Pyramidions
Pyramidions
Diodorus's description of Khufu's pyramid implies the pyramidion and several courses of stones were missing as early as the 1st century BC:
“For the largest is in the form of a square and has a base length on each side of seven plethra and a height of over six plethra; it also gradually tapers to the top, where each side is six cubits (~2.8m) long.” (Diodorus Siculus ‘Library of History’ Book 1, 63)
Pliny's description is similar:
“The largest Pyramid occupies seven jugera of ground, and the four angles are equidistant, the face of each side being eight hundred and thirty-three feet in length. The total height from the ground to the summit is seven hundred and twenty-five feet, and the platform on the summit is sixteen feet and a-half in circuit.” (Pliny the Elder, ‘The Natural History’ Book 36, Chap. 17; translation, J. Bostock, H. Riley 1855-57)
It appears from this translation of Pliny, that in the 1st century AD, the pyramidion was missing from Khufu’s pyramid. Pliny mentions a ‘platform’ at the summit with a perimeter of 16.5 ft (5 m).
Estimation of the size and weight of the missing pyramidion using Pliny's measurement:
~1.25 m square base; ~0.8 m high; weight, ~1 ton.
"From a distance this building appears to end at a point, yet there is a flattened space at the top of about 20 feet (~6 m) square." (from the journal of Lt-Colonel Fitzclarence 1819)
drawing of summit c.1842: link
If the smooth casing was still intact, how did anyone climb up the steep smooth sides of the pyramid to measure the top?
The pyramidion atop a ~146 m high structure with steep (~52 degs), smooth sloping sides (the casing was removed in the Middle ages), weighed over a ton, so anyone attempting to remove it needed to erect extensive scaffolding to reach it.
Perhaps a severe earthquake dislodged it.
Very few pyramidions have survived in the archeological record. One fine example is from the pyramid of Amenemhat III, built near Snefru’s two pyramids at Dahshur. Made of dark grey granite with an angle of incline probably defined by sqd 5 (similar to the lower part of Snefru’s southern pyramid at Dahshur), its inscribed with texts – the opening part of the text on one of the faces reads:
"King Amenemhat sees the beauty of the sun. The face of king Amenemhat is open, he sees the lord of the horizon as he sails in the sky. The soul of King Amenemhat is higher than the heights of Orion, and it unites itself with the Netherworld (dwAt) ......" (A. Piankoff 1969: 5)
The Sun (ra) and Orion (sAH), both mentioned on Amenemhat's pyramidion, are brothers in the Pyramid Texts.
A pyramid defined by sqd 5 has an angle of incline of 54.462 degs. This is the nearest whole number seqed that closely approximates the slope of a pyramid with four equilateral triangles (54.736 degs). At least three pyramids were built with an angle of incline that closely approximates this seqed.
The most popular model however, was a pyramid defined by sqd 5 ½. The basic form of possibly as many as 12 pyramids was defined using this seqed.
Why this particular seqed?
The height of a seqed 5 1/2 pyramid has an interesting mathematical relationship to the perimeter of the base – the radius of a circle with a circumference that is more or less identical to the perimeter of the square base of the pyramid, corresponds to the height of the pyramid.
This may have influenced the choice of a sqd 5 ½ pyramid – it mathematically incorporates a circle within a square / triangular structure.
The sun was the brightest object in their sky that appeared as a circular disk - a circle with a small circle in the centre is the hieroglyphic sign for the sun. The circle motif is also found in the hieroglyphic sign for Duat - a 5-rayed star enclosed within a circle. The royal mortuary complex was also 'enclosed' within the circular horizon, another important aspect of the royal afterlife beliefs.
A Dynasty 5 alabaster offering table with a circle / square theme, from the open court in the sun-temple complex of Niuserre at Abu Ghurab:
The ideogram for 'offering' (R4) repeated 4 times to form a square enclosing the circular centerpiece (representing the sun) of the offering table.
The underside of the pyramidion of Khendjer, has a round tenon in the centre and the triangular profile of the pyramidion inscribed around it - the round tenon is enclosed within the top of the triangle:
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. §151.
The ideogram for Duat (N15) is a 5-rayed star in a circle - an encircled star - just as the dead king who was reborn as a star, who appeared as a star, was encircled by the Duat.
A Dynasty 5 sun-temple complex, has a large offering ‘table’ with a circular stone block that probably represents the ideogram for ‘sun’ (N5), in the centre of a square made up of four large stone blocks, each shaped to form the hieroglyphic sign, R4, the ideogram for ‘offering’. The overall idea of the design is clear – the circular disc of the sun enclosed within a square offering table.
The earliest true plane-sided pyramids were built during the reign of Snefru, and the seqeds
“The pyramid’s date and function as a tomb are in no doubt, despite the fact that the king’s body and all funerary equipment fell victim to tomb robbers and disappeared without a trace. However, its enormous size, the astonishing mathematical properties of its design, and the perfection and accuracy of its construction still invite unscientific explanations.” (Malek 2000: 95)
Exactly what are "the astonishing mathematical properties of its design"?
Petrie noticed that the floor of the sarcophagus chamber was placed at a level where:
- the area of the horizontal section (311.127 x 311.127 = 96800 cubits) = half that of the base (440 x 440 = 196600 / 2 = 96800 cubits)
- the diagonal (440 cubits) = length of a side of the base of the pyramid.
width of the face (311.127 cubits) = half diagonal of base (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)
This suggests 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 level of the sarcophagus chamber can also be determined graphically without mathematics, using a circle / square combination that also determines the basic vertical cross-sectional interior layout of Khufu's pyramid:
Using a circle / square combination to determine the level of the sarcophagus chamber, 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.
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. §151.
The ideogram for Duat (N15) is a 5-rayed star in a circle - an encircled star - just as the dead king who was reborn as a star, who appeared as a star, was encircled by the Duat.
Using 2 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 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 places in the Iunu / Ineb-hedj area:
Chris Tedder, spring 2008
updated autumn 2008