The orientations of the two large pyramids are surprisingly precise – to within a few minutes of north. (They are so similar that it has been suggested the two pyramids were part of the same project). Petrie chose the mean between them as the axis of his plateau survey.
The pyramid of Menkaure has a different orientation 14 minutes east of north. One explanation is that pyramid orientations were determined by sighting on stars, which of course are always slowly moving due to precession. Over a long building period the ‘target’ star will have moved. Menkaure’s aberrant orientation is shown very greatly exaggerated below –
Some have suggested that the still to be fully excavated base of Menkaure is a rhombus or parallelogram. It is probable that the base was designed as a square and the odd shape the consequence of siting the pyramid on the desert surface, as is the case with Khufu’s satellite pyramids. Not knowing the reason for the ‘aberrant’ orientation of the pyramid the best we can do is to assume that the intended (geometrical) plan was orthogonal. We have at least Petrie’s position of the pyramid centre.
For the reasons above some latitude must be given to survey figures. It should be noted however, that Nell and Ruggles found that the east wall of Menkaure mortuary temple has the same orientation as Khufu, which suggests that the builders were capable of maintaining alignments over large distances.
Finally there is the question of error. Egyptian builders were not perfect but very good. Butler estimates errors in built work around 0.1 percent and tunneled work around 0.2 percent.
In his analysis of Khufu, Petrie concluded that the internal architecture had been laid out using a cubit of 0.5237 metres. It is to be noted that his measurement from the north base, horizontally to pyramid centre, is 219.9 cubits and not the expected 220 cubits.
Butler proposed that the base of Khufu, being the longest stone-built quantity available at Giza, should provide the best estimate of the cubit used in the layout, taking the mean base of Khufu as 440 cubits. So this cubit is shorter than Petrie at 0.5236 metres. There is indeed slight variability in the inferred cubit used at different pyramid sites. Whatever the reason for this a further source of potential error is introduced.
In the diagram below Petrie’s measures for pyramid spacing (originally as inches) are shown converted into both the Petrie (blue) and Butler (red) cubits –
Legon was the first to take Petrie’s measures and convert them into cubits of 0.52375 metres (the ‘Petrie’ cubit) and his major finding is widely known – that the pyramids are set out on a square of side 1000 cubits, or a grid of 500 cubit squares. The northeast corner of Khufu and southwest corner of Menkare define a rectangle measuring approximately root 2 (as 1414) by root 3 (as 1732) –
This is essentially a representation of the roots of geometry. But why would the builders assemble many millions of tons of cut stone only to produce a school text book? If they had something to say who were they saying it to, and why? If they were not the hopeless ‘primitives’ of popular imagination perhaps they had a reason
For example Lightbody proposed that Giza builders knew Pi as 22/7 and considered it as a symbol of ‘protection’ – an unwritten concept, associated with the ‘Shen ring’, seen held here by MUT the mother –
In evidence he points to the enclosure of the 3rd dynasty king Djoser –
– the enclosure wall measures 1040 X 530 so circuit of the walls is 3140 cubits.
At Giza he notes that the rectangle enclosing the NE corner of Khufu and the SW corner of Menkaure has a perimeter approximately equal to Pi (1414 + 1732 = 3146) –
Lightbody believes it was Menkaure’s architects alone who had the idea of placing his pyramid so as to achieve this relation.
However, the roots of 2 and 3 relate to a square of 1 (or 1000 cubits at Giza), and Legon found that the diagonal of this square intersects the diagonal of the plan at 1101 cubits, thus locating Khafre south base and making this pyramid also part of the overall plan –
However there is a problem – the east/west dimension is not 1414 but 1417.5 cubits and it has therefore been assumed that this is simply a mistake made by the builders. Thus Tedder, invoking such error, put forward a plan that also included Khafre with a diagonal of exactly 2000 cubits –
– the elegance of this plan is striking, and the builders would surely have been aware of such simplicity. However, if we take the survey measures at face value the scheme does not fit –
Nor does it fit Legon’s overall layout, which requires an east/west dimension of 1417.5 to work –
– according to this plan the west sides of Khafre and Menkaure are separated by 353.5 (1414/4), seen to be derived from the spacing between Khufu and Khafre of 250 cubits.
But the south base of Menkaure is 631 cubits south of that of Khafre. To explain these dimensions Legon put forward a very ingenious scheme –
– a circle of radius 353.5 cubits is ‘squared’ by a square of side 555 cubits (but note 707/555 = 1.2739, larger than the value obtained from Khufu 1.2727).
The situation is shown more clearly here –
– the intersections of the square of 500 cubits and square of 555 define the base of Menkaure (201.5). Notice also that the half base of Khafre is derived from the equation 559 – 353.5 = 205.5. 559 X 4 = 2236 and 353.5 X 4 = 1414 or the roots of 5 and 2.
But the east/west dimension of the plan remains 1417.5 and not 1414 cubits. There is a way to explain this. Legon’s plan is set out in a square of 2000 cubits, or it might be said a grid of 500 cubit units, here expanded to a square of 2500 cubits –
Within this square of side 2500 an equilateral triangle is drawn, of height 2165 (= 5 X 433 ). A double square is drawn on this height, defining the east/west dimension of the plan as 1417.5 –
– one fifth of 2165 = 433 or the height of a triangle with base 250, drawn from the centre of the south side of Khufu and the north east corner of Khafre. The slope of this triangle has its origin in the north west corner of the plan –
– another interesting triangle has a base of 1292 cubits and height of 2238 which is equal to the diagonal of a plan with east/west dimension 1417.5 (drawn in red).
These relations are rather bewildering – a ‘circle-squared’ approximation yielding factors of 19 and 37 and a root three construction to give the diagonal of the plan. Is this serendipity or coincidence? If not, how did the builders manage to integrate the different ways of defining pyramid bases?
This increasing complexity makes it easy to understand the relative lack of interest by scholars in pyramid design.