THE GEOMETRY OF THE KHUFU SHAFTS

In 1993 Gantenbrink set his famous robot to explore the shafts, so far as he was permitted, and announced his preliminary results. He found that the shafts are not perfectly straight and the average angles he gave were as follows :

These figures appeared to give some support to a purely geometric scheme  (1) -

In 1999 Gantenbrink put detailed results on the web (2) which revealed the errors in this scheme : shafts 'a' and 'd' do not converge vertically below the outlet of 'b' ; nor does the major passage line, 'e', intersect 'd' at the level of the outlets of 'a' and 'b' . Shaft 'c' remains an enigma : its precise angle is not known but convergence with 'b' at pyramid height would require 39.3 degrees.

Legon (3) accepted this convergence (at 297 cubits horizontally from pyramid apex) but proposed that shafts 'c' and 'd' are symmetrical, each having a gradient of 17/14 -

Examing these shafts in more detail, it is seen that the northern shafts form a series of bends near to the chambers so as to avoid the Grand Gallery -

The south shaft of the King's Chamber varies near the chamber for no apparent reason. Nevertheless the south shaft is straight (or rather is made up of long straight sections) over much of its length, and the point of convergence of the two KC shafts would seem to have been planned in whole numbers of cubits -

In the case of the Queen's Chamber south shaft, the line of the shaft cuts the central axis 36 to 37 cubits above base. If the builder's had laid out this shaft according to Legon's proposal then it would need to intersect the central axis 35 to 36 cubits above base, in a similar configuration to the line of the northern shaft on the right -

The KC northern shaft  is laid out within a rectangle measuring 77 X 121 cubits. 
From measurements of Gantenbrink's published diagrams, QC south  has a sloping length of 121 cubits (from the centre of the pyramid to its termination by the famous 'door') for a vertical rise of 77 cubits - giving an angle of just over 39.5 degrees. (Measured angle is 39.6). The door is 114 cubits above base and thus equal to the height of base above sea level. Note that the angle proposed is that of the hypoteneuse of a near Pythagorean triangle with sides 363, 280, 231 - so that if a square is drawn on the height of the pyramid and divided by 3 the door is 93.3 cubits horizontally from centre. (It is possible that a similar threefold division determines the junction of ascending and descending passages, as shown to the right).

It has generally been assumed that floorlines governed the layout of internal pyramid features, as exemplified by the vertical positioning of the king's chamber floor. Meanwhile Butler has shown that the passage system of Khafre was laid out east/west with reference to its central plane. Gantenbrink states that important shaft and passage junctions were measured from their ceilings (if this is so it might have some bearing on the meaning of these shafts). In the case of Khufu the floorlines give an interesting result :

The descending passage slope is given by the diagonal of  the double square. The slopes of the Grand Gallery and Ascending passage are significantly less. The mean slope of the latter, if extended, intersects KC shaft outlet level 148.5 (1½ X 99) cubits south of central  plane, and gives weight to Legon's QC shaft scheme (shown on the right). This line passes through the junction of the south wall and ceiling of the King's Chamber.

References

1. Cook, R.J. 1994. 'The Stellar Geometry of the Great Pyramid'. Discussions in Egyptology. 29.

2. See Gantenbrink's web site : www.cheops.org

3. Legon, J.A.R. 1995. 'The Orion Correlation and Air-Shaft Theories'. Discussions in Egyptology. 33.

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