The Geometry of the Air-Shafts

by John Legon

In several articles in the journal Discussions in Egyptology,1 I have argued against the popular theory that the so-called "air-shafts" in Great Pyramid were really star-shafts which the builders aligned towards certain stars in the northern and southern regions of the sky as they crossed the meridian, and have instead supported the more traditional view that these were indeed air-shafts, which could actually have functioned in the case of the shafts that lead out from the King's Chamber, but were more probably symbolic and served for a hitherto little-known aspect of the Osirian 'funerary' cult.

A major component of my argument against the star-shaft idea has been my recovery of the geometrical conception of these shafts, showing that the design constituted a very simple and elegant extension of the geometry of the Great Pyramid as described in my previously-published articles.2 Just as the dimensions of the internal passage- and chamber-system in the Great Pyramid can be shown to have been developed in whole numbers of royal cubits from the geometrical figure of the pyramid itself, so also we find that the alignments of the shafts fall into place as an immediate development of the most elementary pyramid geometry.

Long before Rudolf Gantenbrink obtained new data for the shafts as the result of his exploration using the robot Upuat, Petrie's measurements of the shaft-angles provided clear evidence of a systematic and geometrical design. In particular, Petrie found that the two shafts leading from the Queen's Chamber have very nearly the same angle of slope, so that because of the placing of the Queen's Chamber in the exact centre of the Great Pyramid from north to south, the arrangement of these shafts is symmetrical from north to south. Owing to the random disposition of the stars in the night sky, however, we should not expect that an alignment to a star of cultic significance in the southern sky would have required the same angle of inclination as an alignment to a cult star in the northern sky; and this in itself is a strong argument against the star-shaft hypothesis.

Again, because the Queen's Chamber is situated in the mid-plane of the pyramid, the equality of the angles means that the shafts would have emerged at the same level in the casing on the north and south sides of the pyramid - a design of no consequence for star-shafts. The angles as determined by Petrie are:3

Shafts from Queen's Chamber North South Mean
Angle of Slope, Petrie 37° 28' 38° 28' 37° 58'

The small difference in these angles is easily explained by the difficulty experienced by Petrie when taking his measurements at the lower ends of the shafts, since the sloping sections were only accessible through the initial horizontal sections, each about 20 cm in square cross-section and some two metres in length.

At a first approximation, one obvious requirement which was satisifed by these angles was that the shafts should take the shortest route from the Queen's Chamber to the outside of the pyramid. In the context of the 'ventilation' theory this makes perfect sense, since the shortest distance would have provided the most effective air-flow. The slope of the shafts would also have encouraged the setting-up of convection currents, causing hot spent air to be drawn out of the chamber while allowing cooler air to take its place. Some writers have claimed that the shafts would have been laid horizontally if they had been intended for ventilation; yet not only would this have nullified convection currents, the length of the shafts would have been increased quite considerably. It is true that the shafts from the Queen's Chamber could never actually have functioned as air-shafts, because the channels were blocked at both ends; but this does not detract from their symbolical significance. According to ancient Egyptian conceptions of the hereafter, the representation of an object was sufficient to secure its function through magical means, just as the false-door so often employed in Egyptian tombs was effective for the deceased.

Now to obtain the shortest possible length for the shafts from the Queen's Chamber to the outside of the pyramid, the incline of the shafts had to be made equal to the inverse of the pyramid's casing-angle, or 11 rise on 14 base in place of 14 rise on 11 base. The line of the shafts then intersected the face of the pyramid with an angle of 90°. The mean angle of the two shafts as observed by Petrie corresponds to this profile with a difference of only about ten minutes of arc:

Angle of Slope, for 11 rise on 14 base = 38° 9' 26"

This same basic angle of slope might also have been used for the shafts leading from the King's Chamber, if it were not for the fact that this chamber was set entirely to the south of the pyramid's central axis. If the same angles had been employed, the shafts would have come out at different levels on the pyramid's north and south sides. It was evidently to avoid this outcome that the builders compensated for the offset of the chamber, by increasing the angle of the southern channel while reducing the angle of the northern channel by a similar amount.

The following angles were obtained by Petrie in the upper outlets of these shafts:4

Shafts from King's Chamber North South Mean
Angle of Slope, Petrie 31° 33' 45° 14' 38° 23'

Once again, therefore, the mean angle is close to the value required to give the shortest distance from the King's Chamber to the outside of the pyramid; while the difference in angle brought the outer openings of the two shafts to the same level on the north and south sides, as we will see.
Notwithstanding Petrie's careful work, his observations were limited to relatively short sections of the shafts, and precedence must now be given to Rudolf Gantenbrink's findings for the angles of slope.5 Following some initial divergence at the lower end, Gantenbrink found that the angle of the southern shaft from the King's Chamber was kept constant at just 45° 0', or exactly half the corner-angle of a square. Again after some variation as the construction was diverted around the Antechamber and Grand Gallery, the slope of the northern shaft was found to be 32° 36'. As Gantenbrink has pointed out,6 this angle is just half the pyramid-profile of 14 rise on 11 base, being close to 7 rise on 11 base or 32° 28' 16".
Shafts from King's Chamber North South Mean
Angle of Slope, Gantenbrink 32° 36' 45° 00' 38° 48'

Since both angles can be ascribed to an elementary geometrical origin, Robert Bauval has sought to explain the design as an example of 'sacred mathematics' being used to achieve a religious function; and he has drawn an analogy with the orientation of churches and cathedrals in relation to the eastern horizon.7 This analogy is invalid, however, because the orientation of a cathedral in no way conflicted with the geometrical concepts which were incorporated within the fabric of the building. Similarly, the precise astronomical orientation of the Great Pyramid with respect to the four cardinal points is independent of the geometry of the structure itself. The stellar-alignment hypothesis would have been immeasurably stronger if the angles of the shafts could only be expressed in terms of some arbitrary mathematical ratios, for which no obvious geometrical design could be determined; but when the southern shaft was simply aligned along the diagonal of a square, as Bauval acknowledges,8 how much significance can be attached to any star that happened to pass over the shaft-exit when at culmination?

According to the air-shaft theory, however, the architect was free to determine the angles of slope which best suited his development of the geometrical design, within the broad framework dictated by the need for a short route to be taken to the outside of the pyramid. The measurements now published by Gantenbrink have made it possible to establish this design, revealing a very interesting geometrical conception which explains why the shafts were so carefully constructed. It turns out that the geometry of the air-shafts is a function of the meridian cross-section of the pyramid itself, and can be developed simultaneously with the geometrical placing of the King's Chamber which I have previously described.9 At the same time, the positions of the shaft outlets can be shown to have been worked out in whole numbers of cubits, in perfect harmony with the proportions of the pyramid.

The north and south shafts from the King's Chamber are now reported by Gantenbrink to have both opened in the casing at the height of 80.63 ms ± 4 cm above the base.10 The apertures thus coincided with the level of the 105th course as determined by Petrie (3174.7 to 3176.0 inches above the base, mean 80.65 ms).11 This is exactly 2 × 7 × 11 equals 154 cubits above the base. The level of the outlets was therefore commensurate with both the shaft-profile of 7 rise on 11 base, and the casing-profile of 14 rise on 11 base, placing the outlets at a distance of 154 × 11/14 or 121 cubits horizontally inside the north and south base-lines of the pyramid. Given the side-length of the base of 440 cubits, the horizontal distance across the Great Pyramid at the level of the outlets will be (440 - 2 × 121) or 198 cubits, which is exactly equal to the height of the pyramid from the floor-level of the King's Chamber to the apex, of (280 - 82) cubits or 198 cubits. 9

We can therefore now define the positions of the outlets as indicated in my geometrical diagram, by first marking off the height of the pyramid of 280 cubits along the diagonal of a square constructed on the height. The King's Chamber is thereby placed at the level of 280 ÷ Ö2 equals 198 cubits below the apex of the pyramid, as shown by Petrie's data. By forming a square on this height, with the base of 198 cubits centred on the central axis of the pyramid, the positions of the outlets on the north and south sides of the pyramid are immediately obtained, and the slope of the southern shaft can be drawn along the 45° diagonal of a square.

Now turning to the shafts leading from the Queen's Chamber, it is a remarkable fact that the northern shaft is directed towards exactly the same geometrical 'focal point' as the northern shaft from the King's Chamber. The angles of these shafts were thus obviously derived from one and the same geometrical construction. I have previously pointed out that if the shafts from the Queen's Chamber had been completed, they would have opened through the sides of the pyramid at the level of the 90th course, which is 2711.1 inches or 131.48 cubits above the base according to Petrie's data.11 This course marks one of the great 'stages' in the core-masonry of the Great Pyramid, being noticeably thicker than any of the preceding 44 courses; and it is exactly defined by the centre of the square with the side of (99 + 198) or 297 cubits, which extends from the apex of the pyramid to the focal point of the shafts as shown in the diagram above. The level of the shaft outlets may thus be obtained as (280 - 297/2) or 131.5 cubits over the base of the pyramid, giving a profile of 1 rise on (2 - 11/14) base, or 14 rise on 17 base, with a theoretical angle of 39° 28' 21".

This angle agrees closely with Gantenbrink's finding of 39° 36' 28" for the southern shaft, given a reported uncertainty of 1/5°; while the stated angle of 39° 7' 28" for the northern shaft is said to be uncertain within 2° at present, since only a short fractured section at the lower end could be measured.12 In view, however, of the axial position of the Queen's Chamber, the probability that the two shafts were intended to emerge at the same level on the north and south sides of the pyramid, and the fact that the horizontal sections at the lower ends are the same length, there is no reason to doubt that the northern shaft was constructed with the same angle of slope as the southern shaft, in close agreement with the theoretical requirement.

Shafts from Queen's Chamber North South Mean
Angle of Slope, Gantenbrink 39° 7' 28" 39° 36' 28" 39° 22'

While the King's Chamber was placed at the exact level in the Great Pyramid at which the diagonals of the horizontal cross-section measure 440 cubits, and equal the sides of base, the level of 131.5 cubits for the outlets of the shafts from the Queen's Chamber is that at which the diagonals of the cross-section measure exactly 330 cubits or 3/4 of the sides of base. Again, at the level of 154 cubits now obtained for the outlets of the shafts from the King's Chamber, the diagonals measure just 280 cubits, and equal the height of the Great Pyramid. It should be more obvious than ever, therefore, that the design of these shafts was determined by considerations of geometry, symmetry, and the desire for a coherent dimensional design, and had nothing to do with the conjectured astronomical alignments.

Notes

[1]. J.A.R. Legon, 'The Air-Shafts in the Great Pyramid', DE 27 (1993), 35-44;'Air-Shaft Alignments in the Great Pyramid', DE 28 (1994), 29-34. 'The Orion Correlation and Air-Shaft Theories', DE 33 (1995), 45-56
[2]. J.A.R. Legon, 'The Design of the Pyramid of Khufu', DE 12 (1988), 41-48; 'The Geometry of the Great Pyramid', GM 108 (1989), 57-64
[3]. W.M.F. Petrie, The Pyramids and Temples of Gizeh (London, 1883), 71.
[4]. Petrie, Ibid. 83.
[5]. R. Gantenbrink in R. Stadelmann, MDAIK 50 (1994), 285-294.
[6]. In conversation with the present writer.
[7]. R.G. Bauval, DE 31 (1995), 5-13, 6.
[8]. Ibid., Fig. 3.
[9]. J.A.R. Legon, DE 12 (1988), 41-48, 43.
[10]. I am grateful to Robert Bauval for supplying me with this data. Now see also R. Gantenbrink's web site, www.cheops.org
[11]. Petrie, op.cit., Pl. VIII.
[12]. Gantenbrink, op.cit., 293.



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