Mark: “My model predicts a slope of just under 45 degrees. Petrie made observations in which he determined a mean angle of just over 44.5 degrees for the core masonry which is concordant with my model. Many others have reported a less steep slope. Could Petrie be correct in his assessment of the angle? He was sure it was just less than 45 degrees and not intended to be 45 degrees”

I came across a 20-year-old web page of mine on the web-archive that includes a possible explanation for why the angle of incline of the ‘Red’ pyramid is not 45 degrees (sqd 7), but just under 45 degrees. I copied some of it which I updated, together with some images that illustrate the idea (links at the end):


Concerning the small subsidiary pyramid to the south of the ‘Bent’ pyramid:

”The angle of the casing on a good block at the E.S.E is 44° 34’ (44.57°) …….This is probably the same angle as the North Stone pyramid, which I observed as 44° 36’ (44.60°) on its core masonry. (‘A Season in Egypt’ Petrie 1887: 31)

Petrie measured the angle of the lower part of the ‘Bent’ pyramid and noted the face was slightly convex decreasing from 55.02° lower down to 54.52° further up (mean: 54.77°).
The angle of incline changes to 43°….43.65° for the top part of the pyramid, which is also slightly convex. The convexity of the faces means that the intended angles of inclines are uncertain.

It seems likely the angle of incline of the ‘Bent’ pyramid’s small subsidiary pyramid (44.57°), the top part of the ‘Bent’ pyramid (43…43.65°) and the ‘Red’ pyramid (44.6°) were all intended to have a similar angle of incline of about 44°…44.6°, but why not 45° (sqd 7)?

The angle of incline for the lower part of the ‘Bent’ pyramid (about 54.77°) is close to sqd 5 (54.46°), but why did they choose this particular angle and why a less steep angle for the top part and why a slightly less than 45 degrees angle for the ‘Red’ pyramid?

A possible explanation concerns the large distinctive Orion constellation crossing the southern part of their sky. During the early era of pyramid building, it provided a means of determining a north/south line to orientate the pyramids to the cardinal points. Three stars in Orion: Betelgeuse, Alnitak and Theta 2 Orionis (the middle star in the ‘sword’ of Orion), were in vertical alignment on the meridian due south c. 2560 BC. The east and west sides of Khufu’s pyramid are about 3’ west of north that corresponds to a sighting to the vertical alignment in Orion c. 2580 BC.

The earlier ‘Bent’ pyramid is orientated about 12’ west of north that corresponds to the vertical alignment c. 2630. Betelgeuse the uppermost star in the vertical alignment was 54.52° above the horizon and Alnitak, the middle star, 44.53°. These two angles are similar to the angles of incline for the lower and upper parts of the ‘Bent’ pyramid.

The ‘Red’ pyramid is orientated about 5’ west of north that corresponds to the vertical alignment c. 2590. Alnitak was 44.7° above the horizon, which again is similar to the angle of incline of the pyramid.

Petrie’s mean angle of 44.6° for the ‘Red’ pyramid, which is concordant with your model, is consistent with using the vertical alignment in Orion to determine the north/south direction line. Alnitak, the middle star in the alignment, was 44.7° above the horizon, which closely matches the angle of incline of the pyramid. They may have chosen this particular angle of incline so that the north face aligned with Alnitak that provided a celestial connection to Snefru’s tomb.

The angle of inclines of the ‘Bent’ pyramid, may have been chosen so that the north faces aligned with the two bright stars Betelgeuse and Alnitak in the Orion vertical alignment used to determine the north/south orientation of the pyramid.

Link to diagrams of the alignments: [sites.google.com]

Dates for Dynasty 4 pyramids if orientated with sightings to Orion’s 3-star vertical alignment:

Maidum: -18’, 2662 BC (probably started by Huni, 2637 - 2613 +-50 years)
DS: -12’, 2630 (Betelgeuse: 54.52°; Alnitak: 44.53°)
DN: -5’, 2590 (Alnitak: 44.713°)
Khufu: -3, 2580 BC
Djedefre: -48’, 2825 BC*
Khafre: -4.1’, 2590 BC
Menkaure: +12.7’, 2500 BC

*The substructure of Djedefre’s pyramid is not typical of Dynasty 4 construction that suggests he may have taken over an early dynastic tomb.

The royal pyramids in the Old Kingdom deviate from cardinality, expressed as either a positive angle if the deviation is clockwise or a minus value if the deviation is counterclockwise. The earliest plane sided pyramids up to Khafre’s pyramid, deviate counterclockwise, and from Khafre onwards, clockwise. This pattern implies a precessional induced skew that would result from sightings to the same stars over many years. Orion’s 3-star vertical alignment and orientations of the pyramids follow a similar pattern with dates that are consistent with the dates for Dynasty 4 in the ‘Oxford History of Ancient Egypt’ edited by Ian Shaw: 2613….2498 BC. +-50 years.

The significance of Orion:

Orion was personified by the deity Sah (sAH). In the earliest Pyramid Texts, he is referred to as ‘father of the gods’ who authorised the kings' status in the afterlife. Sah was probably an ancient deity as he is attested on an Early Dynastic seal that has the Sah hieroglyph D61 (sAH), above and behind the national shrine of Upper Egypt - the 'Great House' (pr wr) at Nekhen (nxn) - the "earliest centre of Egyptian royal power and cult centre of Horus as god of kingship “.

An early Dynasty 4  mention of Sah, is an inscription in the tomb of Metjen, Ruler of the House / Palace of 'Great Sah' (wr sAH + 3x 'town' or 'place' det. O49).

'Great Sah' was probably named after Sah, referred to as 'Great' in the Papyrus of Ani, Pl. 15, Chap. 23: "I am Sah (sAH) the Great who dwells with the Souls (bAw) of Iunu.”

A Pyramid Text referring to Sah has the 'town' determinative, O49: "I have gone up on the ladder, with my foot on Orion (sAH) and my arm is in elevation."  Nt 272b (PT 625A) §1763c

In the Middle Kingdom Coffin Texts, a 'Mansion of Orion' (Hwt sAH) is mentioned. The deceased is led by Sirius, and receives Orion’s staff that enables the deceased’s deification:

"My mother is Sirius (spdt), and she prepares my path, she sets up a stairway to this very great plain of Nenmut for my ascent from the Valley of the Mountain of the Sehseh-bird on the north with my river-banks, at the place where Orion (sAH) issues.  I find Orion (sAH) standing in my path with his staff of rank(?) in his hand;  I accept it from him and I will be a god by means of it.  He gives me his staff which is in his hand, and I will cause the strong ones(?) to tremble by means of it.  I will spit on(?) the great ones by means of it, it will make me noble in the presence of Sirius, it will make me secure in the Mansion of Orion.  I ascend and appear as a god, my signs of rank are on me, and I will make the lands of the Fenkhu impotent through them.  I sit on the throne within the shrine and make summons to Orion (sAH) that he should come to me.  O Orion (sAH), come and see me!......."  CT 469 V, 389-391

Chris