Graham: “I think if Chris had used 55 degrees elevation for Betelgeuse he would find that Alnitack was in vertical alignment at an elevation of 45 deg. The date would have to be 2535 BC which is close to Kate Spences's date. Interesting too, the ratio of the angles 55 and 45 degrees is 11 to 9, the same ratio of the heights in the Bent Pyramid - 110 and 90 cubits. !”
Interesting observation!
Betelgeuse 55°; Alnitak 45°: location, Snefru’s southern pyramid c.2534 (Khufu c.2495)
Kate Spence has 2505 for the setting out of Snefru’s southern pyramid - about 80 years later than her 2583 “currently accepted chronologies” date for Snefru and 30 years later than the 2535 date. (radiocarbon dates tend to be earlier rather than later)
In contrast to Spence’s ‘simultaneous transit’ of Kochab and Mizar, Orion’s 3-star vertical alignment dates Snefru’s southern pyramid to c.2630, consistent with Snefru: 2613-2589 +-50 years (Shaw 2000: 480)
Betelgeuse 54.5° and Alnitak 44.5° c.2630 BC (10° apart)
An 11.022 : 9 ratio or for the pyramid, 110.2 : 90, which means a negligible 10cm (3.9 inches) difference in the height of the top part.
As you have mentioned, the numbers 11and 9 crop up in connection with deities:
The Big (or Great) Ennead was in Iunu, and Atum was foremost of the Big Ennead with 4 pairs of deities + Atum = 9
Interestingly, in the Pyramid Texts, 11 deities are also mentioned in connection with the Big Ennead:
[O] Big Ennead in Iunu! May you make Meryre [be firm, may you make this “Perfect Fixture” be firm for] the course [of eternity as the name of Atum, foremost of] the Big Ennead, [is firm]*. Then follows a list of 11 deities:
Shu; Tefnut; Geb; Nut; Osiris; Osiris Foremost of Westerners, Seth; Horus; Sun; Eyes-Forward; Wadjet. P 582, M 226, N 360 (PT 601) Sethe Vol. 2: 377-382; §1661-1671d
*Allen’s note: “This stanza and those that follow refer to the individual members of the “Big Ennead.”
Khufu’s pyramid has an angle of incline defined by sqd 5 1/2 a 14:11 ratio. The number 14 crops up in connection with the moon associated with Thoth.
“The ascendant phase of the moon is sometimes represented by 14 steps, on which the gods of the Great Ennead of Karnak are standing. Each step represents one day of the ascendant phase of the moon: the gods of the Great Ennead penetrate one after another, one each day, into the eye to increase it. The Healthy Eye, presented on a wAd which means ‘healthy’, ‘flourishing’ - is the full moon, ie. the eye healed and recreated by Thoth.” (Dr. A.S. Von Bomhard 1999: 79)
Nefermaat and his son Hemiunu held the titles ‘Overseer of All the King’s Works’ and ‘Greatest of the Five in the House of Thoth’ during the reigns of Snefru and Khufu, so perhaps it's not only solar and stellar aspects that can be explored, but also lunar.
It's quite challenging juggling various astronomical and geometric schemes that may or may not apply to the intended design of the royal mortuary complexes. The object of the exercise is to find a coherent and unified explanation for the designs - to uncover the underlying motivation and inspiration for the architectural elements that are often enigmatic. Many thought-provoking ideas have come to light from posters here that provide possible pieces of the puzzle.
Chris