average azimuths of Old Kingdom pyramids relative to construction dates. Data from Table 2.

The pyramids examined fall into two groups: a) 3^rd Dynasty pyra- mids with large deviations from true north, and b) 4^th - 5^th Dynasty py- ramids whose orientation deviates from north by no more than 50 arcminutes. The groups are separated by a time interval of a few de- cades that elapsed between the establishment of the pyramid of Khaba and the pyramid of Meidum. Therefore, either the orientation method employed by Egyptians changed drastically in this relatively short time span, or two entirely different methods were used for the two groups.

The distribution of the azimuth data (Fig. 1, Fig. 2) permits the following conclusions:

1) The trend-forming pyramids, with one exception (Neferirkare), belong to the 4^th Dynasty. For reasons unknown, the pyramids of Djedefre and Khafre do not follow the general trend of the 4^th Dynasty pyramids.

2) The excellent alignment of the data with the trend line (errors within ±3?; Fig. 1) indicates that: a) the orientation method used was both accurate and precise, and b) the very presence of the azimuth trend serves as an unambiguous indication of the gen- eral correctness of the Turin King List data on the sequence of kings and reign lengths for the Old Kingdom. The azimuths would not form such a distinct trend if this royal reign se- quence were incorrect.

3) The fact that the data appear to track precession suggests that astronomical observations for each of the trend-forming pyra- mids were carried out at a specific moment in time correspond- ing to a short-term recurring celestial event, such as a prominent configuration of two or more celestial objects (for example, their vertical or horizontal alignment).

4) The intersection of the trend line with the x-axis indicates that the celestial object used as a reference point at a certain sky position crossed the celestial meridian due to precessional drift a little more than a decade²⁴ after the pyramid of Khufu was es- tablished.

5) The gradient of the trend line (the angle it forms with the x-ax- is) characterizes the rate of the precessional drift of the refer- ence object. The direction and rate of the precessional drift of circumpolar stars when observed during prominent configura- tions of two or more celestial objects are not the same for dif- ferent positions of the celestial sphere. For Egypt, the rate is in the range of circa ±24??/year. The gradient of the observed trend line of circa +21??/year allows us to limit the range of suitable sky positions on which orienting survey ritual of the pyramids could have fixated. All other positions can be excluded from consideration (Fig. 3).

(Table 2) and the date corresponding to the intersection point of the trend line with the x-axis in Fig. 1.

Fig. 3. The northern starry sky as seen from the Giza Plateau at the end of the 4^th Dynasty. The rate of precessional drift of circumpolar stars is circa +24??/y for the sky position shown (circa 0??/y for a 90° rotation of the celestial sphere; circa -24??/y for a 180° rotation, etc.). The gradient of the trend line in Fig. 1 limits suitable sky positions to only those when the meridian was inside the gray sectors (light gray sectors below the Pole only), which ap- proximately correspond to the rates of precessional drift of circumpolar stars in the range of +21??/y ± 10%. The range of ±10% accounts for the possibili- ty of minor inaccuracies in the chronological data. Stars are shifting relative to the gray sectors over time due to the precession of the Earth's axis. (Adapted from Stellarium 0.18.2).

Unfortunately, the reference object for the single discovered trend cannot be reasonably chosen, since any prominent star from the gray sectors²⁵ could have caused the observed trend in the azimuth data. In the case of a single trend an unequivocal choice cannot be made at all, and any proposal runs the risk of only reflecting subjective biases as to orientation method, as to a specific star (or star pair) used as re- ference point, and as to the absolute dates of the 4^th Dynasty. Howe- ver, the pyramid of Djedefre, cannot be explained with “precession- susceptible” models as an outlier to the trend line, provides a unique opportunity to solve this mystery. The high-resolution graph (Fig. 4) shows that the azimuths of the pyramids of Djedefre, Sahure and Unas align to make a separate trend²⁶ with a gradient similar to that of the main trend. Random similarity of the gradients being unlikely since two (Djedefre, Unas) out of the three pyramids forming the sec- ond trend were accurately measured.

Fig. 4. Two groups of pyramids form two trends with virtually identical gra- dients (about +21??/year for the main trend line (Meidum to Neferirkare); about +20??/year for the second trend line (Djedefre to Unas)). The data are based on Table 2.

The presence of two trends indicates that two different stars (or star pairs) were used as reference objects. The similar gradients of the two trend lines suggest that the orientation procedures for both groups of monuments were carried out in the same position of the celestial sphere, since the rates of precessional drift of circumpolar stars in a certain position of the sky differ very slightly. Therefore, Egyptian surveyors aimed for a specific sky position to orient all the pyramids examined belonging to the 4^th - 5^th Dynasties, except Khafre (see be- low). Both trend lines intersect the x-axis at time points 149 years apart. This represents the time interval separating two consecutive crossings of the celestial meridian by two different reference objects observed in one and the same position of the sky. Based on azimuth data alone these two reference objects cannot be identified, thus clues must be sought in ancient sources.

To summarize, there are parallel trends in the azimuth data, indi- cating the existence of a special sky position in which the 4^th - 5^th Dy- nasty pyramids were oriented using two different reference objects. The similar gradients of these trends make it possible to focus on a narrow range of suitable sky positions where this might have been.

Special position of the sky

Ancient texts or illustrations that describe the process of orienting Old Kingdom pyramids have not yet been found. Inscriptions carved onto temple walls of later periods indicate that the ancient Egyptians carried out a foundation ceremony called “stretching of the cord”²⁷, during which the king and the goddess Seshat ritually set the four corners of the temple based on astronomical observations coupled with unknown manual operations. This ceremony appears to have had a more ancient²⁸, possibly pre-dynastic, origin predating these written records by more than two thousand years. No other foundation rituals are known yet. Thus, it is possible that an identical, or similar, ritual was used to lay the foundations of both temples and pyramids. An analysis of the “stretching of the cord” ceremony is therefore in order. The most complete descriptions accompanying the images of the foundation ritual are written on Ptolemaic period Egyptian temple walls - the Temple of Horus at Edfu and the Temple of Hathor at

Dendera. They contain the following account²⁹:љsp.n.j nb(At) Am(m).j tp sms ?fa.j ?Aj ?na SљAt stj.j ?r.j r nmt(t) an?w sa?.j mAtj.j r Ms?t(jw) skj-a?aw r-gs mr?t.f smn.j ?ss 4 nw ?wt-n?r.k

I have taken the pole; I grip the handle of the mallet; I grasp the measuring cord with Seshat. I turn my sight [lit. throw my face] according to the movement of the stars and I allow my eyes to en- ter into Meskhetiu. The-one-who-lets-the-lifetime-go-by³⁰ is beside his merkhet³¹. I establish the 4 corners of your temple³².

?fa.n.j nb(At) ?na tp sms Am(m).j ?Aj [?na] SљAt dgj.j ?pt r nmt(t) n an?w sb?.n.j m Ms?t(jw) nwj skj-a?aw jp mr?t smn.j ?ss 4 nw ?wt-n?r.k

I have grasped the pole and the handle of the mallet; I grip the measuring cord with Seshat. I observe the course of the move- ment of the stars. I have seen (the Gods of) Meskhetiu. I am [lit. I belong to] the-one-who-lets-the-lifetime-go-by [who] measures [with] merkhet. I establish the 4 corners of your temple³³.

dgj m (p)t r nmt(t) an?w sjA ?ns n Ms?t(jw) smn ?ss nw ?wt-n?r

[The king], observing the sky according to the movement of the stars and recognizing the path of Meskhetiu, establishes the cor- ners of the temple³⁴.

Two important elements in these passages are relevant here:

? If the ancient observer followed the movement of the stars, as- tronomical observations were time-consuming. The observer was waiting for a preselected sky position, started the observa- tions well in advance so as not to miss the right moment. This element confirms the existence of a special position of the sky used to orient the pyramids, predicted by the two trends that emerge from the azimuth data in Fig. 4.

? The object of observation³⁵ was the Meskhetiu (Ms?tjw) aste- rism attested with great certainty corresponding to what is to- day known as the Big Dipper (or Plough) asterism, part of the constellation of Ursa Major [Neugebauer, Parker 1969, 183; Parker 1974, 60]. During the Old Kingdom, the Big Dipper was a circumpolar asterism at the latitude of Egypt with all its com- ponent stars visible throughout the year in a configuration de- pendent on the season and time of day.

C c); Brugsch 1880b, 623]. Image: [Cauville 2007, Pl. 60]; see also [Dьmi- chen 1877, Pl. L].

³⁵ Although the foundation ritual already existed in Early Dynastic times, the mentions of the object of observation known to us belong only to the de- tailed accompanying texts of the Ptolemaic period, since the Egyptians usu- ally omitted details and certain scenes in the descriptions [Karkowski 2016, 112]. The obvious symbolic connection of the seven-petalled symbol of the goddess Seshat with the seven stars of Meskhetiu (“Usually Seshat was por- trayed with a seven-pointed star (although some have likened it to a seven- petaled flower) […] It is certain the Egyptians associated the number seven with the Big Dipper because several portrayals of Meskhetiu - at Dendera, Edfu, Esna, and Philae - surround the picture of the bull's leg with seven stars.” [Krupp 1983, 25]) may indicate that this asterism was the object of astronomical observations throughout the entire time of the usage of the foundation ritual (the depictions of Seshat's symbol date back to the 3^rd Dy- nasty or earlier [Magdolen 2005, 197-205]).

The descriptions given do not mention a particular sky position, nor do they narrow the range of suitable ones, because this circumpo- lar asterism was visible in all its possible positions. They merely mention that the Big Dipper was used to identify it. This does not sur- prise in light of the fact that this asterism played an important role in the mortuary beliefs of the dynastic Egyptians [Nemes 2020; Thuault 2020; Arquier 2020].

However, there are two passages that contain more detail regar- ding the observations:

p? љsr m nhm r?j ?r m a?A Ms?t(jw) s?d ?wt-n?r … mj wn jm ?r-bA?

[The king] stretches the cord in joy, gives the face m a?A Meskhetiu,

and establishes the temple … as before³⁶.

mAA pt sb? an?w r?j ?r m a?A Ms?t(jw)

[The king] … looks at the sky and sees the stars, gives the face m a?A Meskhetiu³⁷.

There are a few interpretations in the literature of the term a?A in the above passages:

? In 1877, the German Egyptologist J. Dьmichen, when discus- sing inscription IV, cites the British Egyptologist P. Le Page Renouf's interpretation of the related term r-a?A ib used in the stellar registers of the Ramesside star clocks. Le Page Renouf was the first to suggest that r-a?A ib denotes a meridian transit or the culmination of a celestial object³⁸. Dьmichen extended this interpretation to the discussed inscription³⁹.

? In 1880, the German Egyptologist H. Brugsch translated a?A of

Meskhetiu as culmination of Ursa Major⁴⁰.

? In 1953, the Czech Egyptologist Z. Ћбba, when discussing in- scriptions IV and V, identified m a?A as the compound prepo- sition m-a?A and translated it as “to/toward” (r?j ?r m-a?A Ms?tjw = turns face to/toward Meskhetiu)⁴¹.

? In 1983, the American astronomer E. Krupp assumed that a?A of Meskhetiu is most likely a particular position of the Big Dip- per⁴².

at the moment of its transit or culmination. The technical expression for this in the Egyptian Calendar now before us is er вk [r-a?A ib], literally `in the middle' ” [Le Page Renouf 1874, 401-402]. According to the modern point of view, the r-a?A ib position in the Ramesside star clocks means “opposite [/in front of] the heart” (ib - heart, center). Since the observer was sitting facing exactly south, the celestial meridian was accurately in front of him and therefore the star in the r-a?A ib position was at its culmination. This in- terpretation is generally accepted to this day [Neugebauer, Parker 1964, ix; Clagett 1995, 61; Leitz 1995, 120; Depuydt 1998, 32]. As for the rest of the positions in the Ramesside star clocks, all iAbj positions are passed before the culmination (iAbj - left, iAbtj - eastern), and all wnmj positions - after it (wnmj, imn - right, imntj - western). Thus, “left” and “right” are used from the standpoint of the observer, sitting facing south, not the target figure, sit- ting facing north, as some scholars suggest.

? In 2001-2008, a group of researchers suggested that a?A of Meskhetiu denotes a specific star of the Big Dipper asterism, however, their interpretation is based on a misreading of this term⁴³.

It should be noted that m a?A in the inscriptions IV and V can be trans- lated in two ways: either a) as the compound preposition m-a?A or b) as the primary preposition m and the noun a?A. Let's look at both cases:

a) The compound preposition m-a?A should be translated as “oppo- site”, “in front of” [Erman, Grapow 1926 (Wb. I), 233, 18-19]. The phrase r?j ?r m-a?A is not attested in the Egyptian texts. Ap- plying this information, we get: “[the king] gives the face oppo- site Meskhetiu”, that is, the observer faces the asterism. Since there is a shorter standard phrase to express the same meaning - r?j ?r r (= give face to/toward something), it is not clear why the ancient carver used the unusual phrase with the discussed com- pound preposition instead of the primary one - r.

b) The noun a?A should be translated as “accuracy”, “correctness”, “straightness” or, possibly, “equality”, “level”⁴⁴. The phrase r?j ?r m is attested in the Egyptian texts with the meaning “pay at- tention to” [Erman, Grapow 1929 (Wb. III), 126, 15]⁴⁵. This information gives us the following translation: “[the king] pays attention to⁴⁶ the accuracy of Meskhetiu”, that is, the observer notices a certain configuration of the asterism which can be characterized as accurate, straight or, possibly, equal.

Based on the texts only, it is impossible to make a definitive choice in favor of one of the two translations. Although the second option is more helpful in the context of this investigation, to date, no information is at hand to know which configuration of Meskhetiu might have seemed accurate, straight or equal to the Egyptians. Since the analysis of the texts accompanying the images of the orientation ceremony yields no further clues, iconographic sources may help.

The earliest known depictions of the Meskhetiu asterism are found on the 9^th - 12^th Dynasty coffin lids from Asyut [Neugebauer, Parker 1960, Pl. 1-29; Pogo 1932, Pl. A-F] (for example, Fig. 5). The inner side of the lids contain diagrams of the diagonal star clocks, where Meskhetiu is depicted in the company of the three deities - Nut, Sah and Sopdet and appears as a bull's foreleg containing seven stars. The central text field of offerings divides the sky, represented by the inner surface of the lid, into two halves (the northern one with Nut and Meskhetiu, and the south- ern one with Sah and Sopdet), which are the prototypes of the northern and southern panels making up the vaults of New Kingdom tombs.

If we consider both halves as schematic representations of the sky, then the southern one appears unrealistic: Sah should be lo- cated in the southern sky, west of Sopdet, not above it. Thus, the rela- tive position and orientation of the deities cannot be regarded as realistic with respect to what was observed in the sky. Therefore, the position of Meskhetiu on the coffin lids does not contain the informa- tion needed, and, most likely, is caused by the design features.

Several centuries later, Meskhetiu appears on the astronomical ceilings of New Kingdom tombs (pictures in chronological order: Fig. 6; Fig. 7; Suppl. Materials, Fig. SM1-SM3).

Fig. 6. Part of the northern panel on the ceiling in the tomb of Senenmut (18^th Dynasty). (After [Wilkinson 1991, Fig. 1]; see also [Pogo 1930, Pl. B-G]). The picture of a bull is accompanied by the text Ms?tjw.

Unlike the coffin lids, in these paintings the deities have different orientations, i.e., vertical, horizontal, and diagonal, thus providing in- formation on the relative positions of asterisms in the sky as the Egyp- tians imagined them. Meskhetiu is represented in the paintings either as an ovoid bull (Senenmut family; Fig. 6) or a whole bull (Seti I fa- mily; Fig. 7). In both traditions it is oriented horizontally in the upper part of the pictures with its head to the left and tip, or tail, to the right.



Fig. 7. Part of the northern panel on the ceiling in the tomb of Seti I in the Valley of the Kings (19^th Dynasty). (After [Wilkinson 1991, Fig. 2]). The picture of a bull is accompanied by the text Ms?tjw.

Comparison of the depictions of the Meskhetiu asterism and two objects associated with it by the Egyptians and used in burial rituals - the foreleg of the sacrificial bull and the ceremonial meskhetiu-adze - demonstrated an interesting pattern detected by J. Relke and A. Er- nest [2003, 64-80, Fig. 5-7, Fig. 9]: a) pictures of the Meskhetiu-bull on the astronomical ceilings depict it in the horizontal position with the tip or tail to the right; b) pictures of the bull's foreleg as a funer- ary offering to Osiris depict it in the horizontal position with the hoof to the right; c) pictures of the ceremonial meskhetiu-adze in the “opening of the mouth” ceremony mostly depict it in the horizontal position with the handle to the right. In all the listed cases, a bull's foreleg, an ovoid bull, a whole bull, or a ceremonial adze were de- picted in a horizontal position with their wide part (head, bowl) to the left, and the tapered part (hoof, tip, tail, handle) to the right. This corresponds to the orientation of the Meskhetiu asterism during the 3^rd - 2^nd millennium BCE at upper culmination - horizontally above the Pole with the bowl to the left and the handle to the right (see be- low). Thus, the upper horizontal position of the bull in the New King- dom paintings is equivalent to the upper horizontal position of the as- terism during its upper culmination⁴⁸.

In general, an asterism is visible best during its upper culmination, and this highest position is uniquely distinct from all other possible ones. Therefore, the ancient Egyptians may have thought of Meskhe- tiu to be in the most important, possibly sacred⁴⁹ position, which was highlighted in the funerary iconography for this very reason.

It is important that during upper culminations of the Big Dipper in the Old Kingdom epoch, the celestial meridian was inside the right dark gray sector shown in Fig. 3. Consequently, the direction and rate of the precessional drift of stars in this position of the sky were suita- ble for explaining the observed trends in the azimuth data of the pyra- mids in Fig. 4. Thus, the iconographic sources allow us to exclude the left dark gray sector in Fig. 3 (as well as the right light gray sector) from the range of candidate sky positions.

However, the culmination of an asterism, as opposed to the culmi- nation of a star, is a long-term event. By contrast, the excellent align- ment of the azimuth data with the trend lines indicates that a short-term recurring celestial event must have been used to orient the pyramids. The question arises therefore, which prominent configura- tion of the stars of Meskhetiu, viewable as a short-term event, took place during its upper culmination and attracted the attention of the Old Kingdom Egyptians? In the Seti I tradition the bull is depicted standing on a horizontal platform (Fig. 7) and seems to be balancing relative to the vertical protrusion, while the whole structure resembles balanced scales⁵⁰. This “balanced” position matches with the above- mentioned epithets “accurate”, “straight”, “equal” or “(scales are) point, his head tilted sharply forward. The arrangement of the seven stars of the Thigh is known to [...] us from the coffins of Asyut; the development of the Thigh to the ovoid bull (head to the left, tip to the right) in the Senenmut and Ramesseum representation, to the whole bull (head to the left) in the tombs of Seti I and the 20^th dynasty leaves no doubt that the left-facing Mascheti [Meskhetiu] is shown in the upper - not in the lower - culmina- tion.” [Pogo 1931, 108]. See also [Pogo 1930, 308-311].

level”. Significantly, a?Ajt - a feminine noun with the a?A root - is translated as “true balancing” [Faulkner 1991, 50]⁵¹.

In consequence of the foregoing, what might this “balanced” Big Dipper have looked like in the sky? Due to the elongated shape of this asterism and the location of its outer stars Dubhe (б UMa) and Alkaid (з UMa), relative to the Pole in the Old Kingdom epoch (they were roughly at the same distance from it), there was a special posi- tion of the asterism during its upper culmination when these two stars were at the same altitude, i.e., were aligned horizontally above the Pole (Fig. 8). In this position, not two, but three brightest stars of the asterism, Dubhe, Alioth, and Alkaid aligned horizontally, and the Big Dipper appeared “balanced” relative to its imagined center, Alioth (е UMa), while the two outer stars represented the balanced scales.

This unique, “balanced” position of the Meskhetiu asterism a) fits well with the depictions of the horizontally located bull, foreleg and adze, because at this moment the asterism extended above the Pole strictly along an imaginary horizontal line with its wide part to the left and tapered part to the right; b) explains why the bull in the Seti I tradition (Fig. 7) is depicted balancing on the stylized scales;

c) matches well with the epithets “accurate”, “straight”, “equal” or “(scales are) level” from the descriptions of the orientation ceremony;

d) is suitable for explaining the azimuth trends in Fig. 4, because du- ring the horizontal establishment of the star pair Dubhe-Alkaid, the celestial meridian was close to the left edge of the right dark gray sector in Fig. 3; e) could be accurately identified in the “stretching of the cord” ceremony using two poles and a cord⁵². There are no other prominent positions of the asterism's stars satisfying the criteria.

To summarize, the analysis of textual and pictorial sources⁵³ sug- gests that the sacred, sought-after position of the sky was the horizontal alignment of Big Dipper's outer stars - Dubhe and Alkaid (Fig. 8). This alignment, to which the “stretching of the cord” rite was targe- ted, “balanced” the asterism for a short moment, equal to a short-term celestial event. It thus served as a distinct time marker to determine the azimuth of a reference object using the merkhet. Since the merkhet was a small instrument, the target in the orientation procedure was most likely a single circumpolar star. Having determined the exact sky position and reference moment, candidate stars that may explain the two observable azimuth trends can now be identified.

The synchronism

The two trend lines in Fig. 4 intersect the x-axis. This means that the corresponding reference stars crossed the celestial meridian when observed in the special position of the sky. Since the position of the celestial sphere during the orientation procedure has now become ap- parent, it is possible to determine the absolute dates when prominent circumpolar stars crossed the celestial meridian in this fixed sky posi- tion.

Star	Apparent magni- tude, m	Year of cros- sing the me- ridian, BCE	Interval between two cros- sings, years
			Thuban	Alioth	10 Dra	Pherkad
Thuban (б Dra)	3.67	2800	-	153	222	484
Alioth (е UMa)	1.76	2647	-	-	69	331
10 Dra	4.58	2578	-	-	-	262
Pherkad (г UMi)	3.00	2316	-	-	-	-

Table 3. Complete list of circumpolar stars (m < 5) that crossed the celestial meridian in the range of 3000-2200 BCE (2600 ± 400 BCE) when observed in the “balanced” position of Meskhetiu (horizontal alignment of Dubhe- Alkaid). Data from Sky Charts 4.2.1⁵⁴.

As mentioned above, the two trend lines cross the x-axis 149 years apart. Hence, of all the combinations of star pairs, Thuban and Alioth, which crossed the celestial meridian 153 years apart, are the best fit⁵⁵. The graph in Fig. 9 illustrates the azimuths of the two se- lected stars, and the others of the Big Dipper, at the “balanced” posi- tion of Meskhetiu over time. This graph spans the time range when the two candidate stars precessionally drifted across the celestial me- ridian.

Fig. 9. Data on time-dependent changes in azimuths of the seven stars of the Big Dipper, and Thuban. All azimuth data here, and in the following graphs, correspond to observations at the Giza Plateau. Insignificant differences in the geographical coordinates of the pyramids can be neglected. Data from Sky Charts 4.2.1.

The graphs in Fig. 2 and Fig. 9 can be combined⁵⁶ to match the two trend lines in the pyramids' azimuth data with those two lines that correspond to the precessional drift of the candidate stars Thuban and Alioth (Fig. 10). Since the dates in Fig. 9 are absolute, and the dates in Fig. 2 are relative, depending on the chosen Egyptian chro- nology, the latter were ignored for this part of the analysis.

The discovered grouping of azimuths indicates that the stars of Meskhetiu, mentioned in the descriptions of the “stretching of the cord” ceremony, were the targets used to orient the burial complexes since the beginning of the Pyramid Age in the 3^rd Dynasty, and the original plan was “one star of Meskhetiu for each one king”. From the reign of Sneferu, an innovator in monumental construction, the atten- tion of the Egyptians was fixated on Thuban, and several 4^th and 5^th Dynasty kings chose for orientation this star again and again; and from the middle of the 4^th Dynasty both orientation patterns existed simultaneously, with Djedefre being the first⁵⁷ to return back to the old pattern. The question arises what caused this fixation on Thuban? Since it was the Old Kingdom pole star, due to its proximity to the celestial Pole (Thuban was closer to the Pole than the present pole star, Polaris, is now - 8? versus 39?), it occupied the position of a “central star”, around which all other stars wander. The ancient priests could perceive the “motionless” central star as the only place that al- lows the soul of a deceased king to safely “moor”⁵⁸ to the rotating fir- mament and be adopted in the sky among the stars.

In any case, Thuban is the best reference star for the main trend pyramids, since: (a) its proximity to the Pole would significantly re- duce the influence of instrumental errors during the orientation ritual [Puchkov 2019, 23, Table 2]; (b) the rise and fall of the construction of the big pyramids can be attributed to its slow drift to and away from the Pole [Puchkov 2019, 57, Fig. 37].

(Fig. 7, Fig. SM1), or a triangle next to it (Fig. 6). This chain, the celestial analogue of a cattle leash, caused Meskhetiu to move (rotate) in the northern sky, therefore the Mooring-post must corre- spond to the center of rotation of the Old Kingdom sky (see [Locher 1985, S153; Polбk 1952, 177-178, Fig. 7a]). Thus, it can be assumed that the “mo- tionless” Thuban was perceived by the Egyptians as (top of?) the immovable Great Mooring-post (mnjt wrt: §863b [PT 458], §872b-c [PT 461], §884b [PT 466], §1366a [PT 553]), to which the soul of the deceased king should “moor”. The triangle in the Senenmut family (Fig. 6) probably corresponds to the location of the celestial Pole during the New Kingdom.

Since several stars are included in the pattern proposed here, we need to check its uniqueness and confirm its validity on an extended input dataset. To begin with, it is necessary to check if the azimuth data of the pyramids correspond to the azimuth data of the stars in the historically expected period. According to the data in Table 3, the reference star for the main trend pyramids in this period would be either Alioth or 10 Dra. However, verification did not reveal any match for the second trend, or the three 3^rd Dynasty pyramids. Continuing, it is necessary to confirm the proposed pattern using the new azimuth data of the Old Kingdom pyra- mids as control data (Fig. 11). The only pyramid available to test in this regard is that of 6^th Dynasty king Teti. This pyramid in North Saqqara has an inexplicably large deviation to the west of north, although the neighboring pyramids of Userkaf, Unas, and Djoser, are more well orien- ted relative to the cardinal points. No accurate azimuth data for the py- ramid of Teti have been obtained, but data for the pyramid temple, usually co-aligned with the pyramid, are

Finally, we have to examine how the accuracy of determining the sacred position of the sky during the orientation ceremony would have affected the observed pattern. Six minutes after the “balanced” position of Meskhetiu (horizontal alignment of Dubhe-Alkaid), its two lower components, Dubhe and Mizar, were aligned horizontally in the Old Kingdom sky. We can check out this position by creating a new graph (Fig. 12). Only six minutes of discrepancy from “true ba- lance” of the asterism strongly affected the position of its seven stars, because they were situated far from the celestial Pole. Therefore, their precessional lines would have significantly shifted, relative to the precessional line of Thuban. Because of this, there would have been no longer a correspondence between Alioth and the pyramids of Djedefre, Sahure and Unas; Phecda, as well, no longer would fit with the pyramid of Sekhemkhet.

The test shows that: a) the match is good for all chronology recon- structions that do not deviate much from the Turin King List data on the reign lengths of the Old Kingdom (see Suppl. Materials, Ta- ble SM2, Fig. SM5-SM6); b) neither consideration of other time ranges, nor any other positions of the Big Dipper within the permissible range allows us to find the reference stars simultaneously for two (three?) trends and single data on the 3^rd Dynasty pyramids. Thus, it can be stated that only the “balanced” position of Meskhetiu and only specific absolute dates provide an explanation for the orientation of the twelve Old Kingdom pyramids, therefore the pattern found is unique.

The following conclusions can be drawn at this stage:

1) The discovery of azimuth trends with similar gradients (Fig. 11) indicates that the pyramids were oriented toward different stars in the same position of the sky.

2) The accuracy of determining the direction in the orientation rites of the 4^th - 5^th Dynasties was approximately constant and amounted from 1-2? for Thuban (due to the low rate of change in the pole star's azimuth) up to 3-7? for Alioth (see Suppl. Ma- terials, Table SM1).

3) The Old Kingdom pyramids, in accordance with the religious beliefs of that time, were oriented to selected circumpolar (“imperishable”) stars, which were perceived as the place of the king's afterlife, while these stars were the goals themselves, and were not used as supporting markers for orientation to an invisible abstraction, that is, to the celestial Pole⁵⁹.

4) The hypothesis about the orientation of the 4^th Dynasty pyramids to true north is a mistake that arose due to the proximity of the direction to true north and the direction to one of the stars (the pole star) chosen by the Egyptians to orient the pyramids.

Analyzing the grouping of the pyramids' azimuths, we found the unique pattern that explains the orientation of a large heterogeneous group of pyramids, but contradicts the expectations of the conven- tional chronology about the dates of the Old Kingdom. In order to reconcile star dates and dates based on written records, a discrepancy of circa two centuries must be explained, or the pattern emerging from this analysis must be discarded as a random coincidence.

V. A brief study of the Egyptian chronology

Knowledge of Egyptian chronology is based on information from Egyptian King Lists, which are either complete but contain signifi- cant gaps in the text (Turin King List), or initially incomplete (Aby- dos King List); and Manetho's figures known to us from later cita- tions, which often differ in detail. Actual reconstructions of Egyptian chronology [Beckerath 1997, 187-192; Shaw 2000, 481-489; Hor- nung, Krauss and Warburton 2006, 490-495] combine relative data on the sequence and duration of the kings' reigns with the basis of the Sothic dates, which can be converted to absolute dates with reason- able accuracy. This approach implies that the periods of Egyptian his- tory are always associated with some uncertainty, which increases with deepening into the past and moving away from the Sothic “an- chor points”, so the dates that reconstructions of the Egyptian chro- nology offer us are estimates.

Information on the sequence of kings and reign lengths of both the Middle and Old Kingdoms is sufficient to roughly reconstruct their chronological structure (relative dates). Unlike the Middle Kingdom, which is pinned to absolute dates more or less accurately due to the Sothic date from the Illahun archive [Parker 1977, 177-184; Rose 1994, 237-261; Krauss 2006b, 448-450], the absolute dates of the Old Kingdom are very approximate because of the significant uncer- tainty in the duration of the 9^th - 10^th Herakleopolitan Dynasties be- fore the beginning of the 11^th Theban Dynasty⁶⁰. Estimates of the length of this period range from conventional 0-50 years⁶¹ up to one or two centuries according to Manetho. All chronologists agree that the 9^th - 10^th Dynasties present a source of uncertainty due to insuffi- cient information as to their length (Table 4). Only Manetho reports the duration of the Herakleopolitan rule, while Eusebius and Afri- canus diverge significantly in estimating the value, providing a weak basis for chronological reconstructions.

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