gizadiagonal

Aspects of Old Kingdom architectural design

Petrie noticed the floor of the sarcophagus chamber was placed at a level where:

- area of horizontal section: 311.127 x 311.127 = 96800 cubits = half base area: 440 x 440 = 196600 / 2 = 96800 cubits

- diagonal: 440 cubits = length of a base side of pyramid.

- length of sides: 311.127 cubits = half diagonal of base of pyramid: 622.254 / 2 = 311.127 cubits

("The form and size being thus fixed, the floor of the main chamber of the building — the King's Chamber — was placed at the level where the vertical section of the Pyramid was halved, where the area of the horizontal section was half that of the base, where the diagonal from corner to corner was equal to the length of the base, and where the width of the face was equal to half the diagonal of the base."  Petrie 1883: 220)

Diagonal of sarcophagus chamber's square level = 440 cubits (length of a base side of pyramid).

Diagonal of upper shaft's outlets square level = 280 cubits (height of pyramid).

The dimensions of the diagonals of these two square levels define the basic proportions of the pyramid. This implies the pyramid designers were familiar with the mathematics and geometry of a pyramid, as is evident in the later MMP Problem 14 that "establishes beyond any doubt that the Egyptians had a standard method for finding the volume of a truncated pyramid.  This would represent the very acme of Egyptian mathematical achievement, except for MMP 10, which some think with justification establishes a formula for finding the area of the curved surface of a hemisphere."  (Gillings 1972: 187-8)

The sarcophagus chamber floor level and the level of the outlets of the upper shafts can be determined graphically using circle  / square combinations:

The relationship between the square and circle inscribed within the square defines the angle of incline of the pyramid:

Relationship of a square to circle = 14:11, the same ratio as the height to half base of the pyramid resulting in an angle of incline of 51.84277 degs.

circle radius = 1;  each side of square around circle = 2;  area of square = 4; area of circle = 3.142

4 / 3.142 = 1.272727...;  height / half base ratio of pyramid = 14:11 = 1.272727... = 51.84277 degs. angle of incline of pyramid.

perimeter of square = 8 (56 palms)

circumference of circle= 6.2857... (44 palms)

8 / 6.2857... = 1.272727... (56 / 44 = 1.272727...) = 51.84277 degs. the angle of incline of the pyramid.

4/𝛑 (22/7) = 1.2727 = 14:11 = sqd 5/2 = 51.84°

Using circle / square combinations to determine the level of the sarcophagus chamber, and the level of the upper shafts's outlets may have satisfied ideological concerns - how to incorporate the circular form of the sun, full moon, duat and horizon / akhet within a square / triangular pyramid design.

The circle motif is found in the ideogram for Duat - the realm of the dead.  From the earliest Pyramid texts at the end of Dyn 5:  "Sah (sAH, 'Orion') is encircled) by the Duat, pure and living in the horizon.  Sopdet (spdt, 'Sirius') is encircled up by the Duat, pure and living in the horizon.  I am encircled by the Duat, pure and living in the horizon.  It is well with me and with them.  It is pleasant for me and for them, within the arms of my father, within the arms of Atum.  W 149 (PT 216) Sethe Vol. 1: 86; §151.

The ideogram for Duat (N15) is a 5-rayed star in a circle - an encircled star - the dead king, reborn as a star, appeared as a star, encircled by the Duat.

Some egyptologists refer to the shafts as model passageways for the ba of the king to ascend to the sky or exit and enter his house of eternity. The outlets for the upper shafts are about 22 cm x 22 cm square. which means that as the ba approached the openings at the top of the shafts,  an area of the sky became visible that grew larger as the ba drew closer to the opening. At some point, the 3-star asterism in the centre of Orion was framed in the square opening with the middle star Alnilam in the centre. When the sun aligned with the shaft it appeared as a circle enclosed by the square opening - further from the opening the sun's disc filled the square, closer to the opening it appeared as a circle in the centre of the square.

A Dyn 5 alabaster offering table with a circle / square theme, from the open court in the sun-temple of Niuserre at Abu Ghurab:

The ideogram for 'offering slab' (R4) is repeated four times to form a square enclosing the circular centrepiece of the offering table.

Other strategic points in the design can be found by using two arcs as an aid to determine the beginning and end of the inclined section of the descending passage:

relationship between the upper north shaft and the entrance/exit in the north face:

The angle of a line drawn from the floor of the entrance into Khufu's pyramid in the casing on the north side of his pyramid, to the same height above the base as the theoretical exit of the floor of the upper south shaft in the casing of the south side of the pyramid, is ~22.5 degs - half the 45 deg angle:

upper shafts

The overall layout of the upper shafts connected to the sarcophagus chamber, was integrated into the basic geometry of the pyramid itself - the angle of incline of the south shaft is based on sqd 7 (the diagonal of the square base of the pyramid), and the north shaft, sqd 11 (the diagonal of the 280 x 440 cubit rectangle that defines the vertical cross-section through the pyramid). 

detail of the layout of the shafts at the sarcophagus chamber

Snefru's southern funerary complex at Dahshur

45 deg, 2000 cubit long diagonal:

Dual Diagonal:

alignments between sacred sites in the Iunu / Ineb-hedj area:

The triangular alignment made up of two back to back 7:24:25 right-angled triangles aligning the pyramid of Netjerikhet, Khem and Iunu (both associated with Netjerikhet), is also the hieroglyph for Sirius, the brightest star in their sky personified by Sopdet (spdt)

Orion (sAH) over the mortuary complex of Horus Netjerikhet as viewed from Djedefra'a pyramid ~52.2 degs to the northwest. When Alnitak and Saiph were in vertical alignment, the angle of the 3-star asterism in the centre of Orion closely matched the angle of the 3-pyramids at Giza.

Chris Tedder, spring 2008

updated January 2020