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Interesting as the relations described might be, we cannot prove that the entire hypothetical project was conceived at the commencement of building – successive kings may have been responsible for developing plans of their own, as for example Lightbody’s idea that the ‘Pi perimeter’ concept governing the positioning of Menkaure was the invention of his architects.

Strictly speaking there are 5 pyramids at Giza, inasmuch as both Khafre and Menkaure are divided into a granite bottom and a limestone top. The granite casing of Khafre is 2 cubits thick, so the limestone part becomes 272 high and 408 wide, and the distance north/south between the limestone bases of Khufu and Khafre becomes larger –

It was Butler who realised the new spacing (in red) reflected the very Fibonacci numbers found in Khufu – 408 : 252 : 440 – multiples of 21 34 55. He went on to analyse the plan of the limestone bases using a cubit of 0.5236m derived from the longest stone-built length, the base of Khufu.

The pyramid of Menkaure has a base of 201.5 cubits. (NOTE) Earlier reports of its slope varied, but Robins and Shute showed it has the same slope as Khufu. The question is how far the granite extended above base. The apparent granite/limestone junction is marked by two arrows in the figure below –

Petrie believed that the granite extended to the top of the sixteenth course. Here are his figures –

Here his figures are shown to scale –

– Petrie thought that, because there are no surviving casing stones above the 16th course, and because the 17th course is a bit thicker, then this represented the transition to limestone casing.

Butler decided that the transition is more likely to have been at the top of the 20th course and the transition from thick to thinner courses. In this case the virtual base of the limestone part becomes 140 cubits.

The most up to date estimate places the base of Menkaure 24 cubits above the base of Khufu. Here is how the two pyramids appear to relate –

– the meridian triangle of limestone Menkaure (89/70) recapitulates the dimensions of Khufu (356/280).

The two pyramids are shown together below as if ‘mirrored’ on either side of Khafre.-

Butler’s plan of the limestone bases is shown below –

With a Menkaure base of 140 an immediate expression for Pi 22/7 appears –

But it is the chambers in the three pyramids that are the key to a unified plan. Here they are shown to scale –

– quite clearly they share certain common dimensions.

If the chambers are lined up along the common passage axis directed to the north, their dimensions are seen to derive from a single comprehensive scheme –

Butler’s important finding is that the proportions of the chambers are given by the proportions of the limestone layout.

Consider the proportions of the granite-lined sarcophagus chambers of Khufu and Menkaure –

Khufu (10 X 20 cubits) is represented in the limestone layout as 660 X 1317, and Menkaure (5 X12.5) as 593 X 1482. In a similar manner the proportions of the other chambers are represented in the layout. For example Khufu’s Queen’s Chamber is derived thus –

The sarcophagus chamber of Khafre (and the antechamber of Menkaure) extend east of the polar passage by 5 cubits. At site scale they extend 315 cubits east of Khufu –

Looking at the underlying geometry it can be seen that the 315 cubit ‘extension’ is given by the alignment on the diagonal through Menkaure –

Using Butler’s cubit the layout becomes a 9 X 11 rectangle –

Here the chambers are all shown packed together –

As a check here are the chamber dimensions compared with the (limestone) layout proportions –

- Khufu KC—–20 X 10——(proportion 2 to 1)
- Layout 1317 X 660—–(proportion 1.995 to 1)
- Menkaure KC–12.5 X 5——-(proportion 2.5)
- Layout 1482 X 593———-(proportion 2.499)
- Khufu QC—–11 X 10———-(proportion 1.1)
- Layout 1387 X 1262———-(proportion 1.099)
- Khafre AC—-20 X 6———(proportion 3.333)
- Layout 733 X 220————-(proportion 3.332)
- Menkaure AC–27 X 7.3—–(proportion 3.699)
- Layout 1702 X 462———–(proportion 3.684)
- Khafre KC—-27 X 9,5——–(proportion 2.84)
- Layout 1702 X 602————-(proportion 2.827)

The correspondences are rather good considering the chambers are mostly laid out in whole numbers of cubits. (The roughly cut ‘store rooms’ below Menkaure King’s chamber are not included in this analysis. Neither is Khufu’s subterranean chamber – this measures 26.87 (27?) X 15.89 (16?) cubits giving the proportion 1.69 (1.688) which might, at an outside stretch, correspond to the layout rectangle 1387 X 806 (proportion 1.72). But this chamber has clearly been abandoned and I have no idea what final dimensions were envisaged, nor what part it had to play, if any, in Khufu geometry as we now know it.)

Of course if Butler is wrong and Menkaure’s limestone base was larger than 140 cubits then the whole chamber scheme falls apart, and with it the conclusion that the pyramids and chambers must have been planned at the start of building. Perhaps a forensic resurvey of the 20th course will resolve the question.

There remain small discrepancies between the two parallel schemes – the austere granite based scheme with its strictly geometric means of expression; and the limestone bases which integrate the three pyramids, such that the Giza group might be seen as a single architectural project. For example, what cubit do we use to measure the layout – Petrie’s or that derived from the base of Khufu? The north/south distance separating the bases of Khufu and Khafre can be 250 or 252 cubits – yet the 2 cubit height of the granite decreases the footprint of the limestone pyramid by only 1.5 cubits. There is a discrepancy here of half a cubit. Apparently the builders do not seem to have been too worried about this.

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