The Mathematics of the Giza Site Plan

Foreword

This piece would not have been written if it weren’t for the inspiring work of John Legon on Egypt’s Giza site which you will find at ‘Egyptology and the Giza Pyramids’. Although his important insights have been around for quite some time (1979), they seem to have been pushed to the background. Nevertheless in my view one of his major points to argue for an overall Giza site plan dimension that would be characterized by a length of 1000√3 and a width of 1000√2 expressed in Royal Cubits, which in numbers is 1732RC (906.88m) and 1414RC (740.37m)respectively, still stands, as I will argue here. In the following I show the solution for his main claim in a ground plan that is not only based on whole numbers all the way, unlike his, but with striking dimension relations and cross-referencing between the three buildings which show us a work of outstanding mathematical beauty and cosmological depth.
Here is his Site plan measured in Royal Cubits of 0.52375m (remember I use 0.5236m, which is though immaterial here):

A1 John Legon site plan Giza

Below is the Great Pyramid (Khufu), in the middle the Second Pyramid (Khafre) and above the Third Pyramid (Menkaure), they are all taken to be squares at the base (unlike Lehner). North-South is upside down in this pictograph.

The Third is the key

I have argued on this website in a rather different context for the measure of the side of the Third Pyramid (TP) to be taken as 198RC = 103.67m (see Orkney and Giza) which makes this Pyramid the mathematical prototype of the Great Pyramid (GP). This may sound arrogant given the fact that no surveyor nor any archaeologist as far as I know ever has come up with this proposal, but given the fact that the professional surveyors give values ranging from 195RC (Lehner) to 201.5RC (F.Petrie) it is obvious that there is for this same pyramid no consensus at all on these lengths. (Why would the last surveyors, Lehner&Hawass, be the best? The site has deteriorated since Petrie’s time, 1880’s, and Lehner gives, what seem to me, rounded figures most of the time. According to John Legon the Lehner data are not valid and plucked from elsewhere).
We are talking here of differences of up to 6.5 RC = c.3.40m in the case of the TP! That is not a few centimetres here or there anymore, as is usually the case, this is really a big difference. My argument for the 198RC (103.67m) base length is mathematical and circumstantial, but also born from the conviction that what we have to do as researchers and scientists in these matters is finding the most logical, simple and reasonable hypothesis, given that we probably will never know precisely what they had in mind, unless at some point papyrus scrolls are found that give the original outlay of the plan. We don’t have those so we must imagine and reason what they could have had in mind when creating these plans, the perennial attraction and weakness of archaeology.
At least Legon’s insight strongly supports the inference that the whole site was a total concept from the beginning, as I have come to think. So that the positions and dimensions of the three pyramids at Giza were set out with the same unit of length. I emphasize this because I use a slightly different unit of length than is custom, that is 52.36cm instead of Petrie’s 52.375cm, it is though within the margin of error he gives (20.62 in +/- 0.005in, 20.615in = 0.5236 2m).

 I have written elsewhere that in my view Pharao Sneferu, Khufu’s father, who built the first three ‘true’ pyramids ever, in his lifetime, was the great architect of Giza and possibly a mathematician himself. His last work, the pyramid of Meidum, a reconstruction, has the same ratios as the Great Pyramid, that is, base 275RC with height 175RC = 11 : 7 (440 : 280 = 11 : 7, seked 5.5). It is also my claim that Sneferu’s lineage took it upon themselves to realize his grand mathematical and cosmological design in stone (except the short reigning Djedefre, son of Khufu, who was succeeded by his brother Khafre, the builder of the Second Pyramid). Even his great-grandson  Menkaure, building the much less impressive Third Pyramid, stuck to the grand design in our understanding. Anyway Sneferu was such an impressive personality and successful ruler that he became deified and widely worshipped after his death. His massive ‘state works’ brought great prosperity to Egypt with people employed on a scale never seen before.

[[Lehner gives at least two slopes a gradient which is virtually the same as the Great Pyramid, namely 51° 49′ 38” compared to 51° 50′ 40”  . (see The complete Pyramids). This is important because it sustains my claim of a 5.5 seked, since the difference is too small to be disregarded.]]

The proof

Let me first give my whole number solution to Legon’s claim of the overall Giza Site dimensions in √2 (1.414 213562…) and √3 (1.732 0508….). I will use Legon’s pictograph and show the three measures I have changed and the consequences.
For the short side I only have to change Legon’s (=F.Petrie’s) 201.5 RC into 198RC to get the perfect 1000√2 = 1414RC because Legon’s anomalous sum of 1417.5 minus 3.5 = 1414  (201.5 – 198 = 3.5), so the correct sum is 440 GP + 213  + 411 SP + 152 + 198 TP = 1414RC, which by definition is 2000 Remen (the Remen according to Petrie in Nature was the land-measuring rod),  the rest is not contested here, we even see the spacing conspicuously as 213 + 152 = 365, the days of the year.
For the long side we only have to change 201.5RC into 198RC again and 429.5 into 433RC, so both Legon’s data change to whole numbers here, and we get  440 GP + 250 + 411 SP + 433 + 198 TP = 1732, which is the required 1000√3, which is virtually 2450 Remen. (see 2450 x 2 = 4900 = 70²)
A strong argument for the correctness of 433RC, which is used by several other researchers and appears also at another place (220+213=433), is that 433/250 = 1.732 which is again √3.
What more do we want? Is this not intricate enough in itself then, and full of meaning?

A1 John Legon site plan Giza

So the question is why do I think the 198RC for the side of the Third Pyramid is what they (or Sneferu) at least had in mind, if not perhaps perfectly executed. The reason is that other values make no mathematical sense at least not as clear as the 198RC x 198RC I suggest, which has clear cross-referencing with the Great Pyramid. (the diagonal of 198² = 280.014… is virtually the same as that of a rectangle of 196 x 200 = 280.028… btw.). My analysis is supported by a dressed pavement at the east side of the TP indicating the level of the 198RC side length, as John Legon made me aware of
It is said that the Third Pyramid was less well built as the others, so it may be slightly sagged here and there, which also makes for the different interpretations and choices the surveyors make in their final verdicts. I claim the Original TP had a seked of 5.5 , while others claim on different bases it would be 4:5, which is only a slight difference of 275:280 = 55:56. It is difficult to refute my claim on the evidence.

This website is about the natural number mathematics of the complementary ratios 7 : 11 : 14 with a π-value of 22/7 (3.142857….) and 9 : 10 : 11 with a π-value of 2800/891 (3.142536….) which taken together produce an irrational Pi: 20√2/9 (3.1426968…, coined Qute here), being the root of a π²- value 800/81 (22/7 x 2800/891) and these differing values then  in relation to circles and squares. I have called it the geometry of square root 2. (extensive elaboration on this website).
Well, Menkaure signifies the basics of this natural number geometry, brought back to the smallest natural numbers (99, 126, 140, 198, 280), in an elegant structure which is the key to understanding the dimensions and symbolism of the GP, that is, revealing the radius and circumference of the Earth.
It is quite well-known that the crux and genius of the Great Pyramid design is that its height signifies the radius of a circle of which the circumference is equal to the total square baseperimeter of the pyramid, like is exemplified in the pictogram below:
(Beware we use here the Pi value 22/7 on which the GP design is based)

Pyramid basics

When we square a circle by circumference and circumscribe this square with a new circle (grey) we will always see that the radii of the two circles relate as 9:10 (see below) This can only be done by using a Pi value of square root (800/81)= √(8 x (10/9)²) = 2.22222…√2 (3.14269680…)

10,9,9 c

Here we see the foundation of the relation between circle and square with equal perimeter. The circles 9 and 10 can be read in many ways because they signify ratios, it means that 9 can be chord length, with 10 as its circle segment, but also that 9 is the radius of circle 9 and 10 the radius of circle 10. What it means is that any circle circumference changed into a square perimeter will have a diagonal which is related to the radius of the original circle as 10 : 9 or 11 : 9.9 or 110:99 etc.  (with official Pi it is 10 : 9.00317…. a trifling difference)
This way we can find several possibilities of rationalising the square root of 2 in natural numbers, here as 140/99, but also 99/70 or doubled 198/140

110 over 99 c

In the following diagram we find the ratios that relate to the seked 5.5 (7 : 5.5 = 14 :11) of the GP, that is the height related to the half side, but also the height related to the half diagonal as 9 : 10 which expresses itself in the angle of the edge of the Pyramid, when we want to match these two ratios in whole numbers we have to multiply 14 x 9 = 126, where follow a half side of 99 and a half diagonal of 140. From these ratios it emerges that a pyramid height of 126RC corresponds to a base side of 2 x 99 = 198RC, with a diagonal of 2 x 140 = 280RC, which are the supposed dimensions of the Third Pyramid. So height 126RC = 65.97m, (Lehner: between 65m and 66m), side 198RC = 103.67-70m  and diagonal 280RC = 146.61m, which is the original height of the Great Pyramid, an obvious cross reference.
But that is not all because the 198RC and 126RC refer respectively to the height of the GP from the floor level of the King’s chamber at 82RC (280-82=198) up to the top and secondly to the height of the GP at the air shaft exit level from 154RC (280-154=126) up to the top at 280RC. Also at air shaft exit level the side of the GP is 198RC, which means that the top of the GP from airshaft exit level is equal to the whole of the mathematical Third Pyramid. The volume of the Third Pyramid is therefore virtually 1 : 11 (10.97) of the volume of the GP, which means that the top of the GP at air shaft level is 1/10 of the rest of the pyramid.

pyramid-geometry-v2-300x225

As regards the Great Pyramid we can express the height in Remen, is 99/70 x 280 = 396 Remen and the half diagonal as 440 Remen so we find also here the desired ratio  440 : 396 = 10 : 9.

All this proves that the Third Pyramid was the mathematical prototype of the Great Pyramid. QED.

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