The Problem of the Bent Pyramid

John A.R. Legon

Adapted from the author's article in Göttinger Miszellen130 (1992), 49-56

In my analysis of the geometry of the Bent Pyramid in GM 116,[1] I gave a brief description of the dimensions based upon the results of the surveys carried out by Flinders Petrie in 1887,[2] and by Josef Dorner within the past decade.[3] In presenting a summary of the numerous survey results for the purpose of discussion, it was inevitable that some details would not be adequately described; but I was content to put forward a simplified scheme because it provided a sufficiently accurate and coherent explanation for the proportions of the Bent Pyramid when taken as a whole.

My treatment of the dimensions has now met with some criticism from Dorner,[4] and I am grateful to him for having given the matter his attention. For the reasons given below, however, I believe that he is mistaken in his interpretation of the construction, and has computed results for some parts of the pyramid which could not possibly represent the original dimensions of the monument.

The Dimensions of the Base
Owing to the destruction of the outer casing around the base of the Bent Pyramid, the lines of the sides must be found in some places by projecting the slope of the casing still remaining higher up the side, down to base-level. At the ends of the north side, however, elements of the corner-edges and corner-sockets are still preserved; and for this reason Dorner considers the length of the north side to be the most accurately determined. He questions the accuracy of his findings for the other sides, and uses the length of the north side in his final statement of the dimensions, even though this length is about 18 cm less than the average.[5]

It appears, however, that the difficulty of fixing the sides by the projection of the casing has been exaggerated, because the positions of the north-east and north-west corners found by Dorner using this method actually correspond very closely to the positions given by the corner-sockets. The coordinates of the corners differ by between only 1 and 4 cm.[6] Even if the length of the north side is the most securely fixed, it cannot be taken as a certain indication of the original lengths of the three other sides, for clearly these sides may have varied in length when laid out by the builders. The average length of the four sides according to Dorner's survey is 189.61 m, or 362 cubits for a cubit of 0.5238 m. This length of cubit is very close to that employed at Giza,[7] and falls half-way between Dorner's cubit of 0.5233 m, and the value indicated by the dimensions of the Satellite Pyramid to the south of the Bent Pyramid. According to Petrie,[8] the sides of this small pyramid measure 52.44 metres on average, or just 100 cubits for a cubit of 0.5244 m.

The Curvature of the Sides
In endeavouring to describe the proportions of the Bent Pyramid as accurately as possible, I took account of the numerous measurements of the casing-angle taken by Petrie, which show a marked convexity in the faces of the pyramid in both the lower and upper sections. I explained this convexity for geometrical reasons as an intentional feature of the design, deliberately brought about by the builders.

Dorner, however, assumes that the faces were meant to be flat; and he excluded from his survey all the points of the casing at the level of the change of slope on the east, west, and south sides of the pyramid, which fell inside his hypothetical plane of the casing on these sides. No points were observed by him on the upper part of the pyramid on any side, and the dimensions of this part were instead calculated by assuming that the steeper casing-angle which had been used generally near the base of the pyramid, could be applied uniformly on each side. In Dorner's opinion, the curvature of the casing which is seen higher up in the sides was the result of a settlement of the foundations, which caused the casing to tilt inwards at some points whilst leaving other points relatively untouched.

Although some cracks which have appeared in the masonry of the pyramid suggest that a movement of the substructure has taken place, however, the magnitude of the subsidence and its effect on the present shape of the pyramid are open to dispute.  Maragioglio and Rinaldi  described the cracking in detail, but at no point attributed the curvature of the sides to this cause.[9] On the contrary, they imply that the curvature was a consequence of a lack of accuracy in construction of the pyramid. [10]

Even the break in the northern entrance passage, which appears to give evidence for a settlement, was interpreted differently by Petrie. In his opinion, "a settlement of 11 inches (28 cm) in such solid masonry, not far from the ground, is impossible, the more so as it would need a uniform settlement of the whole of the lower part of the passage, which should quickly cease at one point, and soon after continue at an equal amount."[11] According to Petrie, the dislocation was caused by a sudden turning up of the lower part of the passage, not a sinking of the upper part;[12] and it appears that the slope was deliberately split into two sections as in the western passage, and in the sides of the Bent Pyramid itself. Even though the curvature on the west side of the pyramid is greater than on the north, and should have been caused by a major settlement passing across the line of the western passage according to Dorner's theory, there is no sign of a large dislocation in this passage.[13]

If Dorner is right, however, then the effect of the settlement can be observed in the pyramid by the difference between the present position of the casing at the level of the change of slope, and the position as computed when the steeper angle at the base is projected up to that level. This steeper casing-angle being about 55° 01' on average, according to Petrie, while the overall mean angle of the lower slope is 54° 48', it can be calculated that the casing at the change of slope, some 47 m over the base, must have tilted inwards by an average of about 25 cm.

Now in my view, such a large shift inwards in solid megalithic masonry is a physical impossibility. Whatever vertical motions may have taken place through subsidence - and these were possibly very slight - a horizontal movement of the casing towards the centre of the pyramid required a corresponding displacement of the internal masonry, which manifestly cannot have taken place. In large-block masonry laid in regular courses, moreover, the tremendous binding forces between the courses effectively prevented lateral movement, even supposing that space existed within the pyramid into which the masonry at the sides could have fallen inwards.

Because a settlement of the foundations cannot, therefore, have reduced the dimensions of the pyramid at the level of the change of slope, the convexity of the sides must have been brought about by the builders, and should be taken into account when computing the dimensions. The upper base-width is thus found to be about 0.5 m less on average than calculated by Dorner, whose larger original dimension would imply that the entire upper portion of the pyramid had decreased in size since the monument was constructed.

The Geometrical Explanation
Now according to Dorner, if the curving in the sides of the Bent Pyramid had been intended, it would be noticed regularly and also at the corners. This is not the case, however, since the builders may have wished to avoid an unsightly curving of the corner-edges, while obtaining the required slope along the sides. This is shown by a distortion in the casing which is most noticeable on the east side where according to Petrie, the north-east corner at the change of slope appears to stick out unduly by about 50 cm beyond the rest of the face at this level.[14] If this had been caused by subsidence, gaping fissures would have opened up in the brittle limestone casing where a part of the face had fallen inwards; when in fact, despite some fracturing of the masonry, the surface of the casing is smooth and the slight curve has clearly been cut into the casing-stones.

This curving, I believe, is a logical result of the geometry of the Bent Pyramid, brought about through the inevitable conflict between a geometrical ideal and the need for practical measures to be used in the construction. As described in my previous article, Petrie's overall mean angle of 54° 48' for the lower section of the Bent Pyramid accurately corresponds to a theoretical slope obtained by raising the diagonal of a square as a vertical to the side - an elementary development from the pyramid's square base for which the precise casing-angle would have been 54° 44' 8".

The question, however, is how this proportion was rationalized, since in modern terms the diagonal of a square of unit side is equal to the irrational square root of two. I have previously suggested [15] that this factor could have been estimated experimentally by setting out a square with a side of 10 cubits, or 70 palms, and measuring the diagonals directly. The exact dimension being 70Ö2 or 98.998... palms, it could have been taken as 99 palms with negligible error.

I have ascribed this proportion of 70:99 to the layout of the Bent Pyramid, taking the minimum enclosure-width of 99 cubits with the semi-base of the pyramid of 181 cubits, to give the formative dimension of (99 + 181) equals 280 or 4 × 70 cubits. I think this relationship is conceptually valid even though the enclosure-width averages about 100 cubits;[16] for the builders would probably have attached as much meaning to the round-figure approximation for Ö2 of 70:100 or 7:10, as to the ratio of 70:99, when the difference was only 1%, and when the numbers 7 and 10 were clearly significant in their own right. The enclosure-width of 99 cubits was actually employed at the north-west corner of the Satellite Pyramid; and it is also found in the middle of the north side of the Bent Pyramid owing to the curious but deliberate curving-out of the base-line on this side.[17] Because of these facts, I think we can claim to have discovered the key to the design of the Bent Pyramid, in which the dimensions of the base and the slope of the sides were geometrically integrated with the width of the enclosure.[18]

That the architects of the Fourth Dynasty were interested in the diagonal relationship of a square is certainly suggested by the fact that both the Satellite Pyramid, and the Northern Stone Pyramid of Dahshur, have casing-angles which approach the 45° diagonal angle. The actual angles fall short of the diagonal to the extent that they can be defined with equal accuracy as either 10 slope in 7 rise, or as 7 slope on 5 base.[19] The variation in the width of the enclosure of the Bent Pyramid, through which the builders were able to express both the geometrical ideal dimension of 99 cubits and the numerical ideal of 100 cubits, has a clear parallel in the form of the Bent Pyramid itself; for the builders often employed the practical slope of 10:7 near the base, while obtaining the geometrical requirement of 99:70 as the mean slope by curving the casing inwards along the sides towards the level of the change of slope. By this means they set the dimensions required for the upper section of the pyramid.

In my earlier article, I equated Petrie's average angles for the lower and upper parts of the lower slope with the profiles of 10:7 and 7:5, which together give 99:70 exactly, and implied that these angles had been applied systematically on each side of the pyramid. As Dorner has correctly pointed out, however, the curving does not occur at the well-preserved north-east corner, nor can it be seen on any part of the north side judging from Petrie's data. On the other hand, Petrie's results for the south side show that the exceptional steepness of the north side was effectively compensated by the use of an angle of slope which is not only much less than the angle of 55° 00' 29" required by the profile of 10 rise on 7 base, but also significantly less than the angle of 54° 44' 13" as required by the slope of 99 rise on 70 base:[20]

North Side South Side
Lower casing-angle: N.N.W. Near W N.N.E. S.S.E. S.S.W
Upper part of slope 54° 59' - 55° 02' 54° 01' 54° 38'
Lower part of slope 55° 23' 54° 59' 55° 02' 54° 40' 54°o 38'

I can see no justification in these figures for Dorner's assumption that the angle at the north-east corner had been applied uniformly to the lower slope of the Bent Pyramid on each side. On the south side, the average angle of 54° 29' places the edge of the change of slope about 60 cm inside the position of the edge as postulated by Dorner, thus confirming my calculation of the upper base-width with a result more than a cubit shorter than that given by Dorner.

On the east and west sides, Petrie's measures show that a slope in some places steeper than 10:7 was used at the base, falling away markedly in the upper parts of the sides below the change of slope:

East Side West Side
Lower casing-angle: E.N.E. E. E.S.E W.S.W W.N.W.
Upper part of slope 55° 12' 53° 44' - 54° 00' 54° 36'
Lower part of slope 55° 12' 55° 20' 54° 46' 55° 02' 55° 04'

On the east side of the north-east corner, the straight alignment of the corner-edge was ensured by continuing the steep angle at the base up to the level of the change of slope - an alignment which was apparently achieved also at the north-west corner.[21] The lines of the corner-edges, however, which the builders may have wanted to preserve, did not control the height of the upper section of the pyramid, which can only be calculated by taking the cross-section through the centre of the sides. The overall mean angle of the east and west sides of 54° 46' again confirms the theoretical requirement of 54° 44' 13", and gives an average base-width for the upper part of the pyramid which is about 60 cm less than Dorner's result.

These observations suggest that for the sake of convenience or numerical significance, much of the lower part of the Bent Pyramid was in fact carried up using the profile of 10:7. Once the level of the change of slope had been reached, however, a correction was necessary to give the required cross-section; and this could have been achieved by marking out the square base for the upper part on the top surface of the lower part, taking the northern upper edge as the base-line. The casing would then have been dressed downwards from the newly-defined upper side-edge on the east, west and south sides, to a height above the base which was allowed to vary, so that the curvature of the casing also varied. The steeper profile remained near the base in many places, and the north side together with the north-east and north-west corner-edges were left untouched. This would explain why the Bent Pyramid is not symmetrical, but has steeper angles on the north side than on the south.[22]

Whether the profile of 10:7 was ever employed on the south side, however, is open to dispute. It is not shown by Petrie's measures, which on this side conflict with the angle of 55° 02' as indicated by Dorner's survey-points situated near to the south-west corner.[23] Petrie's angle of 54° 46' as shown at the lower E.S.E., and that of 54° 40' at the lower S.S.E., suggest that a profile very close to 99:70 was used on both sides of the south-east corner, so that the diagonal corner-edge here sloped at practically 1 rise on 1 base.

It is true that these variations in the form of the Bent Pyramid do not appeal to our concept of geometrical perfection, nor to our expectations of great accuracy in a monument of the Fourth Dynasty. I think, however, that Dorner has expected too much precision from the builders, who were entitled to vary the dimensions if they so desired, and when their reasons for doing this were quite logical. It is not sufficient to ignore these variations, or to assume that they were caused by movements in the structure which can never have taken place. A surveyor should not exclude points from a survey because they do not fit in with his theoretical ideal, but should at least publish the data so that a complete analysis can be made. At present we still have to rely upon Petrie's survey-data for some details; and a new survey, including measurements of the passages and chambers, and studies by geologists and structural engineers, may now be necessary if full justice is to be done to the skills of the architects and builders of the Bent Pyramid.


1. J.A.R. Legon, GM 116 (1990), 65-72.
2. W.M.F. Petrie, A Season in Egypt, 1887 (London, 1888), 27-32.
3. J. Dorner, MDAIK 42 (1986), 43-58.
4. J. Dorner, GM 126 (1992), 39-45.
5. Dorner, op. cit. (n.3), 51, 54.
6. Ibid., 51.
7. Petrie, The Pyramids and Temples of Gizeh, (London, 1883), 179, obtained a mean cubit of 20.620 inches or 0.52375 m.
8. Petrie, op.cit. (n.2), 30.
9. V. Maragioglio and C. Rinaldi, L'Architettura delle Pirimidi Menfite III (Rapallo, 1964), 58-60, 94-98.
10. Ibid., 58.
11. Petrie, op.cit. (n.7), 146.
12. Petrie, op.cit. (n.2), 30.
13. Maragioglio and Rinaldi, op.cit. 66, noted a sinking of 5 cm.
14. Petrie, op.cit. (n.2), 30.
15. J.A.R. Legon, GM 124 (1991), 69-78, 73.
16. See Dorner, op.cit. (n.4), 42-43.
17. The base is 51.85 m from the line of the enclosure-wall according to Dorner's survey-points 34 and 14 and wall-thickness of 1.91 m.
18. See Legon, op.cit. (n.1), fig. 2.
19. Petrie, op.cit. (n.2) 27, 32; Maragioglio and Rinaldi, op.cit.76. The theoretical angle is 44° 25'.
20. Petrie, op.cit. (n.2), 30.
21. Dorner, op.cit. (n.4), 40. Most of this corner-edge is missing.
22. The upper casing-angle was possibly also varied to give a length of the upper corner-edge equal to the total height of the pyramid of 200 cubits, just as the upper vertical height was made equal to the lower slant height of about 110 cubits.
23. Points 9, 10 and 25. Dorner, op.cit. (n.3), 53.


The Problem of the Bent Pyramid

John A.R. Legon

Abstract

In this article, the author replies to the objections put forward by Josef Dorner in GM 126, to his analysis of the geometry of the Bent Pyramid of Dahshur published in GM 116. Reasons are given for disputing Dorner's assumption that variations in the casing-angles of the Bent Pyramid were caused by a settlement of the foundations; and these variations are instead attributed to the intention of the builders to express differing proportions in different sections of the pyramid, according to a geometrical scheme deriving from the relationship between the side of a square and the diagonal.



Additional observations included in a draft of the above article

While the lower part of the pyramid contains four-fifths of the total volume, the pressure bearing down upon the foundations varied according to the height of the overlying masonry, and reached a maximum in the centre where the chambers are located. If the builders had reduced the pyramid's casing-angle when cracks first started to appear in these chambers, then the completion of the pyramid more than doubled the pressure on the chambers, and would probably have caused the chambers to collapse.

The reduction of the casing-angle reduced the eventual volume of the pyramid by only 9%, and made little difference to any settlements that may have affected the outer casing of the pyramid.

First uploaded to the Internet on 10th August 2004


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