Theory and Practice: The Pyramid Angle

Screen shot from the documentary about the theory of Jean-Pierre Houdin.

These are the angle measurements Flinders Petrie obtained from various spots around the Great Pyramid using different methods. At first look, the differences appear small, but are they really? When you convert his measurements to the Egyptian pyramid angle scheme of palm per cubit, all conform to a 5 ½ palms run per cubit (or 7 palm) rise. The granularity of the cubit is not the same as the heximal or decimal metric. That explains why these small differences get absorbed in the cubit system.

Petrie's MeasurementDegreesArc-sec.Dec. sqdper 25 r.c.
Casing Stones north51º 46' 45"51°+2,805‬"51.7792°5.5p/7p31c 5p 1f
51º 49'"2,940‬"51.8167°5.5p/7p31c 5p 2f
51º 44' 11""2,651‬"51.7364°5.5p/7p31c 5p
51º 52'"3,120‬"51.8667°5.5p/7p31c 6p
51º 53'"3,180‬"51.8833°5.5p/7p31c 6p
Entrance Passage Mouth51º 53' 20""3200"51.8889°5.5p/7p31c 6p
North Shaft (K.C.) Mouth51º 51' 30""3090"51.8583°5.5p/7p31c 5p 3f
South Shaft
(K.C.) Mouth
51º 57' 30""3450"51.9583°5.5p/7p32c
Perfect Golden Angle51° 49' 38.2525""2,978.2525"51.8273°5.5p/7p31c 5p 2f
5.5p/7p sqd51° 50' 33.9843""3,033.9843"51.8428°31c 5p 3f
From Ronald Birdsall;

However, it is clear that the builders were not able to keep a constant pyramid angle. The angle across all of Petrie's measurements varies by more than 13 arc-minutes (799 arc-seconds). On the south wall the angle is almost a perfect 52° which is neither a golden angle nor the angle the sqd dictates, nor the angle on the north base. To illustrate how much of a difference during building this makes, I calculated the height a 25 cubit horizontal distance would correspond to at each of these various angles. This is the result:

It varies by two full Egyptian palms, which is 8 fingers, or circa 15 cm or 6 inch. I chose 25 cubits for a specific reason, because when Jean-Paul Bauval and I reconstructed the architectural theme from Petrie's measurements we ran up against a problem with his numbers: Petrie estimated that the lip of the floor of the Entry Passage of the cased Great Pyramid was at a height of 668 inch which is 32.4 cubits. In our model of the pyramid design, we predict a spot x on the pavement 25 cubits south of the north base (or north of the south base) vertically intersected the cased pyramid where the floor of the ascending passage once emerged.

In theory, this would be at a height of 32 cubits, but this would not be consistent with a perfect golden angle or the slightly steeper Great Pyramid's sqd which instead predicts a height of 31 cubits, 5 palms and 2 finger and 3 fingers (31.8 rc), respectively; that's 1 palm and 1-2 fingers short. However, it is consistent with what was actually built on the south side and that was my main goal to show in this blog post. The angles Petrie measured are consistent with a design which directed the builders to achieve a height of 32 cubits at a horizontal distance of 25 cubits, especially so on the south side indicating that the intent was to achieve a certain design length rather than obey the sqd at each individual cubit steps.

The difference on the north side at the Entrance Passage from a perfect height of 32 cubits per 25 run is not insignificant at 1 palm despite the steeper angle there compared what Petrie measured at the base on the ground. The interesting thing however is that this one palm mismatch still correlates with an Entry Passage angle more shallow than predicted by a perfect 2/1 triangle design. In a perfect design application of a Two/One Triangle, a 64 cubit horizontal distance will correspond to a 32 cubit vertical distance because the angle is Inv. Tangent (1/2) = 26.565°. The actual angle of the Entry Passage however is more shallow at 26.461° (26° 27' 41"). What this means is that the angle sags from a perfect Two/One design and that explains why at a spot x on the base at 64 cubits-x-25 cubits the height achieved is less than 32 cubits.

To better illustrate the source of the mismatch, I intersected a golden angle with a Two/One Triangle angle using proportions of 64/32 and 25 as in the Great Pyramid design we proposed. The angles do not intersect at a right angle of course since they are not opposites in a right triangle. The corresponding pairs are 51.827°/31.173° and 26.565°/63.435°. The 32 line (here I scaled down by 1 to 10, so it is 3.2 m in the illustration) overshoots the golden angle line by 0.14 and that means the intersection occurs at 31.86. The angle to intersect the 32 line at that height, i.e. 31.86, has an angle of 26.48° or 26° 28' 48" versus the 26° 27' 41" actually measured which is very close.

So the upshot of all this is: The Entry Passage design was probably based on the Two/One Triangle and the target in the design was 32 cubits from a base of 64 with 25 cubits left to the north base. But since this would have placed the lip of the entry outside of the pyramid, the passage angle was slightly narrowed and the pyramid angle may have been steepened at the courses above the base and below the pyramid's Entry. One fascinating implication here is that the smooth casing may not have been part of the original build and represents a post-modification similar to that done at Meydum.

The one remaining problem is why Petrie measured (or extrapolated rather) 668 inch for the E.P. at the casing. That is still 0.4 cubits higher than 32 (1 cubit at the Great Pyramid is 20.62 inch), a not insignificant mismatch of 8 ¼ inch. The only way to reconcile this is to lengthen the corridor from that which is predicted by a 64/32 cubit design. In the Two/One Triangle, the corridor is the hypotenuse. At 26.565°, a 0.4 cubit vertical increase corresponds to a 0.8 cubit horizontal increase and that means the corridor would not be √(64²+32²) but √64.8²+32.4²). 71.554 cubits (1,475.45 inch) versus 72.449 cubits (1,493.89 inch). This 0.8 cubit horizontal distance would be split roughly 1:1 between the 25 cubits north of spot x and the 64 cubit segment south of spot x, because Petrie reconstructs 524.1 ± .3 inch.

Petrie: Having, then, fixed the original position of the doorway of the Pyramid, we may state that it was at 668.2 ± .1 above the pavement of the Pyramid; 524.1 ± .3 horizontally inside (or S. of) the N. edge of the Pyramid casing;

At a corridor angle of 26.48°, 32.4 cubits correspond to a total horizontal distance of 65.04 cubits and the corridor is 72.664 cubits or 1,498.34 inch from where it begins to where it becomes subterranean past the pavement. However, all these predicted corridor lengths are shorter than what Petrie measured since he places the beginning of the rock at 1350.7 inch down from his reconstructed entry. What this means of course is that the bedrock was above the pavement level, but it is not clear from Petrie's data where the Entry Passage corridor actually crosses the pavement level and how long it is at that point. The first marker are the inserted stones which fill a fissure.

Petrie: For the total length of the entrance passage, down to the subterranean rock-cut part, only a rough measurement by the 140-inch poles was made, owing to the encumbered condition of it. The poles were laid on the rubbish over the floor, and where any great difference of position was required, the ends were plumbed one over the other, and the result is probably only true within two or three inches. The points noted down the course of the passage, reckoning from the original entrance (i.e., the beginning of the rock on the E. side of the roof being 1350.7), are the following:—

 E. W.
Beginning of inserted stones, filling a fissure.
Joint in these stones.
End of these inserted stones.
Sides of passage much scaled, 1 or 2 inches off, beyond here
Fissure in rock
Mouth of passage to Gallery
End of sloping roof (4,137 Vyse, corrected for casing).

3,086 – 3,116



3,066 – 3096
3,825 – 3,856

This fissure is mostly under the pavement (see Plate 9 by Petrie):

Also Petrie does not measure the fissure itself, but where stone is inserted. So the reconstruction of the exact spot where the E.P. crosses the pavement is not straight-forward. This is was one of the most difficult aspects of our analysis.

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