Hi Robin,

Congratulations! your website is stylish, easy to navigate and a useful resource for those interested in this subject - many thanks for sharing it.

No time at present to read it all through - its spring, snow has finally melted, and much to do in the garden before the mosquitos arrive.

I’ve cut and pasted from some old notes of mine that I hope will clarify a brief mention in your ‘Layout Plan’ section concerning the 45° 2000 cubit long dual parallel diagonals, a key component of my proposed overall site layout:

Robin: “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”

My “invoking an error” and the idea of taking survey measures at face value needs clarification!

Accurate survey data is essential for determining if a site layout has an intended overall design, but we also need to allow for building and surveying tolerances and the sleight variations in the estimated value of the cubit.

Petrie has some interesting comments on the relationship between the square and its diagonals in relation to the cubit, which we should bear in mind when assessing the veracity of the 45° 2000 cubit long diagonals:

“Considering how in the Great Pyramid, the earliest monument in which the cubit is yet found, so much of the design appears to be based on a relation of the squares of linear quantities to one another, or on diagonals of squares, it will not be impossible to entertain the theory of the cubit and digit being reciprocally connected by diagonals. A square cubit has a diagonal of 40 digits, or 20 digits squared has a diagonal of one cubit; thus a square cubit is the double of a square of 20 digits, so that halves of areas can be readily stated. This relation is true to well within the small uncertainties of our knowledge of the standards; the diagonal of a square cubit of 20.62 being 40 digits of .729, and the actual mean digit being .727 ± .002….” Petrie 1883:181

According to Glen Dash’s 2017 revised survey data of the Giza site layout, the EW dimension from the centre of the east side of Khufu’s pyramid (G1) to the centre of the west side of Menkaure’s pyramid (G3) is 741.450m and NS, 740.10m, that form a near perfect square.

(EW 500115.15 - 499373.7 = 741.450m; NS 100000 - 99259.9 = 740.100m)

The length of the cubit was to the nearest millimetre, 0.524m. (note: a millimetre is the smallest unit of measure used in modern architectural design).

741.450m / 0.524 = 1414.98 cubits
740.100m / 0.524 = 1412.405 cubits.
Mean: 740.775m (1413.692) ~3cm short of ~1414 cubits, the length of the side of a square with a diagonal of 2000 cubits

Petrie’s estimate for the cubit used for the design of Khufu’s pyramid (G1) is 0.523621m….0.523748m, with a mean value of 0.5237m. (Petrie 1883:178)

The sarcophagus chamber’s floor length yields a cubit of 0,524m:

“Probably the base of the chamber was the part most carefully adjusted and set out; and hence the original value of the cubit used can be most accurately recovered from that part. The four sides there yield a mean value of 20.632 ± .004, and this is certainly the best determination of the cubit that we can hope for from the Great Pyramid.” (Petrie: 1883:81)

(20.632” ± .004” = 0.524m the mean value of 0.52415m and 0.52395m.)

According to Dash’s 1915 survey of G1, a cubit based on a 440 cubit side length of the pyramid’s base had a minimum value of 0.5233m and a maximum value of 0.5237m with a mean value of 0.5235m consistent with a surviving New Kingdom 0.5235m cubit rod from Amenhetep I’s reign (1525 - 1504).

It seems the estimated length of the cubit for Dyn. 4 can vary between ~0.522m and ~0.524m. Reasons for apparent slight deviations from a standard cubit are many. The limitations of ancient building and surveying tolerances need to be taken into account. Scores of wooden cubit rods used by surveyors, masons, overseers of work gangs, master builders etc. in use on major construction projects - surveyors also carried coiled ropes with knots marking cubit intervals - the length of their measuring rods and ropes most likely had sleight differences that affected measurement accuracy over longer distances.

Sleight differences in the length of the cubit are accentuated when converting long distances measured in metres to cubits.

The 1047.614m length of the diagonals is 2000 cubits when a value of the cubit is ~0.5238m - well within the limited range of values for the cubit. If the 2000 cubit line was intended it is surprisingly accurate - all credit to the ancient surveyors using a simple back sight and fore sight that could achieve such accuracy over a kilometre distance and over uneven terrain with a height difference of ~12m.

The length of the cubit is irrelevant when calculating the angle as its calculated straight from Dash’s survey data that uses metres.

The diagonal that defines the positional relationship between G3 and G1 is a 45° 2000 cubit long diagonal from the centre of the west side of G3 to the centre of the east side of G1. The actual angle according to Dash’s survey, is 45° 3’ that deviates only 3’ from 45°.

This 2000 cubit line has a 45° SW - NE diagonal even more accurate than the 45° SW - NE diagonal of the base level of G1 that has a 4’ difference (44° 56’, Dash 2015 season survey). In spite of the 4’ difference, we reasonably assume G1 was designed to have a square base so we should have no problem assuming the SW - NE 2000 cubit diagonal had an intentional 45° angle.

A line connecting the centre of G3 to the centre of the west side of G1 has an angle of 58° 14’ 7” that is accurate to within 2’ 50” of the 58° 16’ 57” ‘golden ratio’ angle. (The centre of Khafre’s pyramid (G2) lies slightly to the west (~68cm) of this line)

Whether by design or as an unintended consequence of other factors, the appearance of the ‘golden ratio’ is also found elsewhere at the site and in Khufu’s pyramid geometry. The difference in height between a 14:11 (sqd 5 1/2) based pyramid (280:220) and a ‘golden ratio’ based pyramid (279.844:220) is ~8cm, a negligible difference considering the pyramid was originally ~14666cm high.

The other parallel 45° 2000 cubit diagonal links the centre of G3 with the entrance to Khufu’s cult complex located on the centre line of the eastern side of the pyramid. Extended NE ~20 x 2000 cubits (44000 cubits / 440 khet) it aligned with the necropolis of the ‘Greatest of Seers’ next to the southwestern corner of the important Iunu temple complex.

The Pyramid Texts mention Iunu, Djedut and Dual shrines. For example:

“You have become akh in the Akhet and stable [in] Djedut. Your arm has been received by the bas of Iunu, your arm has been taken by the Sun. [Your head has been] raised [by the Dual Ennead, and they have put you], Osiris Pepi, at the fore of the [Dual] Shrines [of the bas of Iunu……” P 480
Note: Djedut (Ddwt) - the necropolis of Iunu.

Extending the 45° 2000 cubit line SW in the opposite direction, the bright star Alpha Centauri* could be seen setting over the south western horizon a useful point of light to sight to when setting out the 45° Iunu, G1, G3, star line. The accuracy of the sight line could be refined further by setting out a temporary square with diagonals over the initial star sighted line.

A 45° angle line heading NW from the entrance to Khufu’s cult complex that mirrors the G1, G3 / Iunu star line, aligns with the pyramid of Djedefre (Khufu’s son) named ‘Djedefre is a Shining Star’. This line extended SE also aligned with the bright star Alpha Centauri but this time, as it rose over the south eastern horizon.

(Djedefre (or Radjedef) was probably involved in completing his father’s mortuary complex as variations of his ‘Golden Horus’ name (‘The divine falcons are golden’ ) are found in the boat pit next to Khufu’s pyramid.)

*note: Alpha Centauri, the third brightest star in their sky, is actually a triple star system - two of the stars similar in size to our own and the third star a much smaller reddish star that happens to be the closest star in distance to our own. Alpha Centauri is no longer visible from Giza.

As a consequence of the dual 45° 2000 cubit diagonals defining the positional relationship between G1 and G3, the angle bearing NE from the centre of G3 to the centre of G1 is 52.2281° (52° 13’ 41”).

A mirror line heading NW from the centre of G3 at the same angle aligns with Djedefre’s pyramid ~8 x 2000 cubits to the NW. Both these pyramids are about the same size and much smaller than G1 and G2. Both were partly clad in red granite casing blocks. This ~52.2° line extended SE in the opposite direction aligns with Netjerikhet’s (Djoser's) mortuary complex named ‘Horus is the star at the head of the sky’ (Quirke after Helck), the mother of all royal mortuary complex’s that had a pyramid as its central component.

If the dual 45° 2000 cubit diagonals were intentional, they are key to the initial setting out of the Giza pyramid field. Royal mortuary complexes were an integral component of the royal afterlife, an idealised world laid out with mathematical precision.

Quirke on the Amduat:  "One of the most striking features of these accounts of the unknowable is the mathematical precision, both in the naming and in the actual measurements.  Thus the second and third hours specify the area covered as 309 iteru in length, and 120 in width." (Quirke 2000: 48)

Khufu’s Horus name was Horus Medjedu - “This was adopted by the king at his accession and subsequently used throughout his reign........The name expressed the close relationship between the king and the celestial deity he embodied. As a celestial deity, Horus was remote yet all-seeing, enfolding within his wings the entire cosmos. A god with a powerful symbol and metaphor for their own earthly rule.” (Quirke 1990: 21)

In the Coffin Texts, Horus is revealed as being 1000 cubits long. No matter where in the sky this falcon inspired deity was hovering above the earth, he would always look down from what appeared to be the centre of an immense circular horizon. A circle drawn with a ‘Horus’ radius of 1000 cubits has a diameter of 2000 cubits. At the planning stage the designer drew a circle with a radius of 1 cubit and constructed a square inside the circle so that all four corners touched the perimeter of the circle. The square has sides 1.414 cubits and diagonals 2 cubits long. The surveyors could then scale up the drawing 1000x for the initial setting out of the site.
Specified measurements for buildings and various structures found in Old Kingdom inscriptions are always in whole number cubits and multiples of 10, 100, 1000 for primary large measurements used in architectural design. Four textual examples: 200 x 200, 100 x 50, 7000, 1000 x 440.

The ancient Egyptian surveyors had the skill to measure out long distances - the longest (over 3.6 km) mentioned in the texts, is from the Dynasty 5 sun-temple of Niuserra that specifies a stone structure 7000 cubits long. The stone walled rectangular enclosure of the Dynasty 3 mortuary complex of Netjerikhet has an inside measurement of ~1000 cubits.

I hope I’ve sufficiently clarified why 45° 2000 cubit dual diagonals are consistent with the survey data and with the sparse textual evidence that has survived in the archeological record. It obviously requires clear diagrams to fully comprehend the simplicity hidden away in all the numbers, but my web pages with all the info and diagrams are stuck in Google Sites after they were moved there by Google from their now discontinued Google Pages. I lost some diagrams and pics in the process and I just haven’t the time to sort it all out. However, I have all the diagrams on my computer (I hope) so I could make them available if anyone is interested and send some by email or post them here on the HoM if only I could remember how to.

Some comments by egyptologists:

Miroslav Bárta: "As at Giza, the Abusir pyramids of Sahura, Neferirkara and Neferefra were very likely built (i.e. situated in the necropolis) according to a single master plan….” Miroslav Bárta 'Location of the Old Kingdom Pyramids in Egypt', Cambridge Archaeological Journal 15.2 (2005: 186

"It is therefore appropriate to ask, in a landscape as prospect-dominated as the Nile Valley, which sites and monuments were mutually intervisible and whether their respective locations, horizons and vistas are owed to something more than mere coincidence." ‘The Topography of Heliopolis and Memphis: Some Cognitive Aspects’ David Jeffreys.

"The idea that the distribution of the pyramids is governed by definable ideological (religious, astronomical, or similar) considerations is attractive. After all, if there were such reasons for the design of the pyramid and for the relationship of monuments at one site, why should we shut our eyes to the possibility that similar thinking was behind the apparently almost perverse scatter of the pyramids over the Memphite area? The argument that the Egyptians would not have been able to achieve this had they set their mind to it cannot be seriously entertained." ‘Discussions in Egyptology’ 30, Malek 1994: 101-114

Chris