Adapted from the author's article in Göttinger Miszellen116 (1990), 6572
The recentlypublished survey of the Bent Pyramid of Dahshur by Josef Dorner [1] has provided fresh material towards an assessment of the design of this pyramid, while also enabling discrepancies in the results of the surveys carried out by Petrie [2] in 1887, and by Mustapha [3] in 1952, to be attributed to inaccuracies in the latter survey. The following analysis of the surveydata indicates that the doublesloping profile of the Bent Pyramid, which is usually considered to represent a change of plan, was actually conceived from the outset in harmony with concepts of design which are now emerging as fundamental in the architecture of the FourthDynasty pyramids. Irregularities in the form of the Bent Pyramid, which Dorner attributes to settlements in the coremasonry, are found to reflect the complexities encountered by the builders in the fulfilment of an exceptionally ambitious project. As we have shown in previous articles,[4] the pyramidbuilders of the Fourth Dynasty took great care in the choice of dimensions, and applied geometrical methods to establish relationships between the different elements of their designs. The austere geometrical style of this dynasty sprang up during the reign of Sneferu with a dramatic reduction in the architectural content of the monuments, initially at Meydum and afterwards at the Northern Stone and Bent Pyramids of Dahshur. A great change took place in the concept of the pyramid, for reasons which are still difficult to understand. 
The Dual Profile of the Bent Pyramid
As first suggested in a study by A.Varille,[7] moreover, the possibility that the double profile of the Bent Pyramid had been envisaged from the beginning is strongly indicated in the internal design, by the provision of two independent passage and chambersystems. As acknowledged by Stadelmann,[8] this arrangement must have been planned from the outset and gives the impression that the architect had attempted to create "two pyramids in one". Varille referred to the paired elements within both passagesystems: the two portcullises in the western passage, and the two "trapdoors" in the shaft which rises from the lower chamber. But it is also very striking that both entrance passages were constructed in two sections rising at different angles of slope. The western passage thus begins with an angle of 24° 17', but turns up at an angle of 30° 9' to emerge in the pyramid's face at a height of over 30 m from the ground (see fig. 2). This seemingly needless change of angle clearly anticipated the doublesloping profile of the Bent Pyramid itself, and surely demonstrates the architect's intention to pursue an allpervading theme of duality. The Dimensional Design

Side Length in Metres  Side Length in Cubits  
Petrie  Dorner  Petrie  Dorner  
North  189.57  189.41  361.95  361.64 
East  189.32  189.75  361.47  362.29 
South    189.71    362.21 
West  189.49  189.57  361.79  361.95 
From Dorner's survey, therefore, the mean side is 189.61 metres, or 362.02 royal cubits. Dorner asserts that this base of 362 cubits shows "no sensible planning", and he develops Maragioglio and Rinaldi's theory that the dimension derived from the enlargement of an earlier buildingphase, laid out on a base of just 300 cubits. It is, of course, quite possible that the Bent Pyramid was formed around a nucleus of some kind; but the existence of such a nucleus, if it could be proven, would not in itself be evidence for a change of plan. As we will now show, the base of 362 cubits can be ascribed to a simple geometrical development, following a method established in other pyramids of the Fourth Dynasty. 
It will be seen in fig.1 that the Bent Pyramid is surrounded by a temeno wall, enclosing an open space which Petrie found to be approximately 100 cubits in width on each side. On the south side, a Satellite Pyramid was constructed with a base of just 100 cubits (Petrie: 52.44 m = 100.1 cubits), within a projection of the temeno wall on the exact northsouth axis; but the spacing as shown by the distance from this small pyramid to the Bent Pyramid has a minimum value of 99 cubits, according to Petrie's measure of 51.92 m or 99.13 cubits. Now a spacing of just 99 cubits would combine with the semibase of the Bent Pyramid to give a dimension of (99 + 181) or 280 cubits, which is a plausible startingpoint for the dimensional scheme owing to the division of the cubit into 28 digits. Was the division into parts of 99 and 181 cubits then an error of measurement, or does it reflect a definite purpose, which would explain the pyramidbase of 2 ×181 or 362 cubits? Soon after the construction of the Bent Pyramid, the measure of 280 cubits was used for the height of the Great Pyramid, which was divided into parts of 82 and 198 or 2 × 99 cubits at the level of the 'King's Chamber'. This division could have been planned by marking off the height on to the diagonal of a square, the sides of which would measure 280/√2 equals 197.989, or 198 cubits.[10] A very similar construction may thus be applied to the enclosure of the Bent Pyramid, which can be laid out with four squares each of side 280 cubits (see fig. 1). When the semiside of one square is marked off on the diagonal, elements of 99 and 181 cubits are produced, so that the pyramid's side of 2 × 181 cubits emerges as a simple development from the dimensions of the enclosure. The significance of this geometry is now evident from a further construction, which gives the exact doublesloping profile of the Bent Pyramid together with the division of the height. Petrie and Dorner agree that the casingangle of the lower slope corresponds to 10 rise on 7 base, or 10 palms to the cubit, but only in the lower parts near the base. The theoretical angle is 55° 0' 29": 
Lower Casing Angle of Bent Pyramid  Petrie  Dorner  
Mean Angle of Lower Slope  Lower Part  55° 01'  55° 05' 
Upper Part  54° 36'   
As Petrie's results indicate, there is a marked convexity in the lower slope which Dorner excludes from his surveydata, and also from his analysis of the dimensions: he considers it to have been caused by a settlement of the masonry. A settlement in megalithic masonry could not, however, have produced a progressive tilting inwards of the sides, which reduced the pyramid's crosssection at the level of the change of slope. This crosssection can at the present time only be calculated from the overall mean lower casingangle, which from Petrie's data is 54° 48'. Now as noted by Lauer,[11] a casingangle of 54° 44' 8" gives a profile of √2 rise on 1 base, and inclines the corneredges of the lower slope at 45°, or just 1 rise on 1 base. This profile can be obtained by raising the diagonal of a square as vertical to the base, and might have been sought by merging the initial slope of 10 rise on 7 base into one of 7 rise on 5 base, or 54° 28': Mean of 10/7 and 7/5 = 99/70 = 1.41428... Square root of two = √2 = 1.41421... Petrie's readings suggest, however, that this mean casingangle was effected by an arbitrary dressing down of the casing, rather than by a consistent combination of two angles, so as to obtain the required crosssection at the level of the change of slope. As shown in figure 2, the theoretical angle can be constructed simultaneously with the semibase of 181 cubits from the relation between the side of a square and its diagonal, taking an initial dimension of 280 cubits to give the size of the enclosure. 
The Division of the Height For the height of the Bent Pyramid, the findings of Petrie and Dorner can be summarised as follows: 
Height in Metres  Height in Cubits  
Petrie  Dorner  Petrie  Dorner  
Lower Slope  47.17  47.04  90.06  89.81 
Upper Slope  57.83  57.67  110.42  110.11 
Total  105.00  104.71  200.48  199.92 
It thus appears that the builders selected roundfigure heights of 90 and 110 cubits for the lower and upper sections of the Bent Pyramid respectively, to give a total height of just 200 cubits. We must, however, consider why this division was chosen, and whether it continues the geometrical scheme already outlined. A surprising consequence is that the upper vertical height was thus made equal to the lower slant height. The lower slope being represented by the hypotenuse of a 1,√2,√3 rightangled triangle, it can be computed from the vertical side as 90 ÷ √2 × √3 equals 110.2 cubits, which is then the vertical rise of the upper slope. The exact division of the total height of 200 cubits into parts in the required ratio of √2:√3 gives the following dimensions: Lower height = 89.898 cubits, Upper Height = 110.102 cubits This division, which is accurately represented by Dorner's data, can be achieved geometrically by simply bisecting the baseangle of the Bent Pyramid as shown in figure 2. An isosceles triangle is then created with one equal side giving the lower slant height, while the other equal side gives the upper vertical height. The upper slope can now be completed because both the upper height and the upper semibase have been defined. We have: Upper SemiBase = 181  (89.898 ÷ √2) = 117.432 cubits Upper Slope = 110.10 rise on 117.43 base = 43° 9' 17" The upper semibase is nearly 822 palms or 117.428 cubits, which is about 20/7 times the element of 41 cubits shown in figure 2. Dorner did not include the upper slope in his survey but made use of Petrie's data, and overlooked the distinct convexity, or lessening of angle, between the lower and upper parts. This is shown by Petrie's readings of about 43° 21' for the lower parts of this slope, and 43° 1' for the upper parts, with an evident mean of about 43° 11'. The lesser angle gives 14 rise on 15 base, or 43° 1' 30", as both Petrie and Dorner state, while the steeper angle gives twothirds of the mean profile of the lower slope, or 2√2 rise on 3 base equals 43° 18' 50", as suggested by Lauer.[11] The upper corneredges are then inclined at just 2 rise on 3 base. Although the architect of the Bent Pyramid could have measured off the profile and dimensions from a geometrical figure without any conscious use of the factors √2 and √3, which would imply a knowledge of the Pythagorean theorem, it is striking that these factors appear again at Giza, where the three pyramids are bounded by a rectangle measuring 1000√2 by 100√3 cubits.[12] In fig.3, these dimensions are developed from a rectangle of 1000 by 2000 cubits, which is divided at the south side of the Second Pyramid into parts of 899 and 1101 cubits, or in the ratio √2 :√3, exactly as in the height of the Bent Pyramid. Such a close connection between plans which would have been laid down within a matter of decades suggests that the same individuals were responsible, or had access to a body of knowledge which in all probability, was reserved for those who had been initiated into these undoubted 'mysteries' of their craft. 
Notes 1. J. Dorner, MDAIK 42 (1986), 4358. 2. W.M.F. Petrie, A Season in Egypt, 1887 (London, 1888), 2732. 3. H. Mustapha, ASAE LII (1954), 595603. 4. J.A.R. Legon, GM 108 (1989) 5764; GM 110 (1989), 2734. 5. V. Maragioglio and C. Rinaldi, L'Architettura delle Pirimidi Menfite III (Rapallo, 1964), 98100. 6. See I.E.S. Edwards, The Pyramids of Egypt (London, 1985), 79. 7. A. Varille, A propos des pyramides de Snefrou (Cairo, 1947), 7. 8. R. Stadelmann, Die agyptischen Pyramiden (Mainz, 1985), 94. 9. The cubit derived by Petrie, The Pyramids and Temples of Gizeh, (London, 1883), 179, and also by the present writer. 10. J.A.R. Legon, Discussions in Egyptology 12 (1988), 43, fig. 1. 11. JPh. Lauer, Le mystere des pyramides (Paris, 1974), 306, 342. 12. J.A.R. Legon, Discussions in Egyptology 10 (1988), 3340; see also Discussions in Egyptology 14 (1989), 5460. THE GEOMETRY OF THE BENT PYRAMID John A.R. Legon ABSTRACT Amongst the pyramids of the Fourth Dynasty, the Bent Pyramid of Dahshur stands out because of its unique doublesloping profile, which is often supposed to have resulted from a change of plan. In this article, however, the profile is found to have been derived from the application of geometrical principles, and to have been intended to continue a theme of duality which is shown in many other aspects of the design. A comparison is made with other pyramids of the period, which embody the same geometrical concepts, expressed through closelyrelated dimensions. 
The author's interpretation of the Bent Pyramid was contested by Josef Dorner in GM 126 (1992), 3945. For my response, see: The Problem of the Bent Pyramid First uploaded to the Internet on 10th August 2004. © John A.R. Legon 