Perplexing questions

Just to the south of Giza rises a rocky knoll called Gebel Ghibli (or Quibli) and Lehner has remarked that this spot would have been ideal for a survey location since it is about the same level as the plateau and offers a panoramic view. This is the view from Gegel Gibli at sunrise –

Fig 7.1

Hale found that this spot lies at the apex of an equilateral triangle with base equal to the distance between the centres of Khufu and Menkaure –

Fig 7.2

That this relation is not simply due to chance is supported by Hale’s discovery  that this spot can be defined by other geometric constructions –

Fig 7.3
  • the point of the triangle (in blue)
  • a construction on the distance (676 cubits) separating Khufu and Khafre (in red)
  • a 345 triangle from the centre of Khafre (shaded yellow)
  • a square drawn on pyramid bases (green)

But how precise are these constructions? The situation at the apex point is shown below –

Fig 7.4

 – the key relation would appear to be the geometrical construction between Khufu and Khafre, making it 1314.5 east of Khafre centre, and 1314.5 X 4/3 = 985.9 south of Khafre centre, as below –

Fig 7.5

1314.5 = 5.5 X 239. This is a term in the root 2 series cited earlier –

Fig 7.6

– the successive values for root 2 being 99/70 and 239/169. And  239 + 169 = 408, the limestone base of Khafre.

It would therefore seem that Khafre is the centre of a virtual square of side 2629 (239 X 11) cubits, while dimensions along the diagonal are multiples of 169 –

Fig 7.7

Can this really quite complex scheme really be intentional or is it serenditious? This question remains open.

Yet there remain further examples of such serendipity at Giza. Tedder found that two Phi rectangles may be drawn between the centres of Khufu and Menkaure to the centre of Khafre –

Fig 7.8

However the dimensions of these rectangles are not absolutely precise but rather seem to reflect Fibonacci terms. Other such ratios appear in north/south and east/west dimensions, but not in terms of actual Fibonacci numbers.

In fact a number of Phi rectangles may be overlaid on the plan –

Fig 7.9

(Actually there is a geometrical construction for Phi in the layout, but only if the square of 2000 cubits is aligned on the west side of Menkaure –

Fig 7.10

The serendipituous appearance of these Phi rectangles prompts the question how can the architect have designed the site according to a relatively simple geometric scheme and then managed to superimpose Phi? They might of course be a pure coincidence. Whatever the answer such speculations draw us further away from the fundamental hypotheses of this web page – that Giza was designed a whole and that Khafre lies at the centre of a symmetrical plan.

For discussion

Three pyramid layout plans have been described above –

  1. The orthogonal plan based on plane geometry
  2. The limestone plan which defines the chamber sizes
  3. The plan drawn on the diagonal
Fig 7.11

The first plan begins from unity – that is, a module of 1000 cubits. A ‘virtual pyramid’ of base 1000 cubits and height 1272+ cubits (or ratio 14/11 ) is constructed. Its slope intersects the diagonal of the square at A, thus defining Khufu.

The south west corner of the plan at C is defined by root 2 (red) and root 3 (blue) constructions within the square.

The south base of Khafre is defined by the intersection of the diagonals of the square and the plan at B.

It seems to me there would be little point in satisfying the gods with what amounts to a gigantic exercise in simple geometry.  The orthogonal plan  represents a reference figure – a second plan was achieved by considering only the limestone portions. This plan was used to determine the proportions of interior chambers –

Fig 7.12

Apparently the builders considered this a way of locking everything together. At the same it is a strong argument for the whole plan having been conceived at the beginning of construction.

There is here however an interesting discrepancy – the north/south distance between the bases of Khufu and Khafre is 250 cubits but one and half cubits need to be added if the limestone base is considered. Yet the dimension according to The Fibonacci relation should be 252 cubits. Did the designers ‘fudge’  things ? Were they more interested in analogies at the expense of perfection?

The final plan involved a 45 degree rotation of the orthogonal plan towards Heliopolis and producing a symmetrical plan centred on Khafre. One might speculate that this was seen as some kind of metaphor for rebirth, much like the root 2 placing of the King’s chamber in Khufu’s pyramid.

Here the three plans are shown together –

Fig 7.13
  • A (in black) is the square of 2500 cubits anchored to the north east corner of Khufu.
  • B (in brown) is the chamber plan.
  • C (in blue) is the diagonal plan through Khafre.

When the three plans are superimposed in this way it admittedly presents a rather confusing picture. And yet each of them seemingly has its own rationale. Legon’s proposition provides a basic geometric framework. Butler shows that chamber dimensions mimic pyramid spacings. And Khafre does appear to be the centre of a symmetrical layout.

What prevents a definite conclusion is remaining uncertainty in the dimensions of Menkaure – the pyramid has not been completely excavated and the position of the granite/limestone boundary is still open to question.

But readers will come to their own conclusions – is the hypothesis I have presented a fantasy, a miasma of coincidences? Or is it like the curate’s egg – ‘good in parts’?

At Giza, numbers, multiples of simple primes, and Fibonacci terms, in all probability carried some sort of symbolic or religious significance. But it is one thing to describe these geometric relations and quite another to say what they mean.

But the symmetrical arcs to the corners of Khafre require no numbers and appear more as art than geometry. We remain far from understanding the thinking of Khufu’s architects.

Fig 10.4