Varille ("A propos des pyarimdes de Snefrou," Cairo, 1949) was the first to propose that the division of the Bent Pyramid into an upper and a lower part with different slopes was consistent with a theme of duality that ran through the whole of the pyramid. This theme was strongly suggested to him by the internal design of the pyramid, which contains two sets of passageways and chambers. The fact that were two so-called burial chambers creates a problem for the idea that this pyramid was intended as Sneferu's tomb. Why would he need two tombs?
Legon ("The Geometry of the Bent Pyramid," GM 116, 1990, p.66) has observed that this duality is also expressed in the passageways themselves. The western passageway has two very different slopes, changing from 24* 17', corresponding to a slope of 4:9 (23* 58') to 30* 9', corresponding to a slope of 7:12 (30* 15').
There is also a duality in the options for the various slopes. Estimations for the slope of the mantle of the lower part, for example, vary between 7:5, corresponding to an angle of 54* 28' (e.g., Rossi, Robins and Shute), and 10:7, corresponding to an angle of 55* 00' (e.g., Petrie, Dorner). Of course, the 7:5 angle corresponds to a seked of 5 palms, while the slope of 10:7 breaks the commonly accepted seked system. (For later pyramids, the usual sekeds were 5 1/2 or 5 1/4 palms).
Rossi's and Robins' and Shute's slope of 7:5 seems to be driven by an ideological need that it must correspond to the quarter-palm seked of later pyramids. The slope according to Petrie and Dorner was not ideologically driven but based on its evidential value. Even so, this duality may have been intended, for both these values are very close to a slope of 99:70 which the Egyptians may have known to correspond to sqrt 2. Legon has indeed suggested that this was the intended slope of the mantle and that it was produced by a mixing of 7:5 and 10:7 slopes for the casing stones, but I think this is going just a bit too far.
There is even another duality within the pyramid. The lower part is comprised of two distinct parts, the outer mantle of 55* and a core pyramid of 60*. As Rossi has pointed out, a slope of 60* gives the core pyramid a profile that is an equilateral triangle. Even then, there is an ambiguity surrounding this slope. There is not sufficient data, at least that I have come across, that tells us exactly what this slope is, but like the slope of the mantle, it could have two possible values - either 7:4, corresponding to a slope of 60* 15', or a slope of 12:7, corresponding to a slope of 59* 45'.
Do we have enough evidence to determine which of these is the more likely? Rossi states that the slope of the core is 7:4, corresponding to a seked of 4 palms, but as I have shown, this may be ideologically driven. Isn't it more likely that the slope is 12:7, as this slope is at right angles to the slope of the western passage that passes through it?
