Giza & the starry sky

The Earth turns on its axis in 23 hours and 56 minutes. As a result, all stars appear to turn around the celestial pole in a day. At the latitude of Giza (c. 30° north) the celestial pole stands about 30° above the northern horizon.

Stars within about 30° of the pole never set. They are visible all year and are called circumpolar. All other stars rise and set, and are invisible for part of the year because the sun is then too close to them.

The Egyptians called the circumpolar stars the imperishable stars; the stars that set they called the unwearying stars. That is how the Pyramid Texts put it Allen 2005, pdf p. 20.

The Earth’s axis is tilted (c. 23.4°) relative to its orbit. As a result, the height of the sun changes over the year. At the solstices the sun is at its highest (summer) or lowest (winter), and at the equinoxes day and night are equally long.

At the latitude of Giza (c. 29.98° N) that gives:

Calculated for an obliquity of 23.44°. In the time of the Old Kingdom the angle was slightly larger (c. 24°), so the extremes differ by a few tenths of a degree.

Source: general knowledge

Whether you can see a star depends on its brightness, its height above the horizon (near the horizon the atmosphere dims the light) and the brightness of the sky. A star that sets or rises with the sun cannot be seen.

The most important observation for calendars is the heliacal rising: the first morning on which a star, after a period of invisibility, can be seen above the horizon again just before sunrise. Sirius, the brightest star in the night sky, is invisible for about 70 days every year because it is too close to the sun; after that it reappears in the east Facts and Details, pdf p. 3.

Because the sun moves about 1° through the zodiac every day, each star rises about 4 minutes earlier each day. So the day of the heliacal rising does not shift from year to year, but it does clearly shift across the seasons. Because of precession that day also shifts very slowly, over centuries.

Every star reaches its highest point once a day. That happens on the meridian, the imaginary line from the south point through the zenith to the north point. That moment is called the culmination or the upper transit.

How high that culmination is depends on only two numbers: the latitude of the observer (Giza: 29.98°) and the declination of the star, its distance from the celestial equator. For a star south of the zenith: altitude = 90° − latitude + declination. In 2580 BC Sirius had a declination of about −21°; so it rose no higher than about 39° above the southern horizon.

For the stars around the north pole it is even simpler. In Giza the celestial pole stands 30° high. A star that is p degrees from the pole circles around it and reaches 30° + p at its upper transit and 30° − p at its lower transit. In the time of Khufu Thuban was only 1.2° from the pole and so swung between 28.7° and 31.2°.

Why this matters: a shaft or passage with a fixed angle of slope points to one altitude above the horizon. Because of precession (2.3) the declination of every star changes slowly; so a star reaches that altitude only for a limited time. That lets you date an angle. That is the tool of chapter 12.

Source: general knowledge

Anyone who wants to check the sky of 4600 years ago needs three conventions.

• Sidereal time. The stars rise about 3 minutes and 56 seconds earlier each day than the day before. After a year that adds up to exactly one day. As a result, the same starry sky appears at an earlier hour on a later date: what is visible on 17 July at 04:05 is in the sky at midnight 61 days later.
• Solar time. We count hours with a clock; the Egyptians with the Sun and the stars. In this research 12:00 is the moment the Sun is due south (apparent solar time). Moreover, the Earth’s rotation is irregular: over 4600 years this adds up to about 6 hours (ΔT). That is why we always set an image by the altitude of the Sun and not by the clock.
• Twilight. A bright star is visible when the Sun is about 6° (civil twilight) to 8° below the horizon; a faint star only at 12° to 18°.
• Year numbering and calendar. Astronomers count a year 0: 1 BC is year 0, so 2580 BC is year −2579. Before 1582 Stellarium uses the Julian calendar. In 2580 BC the summer solstice therefore fell on 15 July Julian (24 June in our Gregorian calendar). In this book every date states which calendar is meant.

Source: general knowledge

Besides altitude and azimuth (2.4) we use two other ways to locate a star.

• Ecliptic longitude: the position along the path of the Sun, measured from the vernal point (0°). The Sun is at 0° at the spring equinox, at 90° at the summer solstice, at 180° at the autumn equinox and at 270° at the winter solstice. A star at longitude 90° therefore stands where the Sun stands on the longest day.
• Precession along the ecliptic: the vernal point moves back along the constellations by about 1.4° per century. The star that stood next to the Sun on the longest day in the time of Khufu no longer stands there now.
• The Milky Way: the glowing band of our own galaxy. Twice a year the Sun stands in it. In Egyptian imagery the sky is the body of the goddess Nut; whether she is also the Milky Way is a modern hypothesis (see 5.2 and 14.1).

With ecliptic longitude we can show in chapter 14 in which constellation the Sun stood at the turning points of the year, and which gods belong there.

Source: general knowledge

Five planets are visible to the naked eye. Two of them, Mercury and Venus, are closer to the Sun than the Earth. They never move far from the Sun and so can only be seen in the evening after sunset or in the morning before sunrise. Venus is then called the evening star or the morning star.

Venus follows a fixed cycle: after 8 years it is back at almost the same place relative to the Sun on almost the same date. In 3 of the 8 years it is in the sky as the morning star on a given summer morning.

In the Pyramid Texts the king appears as the morning star, and the morning star is called the sister of Sothis (14.1). The planets themselves also play a role in the scale comparison of 10.4.

Source: general knowledge

Around 10,000 BC the last ice age ends. The temperature rises quickly and reaches a warm maximum that lasts more than three thousand years. In North Africa this goes hand in hand with a stronger summer monsoon: the rain moves north.

The graph shows the global temperature. It does not directly say how much rain fell in Egypt, but it does show that the period from c. 8500 to c. 5000 BC was warm and that the cooling afterwards coincides with the end of the humid Sahara.

Source: general knowledge

Between c. 10,000 and c. 5300 BC enough rain fell in the Sahara for savanna, lakes and rivers. This period is called the African Humid Period. Hippos, giraffes and elephants lived there, and rock walls (among others in the Gilf Kebir and the Tassili plateau) show swimmers, herders and cattle.

Hunter-gatherers lived there, and later herders. At Nabta Playa in the Nubian desert, c. 100 km west of the Nile, stone circles already stood by a seasonal lake around 5000 BC.

Source: general knowledge

The Egyptian year had not four but three seasons of four months each:

• Akhet, inundation: the land is under water and people travel by boat
• Peret, growth: the water recedes, the land is ploughed and sown
• Shemu, harvest: the grain is brought in and stored

The sources give the start date of the season in slightly different ways. That is not an error: the Egyptian calendar had no leap day and therefore drifted through the solar years (see 4.3).

Source: Nile Scribes 2017, pdf p. 3 · Facts and Details, pdf p. 1 · Ancient Society, pdf p. 2

Agriculture came to Egypt via the Near East: wheat, barley, sheep and goats are found in the valley and the Fayum from c. 5000 BC. The Nile made the system almost free:

• The flood leaves silt behind: natural fertilisation.
• The land is divided into basins with small dykes, in which water is held and then drained.
• After the water has receded, the land is sown (Peret); in the dry season it is harvested (Shemu). Nile Scribes, pdf p. 5
• The goddess Renenutet watches over granaries and the harvest. At the beginning of the ninth month, the first of Shemu, a harvest festival was celebrated.

In the flood months farmers could not work. That time was used for state projects: under Khufu, boat crews transported limestone to Giza during Akhet Archaeology 2022, pdf p. 7.

Source: Nile Scribes 2017, pdf p. 5 · Archaeology 2022, pdf p. 7 · general knowledge

How did a village know when the flood would come? The Egyptians combined three observations:

• The moon: a lunar cycle lasts 29½ days. According to the sources the lunar calendar already existed in the Predynastic period (Nile Scribes, pdf p. 8)
• The river: the water level was recorded and the flood came at roughly the same moment each time.
• Sirius (in Egyptian Sopdet, in Greek Sothis): the brightest star disappeared every year for about 70 days and then reappeared just before sunrise. In the period 3000–2500 BC that was at the end of June, shortly before the flood began. (Nile Scribes, pdf p. 4) (Facts and Details, pdf p. 3)

By counting the time between two heliacal risings of Sirius, they found that there were about 365 days in between. That became the basis of their civil calendar. The Egyptians also noticed that the rising of Sirius shifted by about a quarter of a day per year relative to their 365-day year, but the leap day was long in coming.

The Egyptians used more than one calendar: a lunar calendar for festivals and a civil calendar for administration and taxes. The civil calendar has:

• 365 days: 12 months of 30 days and 5 extra days (epagomenal days)
• Three seasons of four months: Akhet, Peret and Shemu
• Weeks of 10 days, so 3 per month; the day has 24 hours (12 of daylight, 12 of night) (Facts and Details, pdf p. 2) (Nile Scribes, pdf p. 1)
• The earliest references to a 365-day calendar date from the 4th and 5th dynasties, around 2600 BC. (Facts and Details, pdf p. 3)

When does the year begin? New Year was called Wepet Renpet, the “opening of the year”, and fell on the first day of Akhet, the first of the month Tekh (Thoth). It was linked to the heliacal rising of Sirius and the arrival of the flood Ancient Society, pdf p. 16. Around 3000–2500 BC that fell at the end of June (Nile Scribes, pdf p. 4); the modern conversion of the civil calendar gives 19 July (Ancient Society, pdf p. 2).

Because there was no leap day, the calendar drifted by about 1 day every 4 years through the solar year: after about 1460 years (the Sothic cycle) the calendar was back in the same place (Ancient Society, pdf p. 15). The first known leap-day rule is in the Decree of Canopus (238 BC), but it was not applied permanently (Facts and Details, pdf p. 3).

General knowledge about the seasons. The astronomical seasons are defined by the solstices and the equinoxes (chapter 2): in the northern hemisphere spring begins at the spring equinox and summer at the summer solstice. Egypt had a different model, linked to the river, with three seasons of 4 months.

The ceiling from the tomb of Senenmut, the architect of Queen Hatshepsut, shows in a single image how the Egyptians combined sky and calendar. We will need two things from it later:

• the rows of decans: stars that are each invisible for about 70 days before they rise again just before the Sun. Those 70 days of “staying in the Duat” form the bridge to the myth of Osiris and Sothis (14.1);
• the northern figures in the middle: the foreleg of the bull (the Great Bear), a hippopotamus goddess and a figure with a spear. These are the imperishable stars around the pole, towards which the northern shafts of the pyramid point (5.6).

The ceiling is much younger than the pyramids (18th dynasty), but the figures and the system of reckoning behind them are older.

Source: general knowledge

Around 3100 BC Upper and Lower Egypt were united under one king. The Naqada III culture in the south spread northwards. The king wore the double crown and was represented as the god Horus.

• Narmer is depicted on the Narmer Palette (Hierakonpolis) as a victor; in the conventional view he is the first king of the 1st dynasty.
• Later writers name Menes (in Herodotus: Min) as the first king who united the country and founded Memphis, close to the later Giza (Facts and Details, pdf p. 4)
• The kings of the 1st dynasty were buried at Abydos. Memphis became the capital of the united country.
• From the 2nd dynasty onwards the “stretching of the cord” is depicted: together with the goddess Seshat the king lays out the corners of a new building, according to later texts by the stars of the Great Bear. This ritual is the basis of the north alignment of the pyramids (13.2) (Belmonte 2005, pdf p. 1)
• Djoser (3rd dynasty, c. 2670 BC) and his architect Imhotep had the first large stone pyramid built at Saqqara.
• Sneferu (4th dynasty) built pyramids at Meidum and Dahshur and was the father of Khufu (Wikipedia: Cheops, pdf p. 1)

Years were counted per king: after his death the count started at year 1 of his successor, for example “regnal year 8, third month of Peret, day 15” (Nile Scribes, pdf p. 3).

Source: general knowledge

Already in the Predynastic period Egyptian stonemasons made vessels of hard stone. In the 1st dynasty granite from Aswan was used as a building material for the first time: the floor in the tomb of King Den at Abydos. In Djoser’s tomb the burial chamber was made of granite.

Royal statues were at first made of limestone and slate. In the 4th dynasty statues appear in granite, diorite and greywacke: stone that is much harder than copper. They were worked with dolerite pounders, sand and a lot of time; copper tools were used for softer kinds of stone (Archaeology 2022, pdf p. 2).

The best-known example is the statue of Khafre. Auguste Mariette found it in 1858 in Khafre’s valley temple. It is made of diorite (anorthosite gneiss), an exceptionally hard and dark stone that was brought c. 400 miles along the Nile. The valley temple contained 23 statues of the king (Wikipedia: Chefren, pdf p. 3) (pdf p. 4).

Source: Wikipedia: Chefren, pdf p. 3 · Archaeology 2022, pdf p. 2 · general knowledge

The earliest gods are local: the falcon Horus of Nekhen, Seth of Naqada, Hathor as a cow, Min of Koptos and Neith of Sais. With unification Horus became the god of kingship: every king was Horus, and after his death he became Osiris.

Around the sun at Heliopolis the Ennead arose, the nine gods: Atum, Shu, Tefnut, Geb, Nut, Osiris, Isis, Seth and Nephthys. According to the myth, Nut, the sky, swallows the Sun every evening and gives birth to it again every morning; her five children are born on the five extra days of the year (4.3). Hathor, the cow, later carries the sun disk between her horns, already in statues from Giza itself (14.2). The Nile itself was worshipped as Hapy, the god of the flood (Nile Scribes, pdf p. 4), and Sirius as Sopdet.

The oldest extensive texts are the Pyramid Texts, from Unas (c. 2350 BC) onwards, but according to the translator the material is clearly older (Allen 2005, pdf p. 15). More about this in chapter 5.

Source: Allen 2005, pdf p. 15 · Nile Scribes 2017, pdf p. 4 · general knowledge

Our most important source for the beliefs of the Old Kingdom are the Pyramid Texts: the oldest preserved literature of Egypt, carved into the chambers of the pyramids. They were first found in the pyramid of Unas and, according to the translator, are clearly older than the first inscription (Allen 2005, pdf p. 15).

Later sources are the netherworld books of the New Kingdom: the Amduat, the Book of Gates and the Book of Caverns. They are a thousand years younger than the pyramids. They show that the image of the nightly journey of the Sun through the Duat, with Osiris at its centre, persisted for centuries; for the Old Kingdom itself we use only the Pyramid Texts.

• Pyramid Texts: 6 royal pyramids, from Unas to Pepi II and Queen Neith (pdf p. 16)
• Amduat: known since Queen Hatshepsut, c. 1470 BC, 12 hours of the night (pdf p. 7)
• Book of Caverns: created in the 13th century BC (pdf p. 1)
• Book of Gates: twelve gates for the twelve hours of the night (Budge 1905)

Source: Allen 2005, pdf p. 15 · Hornung & Abt 2007, pdf p. 7 · Book of Caverns, pdf p. 1 · Budge 1905, pdf p. 1

The texts describe the world as a bounded space between land and sky, within the ocean Nu:

• Geb is the earth, Nut the sky and Shu the air in between (pdf p. 20)
• Nut is depicted as a woman bending over the earth. In the evening she swallows the Sun, at night she carries it in her body and at dawn she gives birth to it again; Osiris too lives “in the womb of Nut” (pdf p. 20). A series of spells of King Teti is even called “spells for entering the womb of Nut” (pdf p. 67).
• By day the sun sails across the sky in the day bark and at night in the night bark through the netherworld, the Duat.
• The second part of the Duat, up to the eastern horizon, is called the Akhet: the “place of becoming effective” (pdf p. 20)
• The path of the sun through the year is called the Winding Canal, about 12° wide Allen 2005, pdf p. 20
• The southern sky is the Marsh of Reeds, the northern the Marsh of Rest; in them lie the imperishable and unwearying stars Allen 2005, pdf p. 20

The burial chamber of the pyramid represents the Duat, the antechamber the Akhet (pdf p. 21) (pdf p. 22).

The Giza plateau is a limestone plateau on the west bank of the Nile, at the edge of the desert and about 15 km southwest of the centre of present-day Cairo. It belongs to the vast necropolis of Memphis, the capital of the Old Kingdom, and lies exactly on the border between the fertile land and the Libyan desert.

On this plateau stand the three great pyramids of the Fourth Dynasty, surrounded by smaller pyramids, mastabas (rectangular tombs of high officials), temples, raised roads (causeways), boat pits and the Great Sphinx. Together they form one coherent, planned building complex: a necropolis laid out within a few generations. The “Giza complex” comprises three large and six small pyramids (Wikipedia: Cheops, pdf p. 1).

The three pyramids lie on a diagonal line from northeast to southwest and decrease in size: Khufu (Cheops), Khafre and Menkaure. On the east side of each pyramid lay a mortuary temple; from there a causeway ran to a valley temple at the edge of the fertile Nile valley.

• The original entrance is on the north side. In the 10th century an Arab treasure hunter broke out a new entrance; around 820 the caliph Al-Ma’mun had a hole made (pdf p. 5)
• A descending passage of c. 1 m² leads to an unfinished subterranean chamber.
• The ascending passage was sealed with granite plugs and leads to the Grand Gallery.
• The King’s Chamber is lined with dark red granite; the sarcophagus (2.27 m long, 1.05 m high) is empty, without lid or remains (pdf p. 6)
• The Queen’s Chamber was never a burial chamber; the names date from the 18th century and rest on speculation.
• Above the King’s Chamber lie five relieving chambers with granite beams of c. 40 tonnes. Builders’ graffiti with the name of Khufu are the only authentic proof that he was the builder Wikipedia: Cheops, pdf p. 6
• In 2017 a large void above the Grand Gallery was discovered using cosmic rays (muons) Wikipedia: Cheops, pdf p. 5

The pyramid was measured in the royal cubit: 1 cubit = 7 palms (palm) = 28 fingers (digit), c. 523.5 mm. In cubits the base is 440 cubits and the height 280 cubits (ratio 11 : 7). (Bartlett 2014, pdf p. 7)

The Egyptians did not express a slope in degrees but as a seked: the horizontal distance per cubit of rise. For the Great Pyramid this is 5½ palms (5 palms and 2 fingers), a slope of 14 in 11 and an angle of 51°50′35″ (Furlong, pdf p. 1).

With these two choices (440-cubit base, seked of 5½ palms) the shape of the pyramid is fixed. We keep theories about special numbers in these dimensions for the second part (chapter 10), where we test them.

How the blocks were raised is not known for certain. What is established:

• According to Herodotus the construction of the Great Pyramid took 20 years; modern research estimates that the number of workers was no more than 20,000 – 25,000 (pdf p. 4)
• The building material came by boat: limestone from Tura, granite from Aswan, gypsum and basalt from the Fayum and wood from Lebanon Archaeology 2022, pdf p. 2
• Enormous quantities of copper were used for chisels and saws; for every centimetre of sawn stone one to four centimetres of copper is lost Archaeology 2022, pdf p. 2
• The mud-and-sand ramp model is the most common; alternatives are the spiral internal ramp (Houdin) and tilting with lever cradles (Mullers et al., 2022) Wikipedia: Cheops, pdf p. 4

The workers of Merer’s boat crew noted in their logbook that they sailed back and forth between Tura and Giza two or three times in ten days Archaeology 2022, pdf p. 2.

The Egyptian year was determined by the Nile: Akhet (inundation), Peret (growth and sowing) and Shemu (harvest), each four months of 30 days (Facts and Details, pdf p. 1). Whoever knew this cycle knew when the land would be under water, when to sow and when to harvest.

The year began in July with the flood and the rising of the star Sirius just before sunrise. The Egyptians noted that Sirius and the sun coincided at roughly the same moment every year, and therefore counted a year of 365 days Facts and Details, pdf p. 1. That gave them a hold for anticipating the flood and planning the agricultural year.

That agriculture produced more than the farmers themselves needed is evident from the construction itself. The workers were paid in beer, bread and grain (Wikipedia: Cheops, pdf p. 4). Cattle from the Nile Delta were brought to the plateau to feed the workers, and by the palace stood archives, a granary and barracks (Archaeology 2022, pdf p. 2). According to modern estimates no more than 20,000 to 25,000 people worked on the Great Pyramid, about 1% of the estimated two million inhabitants of Egypt Wikipedia: Cheops, pdf p. 4.

So one in a hundred inhabitants could be released from producing food for the construction. That is only possible when the other inhabitants produce a food surplus. This is an inference from the numbers mentioned.

Source: Facts and Details, pdf p. 1 · Wikipedia: Cheops, pdf p. 4 · Archaeology 2022, pdf p. 2

A project of this size requires recorded agreements, numbers and instructions. The Egyptians used two forms of writing for this: hieroglyphs in stone for texts that had to last, and ink on papyrus for daily administration.

Near the Red Sea a French-Egyptian expedition found the oldest inscribed papyri found so far. One of them bears one of the royal names of Khufu (Archaeology 2022, pdf p. 6). They include accounting documents and the logbooks of the inspector Merer, who led a crew of c. 40 men and noted daily where he sailed, what he transported and where he slept Archaeology 2022, pdf p. 6 (pdf p. 7). Such a group (phyle) was one of the four units of a larger gang with its own name, such as “The escort team of ‘The Uraeus of Khufu is its prow’” Archaeology 2022, pdf p. 6.

There is also writing on the building site itself: in the relieving chambers above the King’s Chamber there are builders’ graffiti of the workers with the name of Khufu (Wikipedia: Cheops, pdf p. 6). Years were counted by the king’s regnal year (Nile Scribes, pdf p. 3).

For knowledge that must not be lost, stone was chosen. The Pyramid Texts are the oldest preserved religious writings of Egypt and at the same time the oldest literature of the country (Allen 2005, pdf p. 12). They were carved into the burial chambers of Old Kingdom kings, in vertical columns of hieroglyphs. Each spell is usually preceded by a sign for “Recitation” (pdf p. 14): the text was meant to be spoken.

Source: Archaeology 2022, pdf p. 6 · Allen 2005, pdf p. 12 · Wikipedia: Cheops, pdf p. 6 · general knowledge

The Egyptians obtained raw materials from all over Egypt and beyond. For the construction of Khufu’s complex the materials came from the regions below.

Copper was indispensable. With copper picks the stone was quarried, with copper saws it was sawn and with copper chisels about a finger wide it was smoothed. Experiments have shown that for every one to four inches of sawn stone, one inch of metal was lost from the saw blade (Archaeology 2022, pdf p. 2). The enormous quantity of copper led to a search for new sources.

Source: Archaeology 2022, pdf p. 2 · Archaeology 2022, pdf p. 3 · Wikipedia: Cheops, pdf p. 3 · Wikipedia: Cheops, pdf p. 6

The Nile was the highway of the kingdom. Almost all building material came to the plateau by water: limestone from Tura, granite from Aswan, gypsum and basalt from the Fayum and wood from Lebanon (Archaeology 2022, pdf p. 2). Workers from all over the country and cattle from the Delta also came by boat.

To let ships sail up to the edge of the plateau, a network of artificial waterways and canals was dug, with an entrance gate to the lake of Khufu, the Ro-She Khufu. By the lake lay the harbour, the valley temple, a palace, archives, a granary and workers’ barracks (pdf p. 1) Archaeology 2022, pdf p. 2.

Merer’s logbooks show how this worked. His crew of c. 40 men loaded limestone blocks at Tura, slept at the lake of Khufu and sailed on the next day to Akhet Khufu, the pyramid. In ten days they made two or three such round trips Archaeology 2022, pdf p. 2 (pdf p. 7). The transport time depended on the seasons: in the flood season (c. June to November) heavy loads could be moved; after the water had receded the crew went to the Delta or to the Red Sea Archaeology 2022, pdf p. 7 Archaeology 2022, pdf p. 8.

Ships were also part of the burial. Next to the pyramid of Khufu a very well-preserved boat was found, and five rock pits around the pyramid contained ships. The tradition of giving kings ships already existed in the 1st and 2nd dynasties (Wikipedia: Cheops, pdf p. 2).

Source: Archaeology 2022, pdf p. 2 · Archaeology 2022, pdf p. 7 · Wikipedia: Cheops, pdf p. 2

The Great Pyramid is a work of precision. A base of 13 acres (c. 5.3 ha) is laid out so flat and square that the sides hardly differ (Bartlett 2014, pdf p. 3). The dimensions are not accidental: the design is built up from royal cubits and the seked, a fixed ratio for the slope (see 7.4).

• Original height: 146.6 m; volume c. 2.6 million m³ (Cheops p. 2)
• Blocks: c. 2.3 million, from 2.5 to 80 tonnes (Bartlett p. 3)
• Difference between the four sides: at most 4.4 cm over c. 230 m Bartlett 2014, pdf p. 3
• Joints: c. 1/50 inch (0.5 mm) between the blocks Bartlett 2014, pdf p. 3
• Deviation from true north: only 3′6″ Bartlett 2014, pdf p. 3

Construction took about 20 years (Cheops p. 4) and required a steady building rate of on average one block of average size every 2 to 3 minutes in a working day of 10 hours Bartlett 2014, pdf p. 3.

Source: Bartlett 2014, pdf p. 3 · Wikipedia: Cheops, pdf p. 2 · Wikipedia: Cheops, pdf p. 4

Quarrying, transporting and placing material without the work coming to a halt requires organisation. The pace of construction and the structure of the work gangs make clear that there was an administrative apparatus with planning, bookkeeping and supervision.

The numbers are estimates. Both Bartlett Bartlett 2014, pdf p. 3 and Wikipedia Wikipedia: Cheops, pdf p. 4 assume a construction time of c. 20 years; we leave the exact years aside here.

Source: Wikipedia: Cheops, pdf p. 4 · Archaeology 2022, pdf p. 6 · Bartlett 2014, pdf p. 3

Belief in a life after death required a preserved body. In the Pyramid Texts the soul (ba) remains connected to its own body: the ba unites with its own Osiris, the mummified body in the tomb, and so gains the power to become akh (Allen 2005, pdf p. 19). The spells in the tomb of Unas contain a series of “Spells for the protection of the mummy” (pdf p. 28).

In the offering texts natron occurs, a natural salt: it is mentioned in pellets from different regions of Egypt, with reference to gods (pdf p. 30). Natron is also the substance with which the body was dried (general knowledge).

The sources in this library do not describe the technique of mummification itself; the details above are general knowledge.

Source: Allen 2005, pdf p. 19 · Allen 2005, pdf p. 28 · Allen 2005, pdf p. 30 · general knowledge

Already in this early part of the research it is clear that the Egyptians knew the sky well.

• The Great Pyramid is built so that its four sides face the four cardinal directions. Exactly how this was done is not known; the Pole Star did not indicate north at the time. According to research, the position of the sun in different seasons may have been used (Wikipedia: Cheops, pdf p. 3)
• The deviation from true north is only 3′6″ (Bartlett 2014, pdf p. 3)
• The pyramid is called Akhet Khufu, “Horizon of Khufu” (Archaeology 2022, pdf p. 1)
• The 365-day calendar was linked to the rising of Sirius just before the sun (Facts and Details, pdf p. 1)
• The Egyptians noticed that Sirius and the sun shifted by c. 6 hours (a quarter of a day) every year and therefore knew that a leap day was needed Facts and Details, pdf p. 1

We test the further alignment of the buildings with the sun and the stars on special dates in the second part.

Source: Wikipedia: Cheops, pdf p. 3 · Bartlett 2014, pdf p. 3 · Archaeology 2022, pdf p. 1 · Facts and Details, pdf p. 1

The Pyramid Texts describe the world as a whole: every day the sun travels from birth to death and is reborn at sunrise with the power of Osiris. The king is taken up into this process (Allen 2005, pdf p. 19). See chapter 5 for the mythology.

The construction is the translation of this belief into stone. A pyramid complex consists of parts that each play a role in the burial and the offering cult, as the table below shows.

No Pyramid Texts have been found in the Great Pyramid (general knowledge). The oldest preserved texts are in the pyramid of Unas, the last king of the 5th dynasty (Allen 2005, pdf p. 26).

Source: Allen 2005, pdf p. 19 · Allen 2005, pdf p. 26 · general knowledge

A movement in the stars or a number in a pyramid is only a clue if you know how likely it is that it turns out that way by chance. That is why every claim in this part gets the same five questions:

• 1. What exactly is claimed? One sentence, in advance, without adjusting it afterwards.
• 2. What is the error? The difference between prediction and measurement, in percent or degrees, plus the measurement uncertainty of the source itself.
• 3. How much freedom did the researcher have? Were the scale factor, the year, the star or the horizon altitude fixed in advance, or was there a search until it fitted? Every choice that can be made afterwards reduces the value of the outcome (the “look-elsewhere effect”).
• 4. What does the other side say? The strongest counter-argument is always included.
• 5. Can it be checked? Astronomical claims get a Stellarium screenshot with all settings, so that everyone can call up the same image.

The outcome is classified into five evidence levels. A claim can never score higher than the weakest link in its chain.

The letters return in this part with every claim and in the overview of 15.1.

Source: Bartlett 2014, pdf p. 4 Orion correlation theory, pdf p. 2

Settings for all Stellarium screenshots

• Location: latitude 29.9753° N, longitude 31.1376° E (the Sphinx), altitude 25 m. Name “Giza Sfinx”.
• Calendar: before 1582 Stellarium automatically uses the Julian calendar and counts the year 1 BC as year 0. So 2580 BC is year −2579, and 2579 BC is year −2578.
• Time: in Configuration → Tools set the time zone to LTST (local true solar time) if your version of Stellarium has that option. The times in the tables are solar time.
• The Sun as clock: the Earth’s rotation is irregular; over 4600 years this adds up to about 6 hours (ΔT). So do not set the time by the clock but by the altitude of the Sun: “sunrise” is the Sun at −0.27° (with atmosphere, upper limb on the horizon), “twilight” is the Sun at −8°.
• Atmosphere: on for sunrise images; off for measuring star altitudes. The calculated altitudes are geometric (without refraction); with atmosphere Stellarium adds up to c. 0.6° near the horizon.
• Landscape: preferably flat without buildings. Turn on the grid and constellation lines only if the test says so.
• File name: every screenshot has a fixed ID (e.g. K3). Save it as img/stellarium/K3.jpg; the placeholder on this page is then replaced automatically.

All celestial calculations in chapters 11 to 14 were done by ourselves, with open data and fixed rules laid down before the calculations.

Stellarium may differ from our results by a few tenths of a degree because of refraction, ΔT and the time to the minute.

Source: Vondrák et al. 2011 · Petrie 1883, §44 · Petrie 1883, §56

Two theories explain the shape of the Great Pyramid: the π theory (the perimeter of the base is the circumference of a circle with the height as radius) and the golden-ratio theory (the cross-section is a Kepler triangle with ratio 1 : √φ : φ). Both fit to within 0.1%.

Base 230.363 m and height 146.515 m according to Bartlett (2014); the royal cubit is 0.5236 m = 7 palms = 28 fingers. The height is a reconstruction, because the top is missing.

What does the other side say? The two numbers are 0.14 m apart (146.51 m versus 146.65 m), much less than the uncertainty in the height of a pyramid without its top. So the measurement cannot choose between π and φ. The simplest explanation is a seked of 5½ palms, i.e. the ratio 14 : 11 (280 : 220 cubits). That gives π and φ as by-products within 0.06%. Ramirez shows that π and φ are linked via π ≈ 4/√φ (deviation 0.1%), so whoever sees one automatically sees the other.

Bartlett does not reject either theory, but argues that the shape also follows from the Egyptian canon of human body proportions.

Source: Bartlett 2014, pdf p. 4 Bartlett 2014, pdf p. 5 Bartlett 2014, pdf p. 7 Bartlett 2014, pdf p. 8 Ramirez 2018, pdf p. 1 Ramirez 2018, pdf p. 2 Furlong, pdf p. 1

The Pythagorean theorem and the 3-4-5 triangle appear in two places: in the pyramid of Khafre and in the King’s Chamber.

Khafre. A height of 274 cubits and a half base of 205.5 cubits give exactly 4 : 3 (274 / 205.5 = 1.3333). The slope is then 53.13° and the seked 5¼ palms. The Dutch Wikipedia gives 53°13′; the difference from the perfect 3-4-5 angle is 0.03°.

Ratio width : length : height = 2 : 4 : √5; all diagonals are then whole numbers or multiples of √5.

What does the other side say? The dimension 10 × 20 cubits is ordinary. The height of √125 cubits (11.18 cubits, measured 5.85 m) is exactly the height needed to make the space diagonal 25 cubits (25² = 10² + 20² + 125). That may be a deliberate choice, but it may also be a chance result of a room about 11 cubits high: the measured height (11.17 cubits) differs only 0.3% from 11.18 cubits, and a measurement error of a few centimetres does not change the picture. Nor is it recorded anywhere that the builders ever wanted the space diagonal as a dimension. An Egyptian arithmetic problem with a 3 : 4 ratio (papyrus Berlin 6619) is known from the Middle Kingdom, more than six centuries later; no mathematical text survives from the Fourth Dynasty. Papyrus Berlin 6619 (Wikipedia)

Source: Wikipedia: Chefren, pdf p. 1 Furlong, pdf p. 1 Bartlett 2014, pdf p. 7

The Great Pyramid is said to be a scale model of the Earth at 1 : 43,200: the perimeter of the base is then the circumference of the Earth and the height is the polar radius. The numbers agree to within 0.7%.

The infographic calculates with 1760 cubits × 0.5236 m = 921.5 m, but under “1760 royal cubits × 43,200 ≈ 40,075,017 m” gives a value that is incorrect: 1760 × 0.5236 × 43,200 = 39,810 km. For 40,075 km a base of 927.7 m (1771.7 cubits) is needed.

What does the other side say?

• Circular reasoning. The number 43,200 is half a day in seconds (86,400 / 2). The two “derivations” (scale and one second of Earth rotation) are therefore the same equation and do not count twice. The second (1/86,400 of a day) did not exist in the Old Kingdom; that division comes from Babylonian and Hellenistic times.
• Free scale factor. With a scale factor that may be freely chosen, any building fits some planet or measure. 43,200 was chosen afterwards so that the perimeter fits; the height is then the second measurement that must fit independently, and it does so with −0.43%.
• Size of the Earth. The circumference of the Earth was only determined around 240 BC by Eratosthenes. An earlier measuring project would have required surveyors to measure an arc of many degrees; there is no text about it.
• Error in the infographic. The rotation speed of 463.8 m/s applies at the equator. At the latitude of Giza it is 401.8 m/s; the link with the half perimeter (460.7 m) therefore only holds at the equator.

Source: Jelitto 2015, pdf p. 13 Bartlett 2014, pdf p. 7

Jelitto (2015) suggests that the three great pyramids represent the three inner planets: Khufu = Earth, Khafre = Venus, Menkaure = Mercury. Venus and Mercury are the “morning stars”: they never move far from the Sun and stand above the horizon just before sunrise.

Size (scale 1 : 43,200)

Planetary dimensions: equatorial circumference Earth 40,075 km, Venus 38,025 km, Mercury 15,329 km; polar radii 6357, 6052 and 2440 km. Error = (pyramid − planet) / planet.

Mutual distance (scale 1 : 86,400,000)

Herman (2020) gives 482 m and 897 m as measurements in the image; the errors are then −4% and −1%. With Petrie’s figures they are slightly higher.

What does the other side say?

• Two different scales. For size the scale is 1 : 43,200, for distance 1 : 86,400,000 (2000 times smaller). There is no model with a single scale.
• Menkaure does not fit. For Mercury the actual perimeter of 414 m had to be replaced by 385 m; even then an error of +8.5% remains. Only Khufu (−0.7%) and Khafre (−2.2%) fit well.
• Moving targets. The Earth–Venus distance varies between 38 and 261 million km. The model uses the minimum distance; that is a choice.
• Planetary dimensions. The diameters of Venus and Mercury could only be measured with the telescope (17th century and later). Deliberate dimensioning requires an explanation of how the builders knew this. There is no source; the comparison remains a modern observation.

Jelitto also calculates when Mercury, Venus and Earth ever had the same mutual angles as the pyramids on the plateau (Petrie distances). This astronomical calculation shows that such configurations exist; they lie far outside the 4th dynasty and therefore give no date.

Source: Jelitto 2015, pdf p. 13 Jelitto 2015, pdf p. 14 Allen 2005, pdf p. 448

Bauval’s Orion correlation theory (1989) says that the three great pyramids mirror the three stars of Orion’s belt: Khufu = Alnitak, Khafre = Alnilam, Menkaure = Mintaka. Orion is Sah, the soul of Osiris; in the Pyramid Texts the king is linked with Osiris and Orion at his death.

Orion positions: Alnitak 05h40m45.5s −1°56′34″, Alnilam 05h36m12.8s −1°12′07″, Mintaka 05h32m00.4s −0°17′57″. Belt and Nile are mirrored: the Milky Way is seen as a celestial Nile.

What does the other side say? The mutual distances do not match: the ratio differs by 9.6% and the bend in the row is 11.5° versus 7.5°. The belt is not a copy of the ground plan, but a rough resemblance of three points in a row with a bend to the same side. With three points and a few freedoms (mirroring, rotating, scaling) such a resemblance is not rare. Moreover, the astronomers Krupp and Fairall found with planetarium equipment that in the period named by Bauval and Hancock (10,500 BC) the belt makes an angle of 47–50° with north, versus 38° for the pyramids. In our own calculation (Vondrák precession) these two angles only agree around 12,000 BC; that is therefore a date the theory needs, not a date the theory produces.

What remains? The connection with Orion lies not in the shape but in the altitude: the belt passes through the sky at 45° above the horizon, exactly the angle of the southern shaft of the King’s Chamber. This is tested with Stellarium in 12.1.

Source: Orion correlation theory, pdf p. 1 Orion correlation theory, pdf p. 2 Jelitto 2015, pdf p. 13 Allen 2005, pdf p. 449 Wikipedia: Cheops, pdf p. 3

Was the Sphinx with its causeway (Khafre’s raised road) built as a calendar that marks the place of sunrise on the horizon? The working hypothesis of this research is: the solar year begins at the summer solstice of 2580 BC (15 July, Julian), when the Sun rises furthest north. The Egyptian New Year (Wepet Renpet), the first morning on which Sirius can be seen again, falls two days later: 17 July (M2 in 14.1b). The year 2580 was chosen because it falls in the middle of Khufu’s reign (c. 2589–2566 BC); it is a reference year, not a result.

This produces a series of fixed moments, calculated below. The azimuth of the Sun at rising is measured from north (east = 90°).

Calculated for the Sphinx (29.975° N) with a flat horizon and the moment of sunrise “upper limb on the horizon”. The dates are ±1 day: find in Stellarium the day on which the azimuth of the Sun at rising equals the table value. Sensitivity: with a causeway direction of 103.5° or 105° the first passage shifts to day 120.6 or 124.2 respectively and the second to day 238.7 or 234.9 respectively.

If we count from New Year (17 July, Sirius) instead of from the solstice, two days are subtracted: K2 day 88.3 (−1.7 d relative to day 90), K3 day 119.8 (−0.2 d relative to day 120), K4 day 176.8 (−3.2 d), K5 day 235.5 (−4.5 d relative to day 240) and K6 day 269.1 (−0.9 d). The first causeway passage then fits the end of Akhet even better, the second less well. Day numbers elsewhere in this book are counted from the solstice, unless it says “after New Year”.

What stands out?

• The Sphinx faces the equinox point. Its gaze is at 90°; the Sun stands there on the day of the autumn and spring equinox (azimuth 89.5°, a difference of 0.5° due to refraction and the position of the solar limb).
• The causeway lies at 104° (map value 104–105°) and the Sun stands there on 13 November (K3) and 9 March (K5). K3 falls at 0.3335 of the year, so at ⅓.
• The civil year fits. The Egyptian year has 12 months of 30 days (360) plus 5 days, in three seasons Akhet, Peret and Shemu of 4 months (120 days). The first causeway passage falls 1.8 days after the end of Akhet (day 120), the second 2.5 days before the end of Peret (day 240). The quarter points of the civil year (days 90, 180, 270) fall within 1.3 days of the equinoxes and the winter solstice.
• ¼, ½ and ¾ of the real year do not fall on the solstices. Because the Earth’s orbit is not circular, the seasons are unequal: 0.247, 0.489 and 0.742 of the year. The hypothesis “a quarter year = autumn equinox” therefore holds to within 1 day (0.247 versus 0.25), “half year = winter solstice” to within 3.9 days (0.489 versus 0.5) and “three quarters = spring equinox” to within 2.8 days.

What does the other side say? (1) The direction of 104° is a map value; a deviation of 1° shifts the dates by 3 days. (2) The horizon around Giza is not flat (Mokattam hills in the east), which changes the azimuth of rising by a few tenths of a degree. (3) A Sun shining along a long road does not make a calendar: there is no text from the Old Kingdom that describes this. (4) In a 365-day year without a leap day the civil year slips 1 day every 4 years; over a lifetime (60 years) that is 15 days. The calendar can therefore only serve as fixed reference points.

Source: Ancient Society, pdf p. 1 Ancient Society, pdf p. 2 Nile Scribes, pdf p. 3 Nile Scribes, pdf p. 4 Facts and Details, pdf p. 1

At each of the six solar points, different stars stand above the horizon just before sunrise. The question is whether this allows a link between stars, calendar and mythology: was a star of an important god in the sky at New Year?

Azimuth from north; altitudes are geometric without refraction. D7 equals D1 (year full circle). Each star is given with its position in twilight; the sun itself has not yet risen.

The strongest link: Sirius at New Year

The Egyptians began their year at the heliacal rising of Sirius (Sopdet): the first morning on which the star is visible again shortly before sunrise. That coincided with the beginning of the Nile flood. In the Pyramid Texts Sirius is linked with Isis and Orion with Osiris; the king ascends to both.

The four lines are common limits for the visibility of a bright star (depth of the Sun and altitude of the star). They are calculated without an atmospheric model.

In 2580 BC that rising therefore fell 1.4 to 4.4 days after the summer solstice: the heliacal rising of Sirius and the start of the solar year coincide to within a few days. That gap grows by about 0.75 days per century (with Sun −8° and star 3°: −4.2 days in 3580 BC, 0 days around 3100 BC, +3.4 days in 2580 BC and +7.2 days in 2080 BC). So Sirius and the solstice coincided best around the time the calendar came into being, not exactly in Khufu’s time. The Gregorian date (25 June) fits the remark that Sirius rose in “late June” between 3000 and 2500 BC. The model was also checked against the Sirius calculation of Vodolazhskaya et al. (2767 BC), with which it agreed.

Four corner stars

In Persian tradition these four are the “royal stars” (the guardians of the four corners of the sky). That is a much younger tradition than the Fourth Dynasty.

Which constellation at the winter solstice? Fomalhaut is the bright star in the mouth of the Southern Fish (Piscis Austrinus), the small constellation below the feet of Aquarius; it stands 20° below the ecliptic. Directly above it, at the same longitude, stands Sadalmelik in Aquarius; that is where the Sun itself stood at the winter solstice of 2580 BC (0.1°). The Pisces of the zodiac and Aries lie further away: the vernal point only moved into Aries around 1900 BC and into Pisces around AD 70 (2.3). Aries (Hamal) did stand 15° high above the Sphinx on the morning of the spring equinox (D6). More about the Sun at the turning points and the gods that belong there in 14.2, and about the two mornings on which the Sun stands in the Milky Way in 11.5.

What does the other side say? (1) For the Old Kingdom only Sirius, Orion and the Great Bear (the “Foreleg”) are reliably established as Egyptian constellations; all other readings are later or uncertain. (2) Every day there is some bright constellation in the eastern sky: any desired combination can be found with a little searching. (3) The “four corner stars” stand at the seasonal points in 2580 BC because precession put them roughly in that place at the time; the same would apply to any building from the period 3000–2000 BC.

Source: Heliacal Rising of Sirius, pdf p. 3 Nile Scribes, pdf p. 4 Vodolazhskaya et al. 2015, pdf p. 1 Allen 2005, pdf p. 448 Allen 2005, pdf p. 449 Allen 2005, pdf p. 452 Ancient Society, pdf p. 1

Two further effects: the shadow effect on the pyramid at sunrise (equinox) and the sunset in the akhet (summer solstice), the hieroglyph of the Sun between two hills that denotes the horizon.

Shadow at the spring equinox

At the equinox the Sun rises at azimuth 89.5° and the four faces of the Great Pyramid are aligned to the cardinal directions to within a few arcminutes. The Sun then shines almost exactly along the north and south faces. The Great Pyramid has concave faces (the so-called indentation): the faces are bent inwards in the middle, so that the strip of light is split in two halves. Such a split of light and shadow is reported by observers (not in the pdf sources). It cannot be seen in Stellarium, because Stellarium does not draw buildings; for this a photo or a 3D model is needed.

Sunset as akhet

The Egyptian name of the Great Pyramid was Akhet-Khufu, “horizon of Cheops”. Seen from the Sphinx, the Sun sets at the summer solstice exactly in the gap between Khufu and Khafre. That is the shape of the akhet sign, and it is also the start of the season Akhet, the season of the flood (11.2, K1).

Position of the Sphinx and the pyramids in degrees and metres from map coordinates (accuracy c. ±1°). Khufu lies 545 m from the Sphinx (azimuth 322°, altitude of the top 15°), Khafre 663 m (278°, 12°), Menkaure 995 m (248°, 4°).

What does the other side say? The builders did not stand at the position of the Sphinx as a fixed viewpoint; the Sphinx is older than the gap (or contemporary) and the pyramids served a different purpose. A gap between two mountains is a natural image that occurs in many places; what is special is only that here it falls at the most visible moment of the year.

Source: Wikipedia: Cheops, pdf p. 3 Allen 2005, pdf p. 6 Allen 2005, pdf p. 7

In 14.1 (M5) we saw that the Sun stands in the Milky Way again on two mornings of the year: 10 May and 8 November 2580 BC. Here we put those two mornings side by side and look at who is in the sky then. What stands out is that one morning shows the figures of life and the other those of death, the Nile and the new beginning, and that in both cases a counterpart in the north or south keeps the balance. On 8 November the Sun also rises almost above the causeway, the line that also carries the calendar of 11.2.

The images (11.5b) were made with Stellarium at Giza, shortly before sunrise, with the Egyptian sky culture switched on. The figures are therefore drawn as Stellarium shows them and remain an assumption (see the other side below). The table shows for each figure how firm the link with an Egyptian god is.

10 May: the life that is about to flourish

On 10 May the Sun rises at about 77°, 27° north of the causeway line (104°). This morning therefore does not belong to the calendar of 11.2; it is its counterpart. Venus and Jupiter stand close to the Sun. In the east, around the Sun, stand Taurus, Aries and Pisces, and below the horizon Osiris waits: Orion remains invisible until 19 June, Sothis until 17 July (14.1). These are figures that in the later Egyptian world of gods belong to growth and fertility: Apis and Hathor (Taurus), Khnum who forms living beings (Aries) and the fish as a sign of rebirth. It is also the moment when the year runs towards its beginning: the Nile is low, and in just over two months Sothis returns with the new year and the rising water (17 July, M2 in 14.1b).

In the south stands Aquarius, the figure of Hapy and the water. Sadalmelik (α Aquarii) then stands about 45.5° high at azimuth 197°. That is the altitude of the southern shaft of the King’s Chamber (45°13′40″, 12.1), 17° from the south and thus within the 25° that we use in 12.3 for stars off the meridian. Sadalsuud stands at 40.6°, 2° above the southern shaft of the Queen’s Chamber (38°28′). The Field, the square of Pegasus, stands higher (47° to 67°); in Stellarium it is called “Veld (Som.)”, so it is Sumerian and not an Egyptian figure. The rising water and the purifying waters of later temple scenes, in which the king is purified with water, belong in our reading to this place. The altitudes are our own calculation (Vondrák precession, ±0.5°) and must be measured in the image.

In the north stand the Guardian (Arcturus) upright and the fox on the Thigh of the Bull, the Meskhetiu or Foreleg. We read the fox as a jackal, Anubis, the god of embalming. The Falcon, Horus, still hangs at an angle above the horizon. In this image life stands in the east and south and death in the north: a balance.

8 November: death, the Nile and new life

Half a year later the roles are reversed. Osiris (Orion) and Sothis set in the west, into the Duat; in May they were below the horizon in the east. Scorpius (Selket, who watches and mourns over the dead) stands in the east near the Sun, and on 13 November (K3) Peret begins: the seed goes into the earth. In the south the Mother Goddess Isis sits exactly on the meridian, with the Milky Way behind her and Khenti Irti, Min-Pan, Anuket and Satis around her. Anuket and Satis belong to Elephantine, in the south where the Nile enters Egypt. That the Milky Way was a celestial Nile is a later and uncertain reading (E), but the position is striking: the river of the sky runs past the goddesses of the Nile. In the north the Falcon, Horus, stands upright and directly opposite Isis; the fox hangs upside down below the Thigh. Mother in the south and son in the north keep each other in balance, with death in the west and east and the river in between: the family of Osiris in four directions.

The causeway as a marker

On 8 November the Sun rises at about 102°, 2–3° north of the causeway line (104–105°). The Sun reaches the line itself, 104.0°, five days later, on 13 November (K3, ⅓ of the year, start of Peret). In M5 the Sun already stands at 105.00° at 06:42, exactly in the Milky Way. In the same week the causeway thus marks two things: the start of the season in the calendar of 11.2 and the morning on which the Sun enters the Milky Way again, while Isis, Horus, Osiris and Sothis take their places in the sky. That may be chance, but it may also be deliberate: the causeway would then be not only a calendar but also a sight line to this moment in the stars. The May morning has no marker at Giza that we know of; that counterpart can only be seen in the sky. It is therefore a reading, not proof.

The Sokar festival: 26 Khoiak

Sokar is the falcon god of the necropolis near Memphis. He is depicted as a mummified falcon, and his title “He of Rostau” refers to the entrances of the netherworld. His festival fell on the 26th day of the fourth month of Akhet (Khoiak), with preparations on the 25th. At the festival the earth was hacked up and cattle were driven, people wore garlands of onions and an image of the god sailed around on the henu bark, a boat with a prow in the shape of an oryx. Later, together with the Osiris festivals, it grew into the Khoiak festivals, which ended on the 30th with the raising of the djed pillar, the resurrection of Osiris. The Sokar festival itself is already known in the Early Dynastic period and appears in the festival lists of the Old Kingdom (the Palermo Stone, the calendar of the sun temple of Nyuserre and tombs at Giza and Saqqara). The date 26 Khoiak, however, is only recorded for the New Kingdom (the calendar of Ramesses III at Medinet Habu). For Khufu’s time it is a projection (level E).

With New Year on 17 July 2580 BC (I Akhet 1, see 11.2), 26 Khoiak is day 115 after New Year: 9 November (Julian; 19 October Gregorian). That falls in the middle of the week of the two mornings. Shifting by one day because of the change of day (sunrise or midnight) does not essentially change that picture.

In the image of 8 November the Falcon, Horus, stands upright in the north opposite Isis, while Osiris goes into the Duat in the west. Sokar is also a falcon: the god of the necropolis of Memphis, later merged with Osiris (Ptah-Sokar-Osiris). The festival belongs with hacking up the earth and sowing at the moment Peret begins; in the sky, the image of death in the east and west and life standing upright in the north fits it. The causeway would then mark not only the start of Peret (11.2) but also the close of the Khoiak festival. That the henu bark was dragged over Khafre’s causeway is our own idea: there is no text that says so (level E).

The Falcon in Stellarium is a choice of the sky culture and not a proven Sokar. The probability that a random date falls within five days of K3 is about 3%, but there are many festivals and many markers in the sky; that is why it remains level E. Sources: Wikipedia (Sokar), ANEFest (Festival of Sokar; Festivals of the Old Kingdom) and Bielesch 2011 (The Khoiak Festival). All known festivals are listed with their Julian date in 11.6.

Level: D = our interpretation of a figure from the Stellarium sky culture; E = speculation, the sources are younger than the Fourth Dynasty. The altitude of Sadalmelik comes from our calculation.

What does the other side say? (1) For the Old Kingdom only Sirius, Orion and the Great Bear (the Foreleg) are reliably established as constellations (11.3); Guardian, Min-Pan, Khenti Irti, Satis, Pisces, Aries and Aquarius are later or a choice of the maker of the sky culture. (2) Every morning sky contains dozens of figures; whoever has enough figures finds a counterpart for every attribute. That is why we take the figures as they appear in the image and add nothing. (3) The attributes of the gods come largely from later sources. (4) The May morning does not lie on the causeway, so the causeway marks only one of the two mornings. (5) The date of the Sokar festival (26 Khoiak) comes from the New Kingdom; for Khufu this projection cannot be checked.

Source: own calculation (Vondrák precession, sun according to Meeus) and Stellarium (Egyptian sky culture); see also 8.7, 11.2, 11.3, 12.1, 14.1 and 14.2.

The four shafts from the King’s Chamber and the Queen’s Chamber run upwards at a fixed angle. We use the measurements of Petrie (1883): King’s Chamber south 45°14′, King’s Chamber north 31°33′, Queen’s Chamber south 38°28′ and Queen’s Chamber north 37°28′ Petrie 1883, §44 Petrie 1883, §56. Not because they are more precise than later measurements (the shafts are not straight over their length), but because Petrie measured on site with one fixed method. Which stars fit them?

Altitude at upper transit through the meridian (southern shafts above the south, northern shafts above the north), 29.975° N, geometric, without refraction. Precession according to Vondrák (2011), proper motion from the Bright Star Catalogue. Thuban reaches the angle twice, because around 2800 BC it stood almost exactly on the pole and then turned away again.

What stands out?

• The three datable shafts together point to the time of Khufu. Mintaka reaches the southern angle in 2606 BC, Sirius the angle of the Queen’s Chamber in 2725 BC and Thuban the northern angle in 2519 BC. Each on its own is off by decades; together all three are within 0.5° in the years 2608–2572 BC, at the beginning of Khufu’s reign (c. 2589–2566). The smallest combined deviation falls in 2584 BC.
• Thuban gets a role. With the Upuaut angle of 32.5° Thuban only fitted in 2356 BC. With Petrie’s 31°33′ the pole star of the time stands almost exactly on the angle of the King’s Chamber northern shaft in the time of Khufu (−0.34°).
• Sirius fits better. With 39.5° Sirius only reached the angle in 2405 BC; with Petrie’s 38°28′ that is 2725 BC, and in 2580 BC it stands 0.47° above it.
• Orion’s knee fits too. Saiph (κ Ori), the star at Orion’s foot, reaches the angle of the Queen’s Chamber southern shaft in 2486 BC and stands 0.44° below it in 2580 BC. With Sirius (until 2572) and then Saiph (from 2594) that shaft is filled throughout the 26th century: Isis and Osiris take turns.
• The northern shaft of the Queen’s Chamber drops out. At 37°28′ no bright star stands on the meridian in the 3rd millennium BC: Kochab is 2.3° and 5 Dra 1.6° too high. 5 Dra, which kept coming into view with the Upuaut angle of 39.1°, thus drops out of the hits. According to all measurements this shaft has a bend; Petrie only measured the lower part.

And if Petrie’s own correction is right? Petrie writes that the average slope over the whole length is probably not greater than 31° and 44½°. With those angles the belt reaches the southern angle between 2749 (Mintaka) and 2596 BC (Alnitak), and Thuban the northern angle in 2617 BC. Then too both shafts point to the century of Sneferu and Khufu.

The broad test: 33 stars, four shafts

So as not to look only at the stars that suit us, we fixed a broad list before calculating: every star that other researchers link with the shafts or the orientation (Trimble and Badawy 1964, Bauval 1993, Spence 2000), all stars of Orion (Osiris), the seven stars of the Great Bear (the Foreleg), the northern stars from a first exploration of this research and every other star that appears elsewhere in this book. Together 33 stars. Every star was tested on every shaft on its side of the sky. Only then do we look at which hypothesis fits.

KS = King’s Chamber south (45°14′), QS = Queen’s Chamber south (38°28′), KN = King’s Chamber north (31°33′), QN = Queen’s Chamber north (37°28′). Every star was tested on the shafts on the side of the sky where it crosses the meridian. In bold: stars that reach a shaft between 2750 and 2400 BC, and Mizar. The last column also counts stars that do not stand directly above the shaft: they reach the altitude of the shaft somewhere within 25° of the shaft direction, as with the star clock (12.3).

What the last column shows: on the meridian the King’s Chamber northern shaft has only one star (Thuban), because at 31°33′ only a star close to the pole can stand due north. Off the meridian, 14 of the 33 stars reach that altitude, including Kochab, Pherkad, five stars of the Great Bear and Mizar, most of them throughout the period. On the meridian Mizar never reaches a northern shaft (at best 2.5° above the Queen’s Chamber northern angle, in 3385 BC), but off the meridian it reaches the altitude of both northern shafts once a day from 4500 to 500 BC. So off the meridian, a star at the altitude of a northern shaft says nothing about the year; only the moment at which all shafts have a star at the same time can do that (13.3).

What the broad test shows. From 4500 to 500 BC three of the four shafts have a star at the same time (±0.5°) in only two periods: 2608–2432 BC (Orion’s belt, Sirius or Saiph, and Thuban) and a short window in 3740–3660 BC (Bellatrix, the belt in the Queen’s Chamber southern shaft and Edasich). All four at the same time does not occur at ±0.5° in the whole precession cycle; at ±1° only in 2464–2342 BC. So the point stands, even if we include all the stars examined: together the shafts point to Khufu’s century, and above all to Orion.

And off the meridian? If we count all stars that reach the altitude of a shaft, the northern shafts almost always have a star, and in 67% of the years two shafts are filled at the same moment (±0.5°). All four at the same moment remains rare: at ±0.3° only in 2460–2388 BC in the whole cycle (the belt, Saiph, Kochab and Phecda or Megrez), at ±0.5° in six periods, the longest of which is 2546–2264 BC. In Khufu’s reign itself there are three at once; four from 2546 BC (at ±1° already in 2580 BC, see the star clock in 12.3). The details are in 13.3.

How wide is such a window?

Only the windows from the 4th millennium BC onwards. In the entire precession cycle of 26,000 years the combinations on the meridian occur only in this one period (13.3). The last column: the star reaches the altitude of the shaft off the meridian, within 25° of the shaft direction. Off the meridian Kochab and Mizar reach the King’s Chamber northern shaft throughout the period; on the meridian never. All four at the same time at ±0.3°, off the meridian: only 2460–2388 BC.

Explanation: on the meridian a window is a few centuries wide and moves with precession; off the meridian a window for a southern star becomes two to four times as wide, and for a northern star often thousands of years. That is why the meridian remains the test that can date. Off the meridian, only the moment at which all four shafts have a star at the same time gives a date.

What does the other side say?

• Shafts are not telescopes. They were closed at both ends (the Queen’s Chamber shafts with small stone doors, the King’s Chamber shafts by the casing) and their slope changes along their length. An angle is then a design choice, not a sight line. That is a perfectly good reading: the builders may have chosen the angle to match the altitude of a star, without anyone ever looking through it.
• Choice of star. Hence the broad test above: 33 stars, fixed in advance. With random bright stars two shafts are often filled at the same time (13.3), but three at once only in the two periods mentioned.
• Measurement differences. Petrie and the robot Upuaut differ by up to 1° (King’s Chamber north 31.55° versus 32.5°; Queen’s Chamber 38.5° versus 39.5°). Each degree shifts a date by 100 to 250 years. The finding above therefore depends on the choice of Petrie.
• Windows, not dates. A star changes its altitude by only c. 0.005° per year. Even with Petrie’s angles a shaft dates to within decades, not to a single year.

Source: Petrie 1883, §44 · Petrie 1883, §56 · Allen 2005, pdf p. 43 · Allen 2005, pdf p. 59 · Allen 2005, pdf p. 118

On the meridian only the northern shaft of the King’s Chamber fits a star: Thuban, the pole star of the time (12.1). For the Queen’s Chamber an indirect story is needed: the shaft does not point at one star at one moment, but at the region of the stars that turn around the pole and never set.

I1, I4 and I5 come from the texts; I2 and I6 from Petrie’s measurements and our own calculation. Levels A–E as in 9.1.

How straight are the shafts themselves?

A shaft is not a ruler: it starts horizontally, bends upwards and sometimes changes direction. The measured angle depends on where you measure.

Petrie: Petrie 1883, §44 Petrie 1883, §56. Upuaut: summarised by Alison Alison and Wikipedia: Star shaft. The difference between the two measurements is up to 1°; that determines which star fits.

How precise could the builders be?

The Great Pyramid is aligned to north to within less than 3 arcminutes (0.05°); the pyramid of Khafre to within c. 6 arcminutes (Spence 2000). That is twenty times better than 1°. For the shafts this means two things:

• For a building, 1° is not a technical limit. Whoever aligns a pyramid to 0.05° can lay a straight section of passage to 0.1°. But a shaft that has to go around the Grand Gallery is not a straight section; for the Queen’s Chamber northern shaft ±1° is therefore within the uncertainty.
• So a deviation of 1° is not a “near hit” either. For the three shafts for which Petrie gave a clear average slope, we test at ±0.5°. At that margin Orion, Sirius and Thuban remain, and only around 2590 BC at the same time.

Source: Allen 2005, pdf p. 20 · Allen 2005, pdf p. 43 · Spence 2000, pdf p. 1 · Spence 2000, pdf p. 3 · Berio 2014, pdf p. 42 · Petrie 1883, §44 · Petrie 1883, §56

The idea: write down every moment at which two or more stars stand on a shaft angle at the same time, and see at the end of the year whether there is a pattern. Below is that count for 2580 BC, recalculated with Petrie’s angles.

Four moments with a fixed sky

Geometric altitudes, 29.975° N, 2580 BC. A star counts if its altitude is within ±1° of Petrie’s angle and its azimuth within 25° of the shaft direction.

Midnight: windows in which two or more shafts have a star

KS = King’s Chamber south (45°14′), QS = Queen’s Chamber south (38°28′), KN = King’s Chamber north (31°33′), QN = Queen’s Chamber north (37°28′). Tested with all 33 stars chosen in advance in 12.1. Only nights in which at least two shafts have a star at the same time (116 of the 366 in total). The periods in bold contain a southern shaft. The full list per day is in sterhoogtes-2580vChr.csv.

What do you see at the end of the year?

• The windows reveal the time of year. The southern shafts are filled only twice a year, shortly after one another: Orion from 2 to 21 November, Sirius from 25 November to 17 December. Whoever read the shafts at midnight read them in the first half of Peret, directly after the flood. In exactly those weeks the Sun rises above the causeway (13 November, 11.2).
• Kochab is in it. In the Orion window Kochab, the brightest star of the Little Bear, stands on the King’s Chamber northern shaft from 6 to 17 November, together with Megrez or Phecda (Great Bear) on the Queen’s Chamber northern shaft. With the Upuaut angles (7.3) 5 Dra stood there.
• 5 Dra has disappeared from the Orion window, but not from the year: at the end of September, in mid-October and in the Sirius window it stands at 37.8–38.4° at midnight, just within the 1° margin of the Queen’s Chamber northern shaft.
• Thuban fills the spring. From the end of February to the end of June Thuban stands on the King’s Chamber northern shaft at midnight: the pole star was then at its highest at midnight.
• All four at the same time: 11–15 November. With the broad list Orion also fills the Queen’s Chamber southern shaft: Saiph, Orion’s knee, stands at 37.6–38.1° at midnight. In those five nights every shaft has a star: the belt (King’s Chamber south), Saiph (Queen’s Chamber south), Kochab (King’s Chamber north) and Megrez (Queen’s Chamber north). Sirius and the belt together never happens: they are 1.5 hours apart on the meridian.

Is this chance? The check

At midnight there is no pair at all from 25 June to 9 September and from 19 December to 20 February. But the base probability is high: at a random moment of the day there is a star on the King’s Chamber northern angle 60% of the time, on the Queen’s Chamber northern angle 52% of the time and on both 25% of the time. At ±2° that is 76%.

The same count, margin and azimuth; only the year differs. 2780 and 2280 BC have been added. Counted with the northern stars from the list of 33 (12.1): the seven of the Great Bear, Kochab, Pherkad and Polaris, Thuban, 5 Dra, Edasich and Aldhibah, and Vega, Deneb and Arcturus (which never stand in the north at these altitudes). The last column counts all 470 stars to magnitude 3.9 as a check.

What the count shows. With Petrie’s angles the peak of the northern pairs lies not in 2580 BC but around 2340 BC (153 nights). That is due to the Great Bear, which in those centuries passed through the altitude of both shafts at midnight. So the count of midnights does not date the construction: it mainly measures how crowded the northern sky is at that altitude. The dating lies in the meridian test of 12.1, where the three stars chosen in advance fit together only around 2590 BC.

What does the other side say?

• Search effect. With 33 stars, a margin of ±1° and an azimuth window of 25° there is often some star. A margin of 0.3° or a smaller list of stars makes the hits rarer. Those requirements are fixed in advance here.
• North and south are not independent. The Orion window exists because Orion stands on the angle; at a 25% base probability the northern stars add little.
• Midnight is a choice. At another hour all windows shift through the year by c. 15 days per hour. Midnight is, however, the natural reference point of the night.

Source: own calculation (Vondrák precession, proper motion, Bright Star Catalogue)

With the Upuaut angles (7.3) 10 November was the finest moment: Alnilam, Kochab and Phecda in three shafts. With Petrie’s angles that moment falls one day earlier, and it is sharper. The four directions of this night are in 12.4b (PT1). On 9 November 2580 BC at 00:09 (day 115 after New Year, 117 after the solstice; 26 Khoiak, the day of the Sokar festival, 11.5) we have:

Geometric altitudes at 29.975° N. The moment is 9 minutes after midnight and 4 minutes before Alnilam’s transit through the meridian.

And the fourth shaft? At the same moment Saiph, Orion’s knee, stands at 37.61° (azimuth 172°): 0.86° below the Queen’s Chamber southern shaft. All four shafts then have a star within 1°. Two nights later, on 11 November at 00:10, all four are within 0.63°: Mintaka 45.29° (+0.06°), Saiph 37.84° (−0.63°), Kochab 31.16° (−0.39°) and Megrez 38.09° (+0.62°) (SN10). Three of the four stars come from the lists of Trimble, Badawy and Bauval or from Orion itself; Megrez belongs to the Foreleg.

That is about ⅓ of the year after the solstice (day 117 of 365, day 115 after New Year; the exact third after the solstice falls on day 121.7, the day on which the Sun rises above the causeway, 11.2).

And New Year itself?

At the moment Sirius appears on the horizon at New Year, 17 July 2580 BC (04:05, Sun at −10.3°), the north shows:

The altitudes at the rising of Sirius are the same every day; only the hour and the position of the Sun shift (at the solstice, 15 July: 04:13, Sun −8.8°). Thirteen minutes later, when Sirius is clearly visible for the first time (04:18, Sun −8°, Sirius 2° high), Dubhe (33.2°) and Kochab (38.6°) are 1–1.7° off; only Pherkad remains just within the margin (−0.96°). The four directions of the solstice are in 12.4b (PT2).

With the Upuaut angles (7.3) Kochab and Dubhe both stood on their shaft at the rising of Sirius. With Petrie’s angles the picture is slightly different: Dubhe remains (+0.6°), and Kochab’s place is taken by its neighbour Pherkad. Both form the “bowl” of a dipper: Dubhe of the Great Bear, Kochab and Pherkad of the Little Bear.

What makes this moment strong

• Three shafts in one image, with the best-known northern star of the time (Kochab) to within a hundredth of a degree and the Great Bear to within a tenth of a degree.
• The timing: midnight, about ⅓ of a year after the rising of Sirius, in the weeks in which Orion stands on the meridian at midnight.
• The check: 2000 years earlier or later the image is completely gone (SN7-V and SN7-L).

What the other side says

• The stars were chosen afterwards. That precisely Kochab and Megrez fit at the same time has a probability of 2.4% (±1°) or 0.7% (±0.3°), but only if they had been designated in advance. Afterwards you choose from all possibilities: that some pair of northern stars fits happens 25% of the time (±1°).
• The date adds little. With Orion on the meridian the whole sky is the same every day; 9 November is merely the day on which that happens at midnight.
• Check over time. Orion on its angle (±0.5°) gives 2705–2344 BC; throughout that period some pair of northern stars always stands on the angles together with Orion. So the northern stars do not refine the dating; Orion, Sirius and Thuban do the work.

Own calculation for 2580 BC, 1440 moments over one sidereal day.

Source: Allen 2005, pdf p. 43 · Spence 2000, pdf p. 1