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by H. PeterAleff

 

 

  

Footnotes :

 

 

 

1 Alexander Badawy: "A History of Egyptian Architecture", Vol. 1: "From the Earliest Times to the End of the Old Kingdom", Histories & Mysteries of Man, Ltd., London, 1990: see page 155 for summary of angles, page 151 for Queens of Pepi II, pages 137and 138 for Khufu ascending corridor and air shafts.

For consistency, I changed his Greek king names to the Egyptian ones:
Cheops = Khufu; Chephren = Khafre; Mycerinos = Menkaure

 

 

 

 

2 Dieter Arnold: "Building in Egypt -- Pharaonic Stone Masonry", Oxford University Press, New York, 1991, page 217 middle.

The angle of Senwosret’s burial passage is not stated; the illustration, which may or may not be to scale, shows it as about  25½0 versus 18½0 for the construction passage.

 

 

 

 

I.E.S. Edwards: "The Pyramids of Egypt", first published 1947, edition consulted Penguin Books, London, 1993, pages 74; 90; 81; 100 to 106, 134 and 135; 168.

 

 

 

 

a: George B. Johnson: "The Red Pyramid of Sneferu: Inside and Out", KMT, Fall 1997, page 21.

 

 

 

4 b: Zahi A. Hawass: "The Pyramids of Ancient Egypt", The Carnegie Museum of Natural History, Pittsburgh, 1990, angle computed from "72.2 feet long, vertical depth 32.8 feet" on page 24; Flinders Petrie had reported 260 32" in "The Pyramids and Temples of Gizeh", London, 1883, Section 30, as posted at users.net2000.com.
au/~fmetrol/
petrie/c6b.html

 

 

 

 

4 c: Mark Lehner: "The Complete Pyramids", Thames and Hudson, London, 1997:
Khufu descending passage page 112; Djedefre page 121; Menkaure page 135; Mastabat el-Fara’un page 139;
Khufu "Queen’s airshaft" page 67 right.
satellite pyramid angles pages 172 and 173;

 

 

 

4 d: George Hart: "Pharaohs and Pyramids", The Herbert Press, London, 1991, page 93.

 

  

 

 

5 As cited from the "Chronology" Appendix in Marshall Clagett: "Ancient Egyptian Science", American Philosophical Society, Philadelphia, 1995. Volume 1, pages 629 to 640; see pages 630 and 638.)

 

 


 

 

 

  

 

  

  Volume 2: Apparent links

 

to the Egyptian stone pyramids   

 
 

Primetitle.gif (26061 bytes)

Slopes of preferred mummy paths and major prime diagonals

You are on page    1   2   3   4   5   6   7   8   9   10   11   12

The next group of number- pyramid coincidences appears to be due not to our earthlings’ dumb luck but to rational human planning, and instead of waiting for them in that far and far- fetched future, we find them in a less distant and still very tangible past.

In most ancient Egyptian stone pyramids, the main passages into the tomb chambers rise at about the same slope. This slope closely parallels that of the most prominent prime- solid diagonals which traverse the "masonry" of the triangular- number and square- to- square  number pyramids and continue into the "air" beyond.

The similarities suggest that these stone passages may have been meant to imitate the diagonals of primes, as perfect metaphors for the "stairways to heaven" offered in many Egyptian funerary texts to pharaonic souls.

The architect and Egyptologist Alexander Badawy drew attention to the fact that many of these burial passages in the major pyramids have approximately the same slope1:

"The ascending corridor had generally a gradient of about 26.50, which represents a ratio of  h: l  = 1: 2."

Despite the great variations from pyramid to pyramid in most other aspects of their designs, the general direction of these passages is so consistent, within the range of measuring and builders’ errors, that it suggests most of these corridors were meant to have that numerically simple slope.

It is clear that the builders did not chose this relatively steep slope for their convenience because the technical challenges of building those diagonal slides through horizontal masonry made their job much harder.

That this slope was not designed to make life easier for them is further demonstrated by tomb builders from a few centuries later who kept the openings for construction access separate from the passages they prepared for the last voyage of the mummy and its soul.

When these crews used sloped construction openings, the angle they chose for their work ramps was much shallower than that of the burial passage. The Egyptologist DieterArnold describes these access slopes in his book on "Building in Egypt -- Pharaonic Stone Masonry":

"Numerous royal and private tombs of the Middle Kingdom have two shafts. One of them is considered to be a construction shaft. (...) Being wider than the burial shaft, it was apparently used for pulling up the waste material and for lowering casing blocks and sarcophagi. The second, smaller shaft starts from a place outside the mastaba and was kept open for the introduction of the burial.

The pyramid of Senwosret I had a preliminary entrance corridor for cutting out the cave of the burial chamber and for lowering the blocks for the crypt. This entrance cut was filled in when the final tomb passage was built at a much steeper slope and so narrow that only the mummy of the king could be moved in."2

The angles of the final passages were thus clearly not picked for practical purposes.

This should not surprise us because the pyramids themselves were not built for practical reasons to begin with.  They seem a rather inefficient use of stone and labor for burying a single corpse in each, if any, and theories that they were meant to store grain for the seven years of famine predicted by the biblical Joseph collapse upon inspection.

The reasons for raising these stone heaps were more likely symbolic, and the building of many among their passage slopes to specific number ratios suggests, in turn, that numbers played an important role in the designers’ beliefs about the function of their project.

It may therefore be more than a mere coincidence that the once most popular passage slope also happens to be the slope of the major and most eye- catching continuous strings of primes in the square- to- square number pyramid, and that the overall shape of that array evokes the approximate silhouette of most major stone pyramids.

Moreover, most exceptions from that 1 : 2 slope match a similar set of prime- solid diagonals in the triangular- number pyramid, and the sides of that array rise at the same angle that became the standard outer slope for satellite pyramids.

Combine those similarities with the ancient symbolic importance of numbers, and it becomes more likely than not that the hypothetical number- pyramid- constructing scribe whom I made up in the preceding chapters was once real and alive.

His or her work may well be the reason why the stone pyramid builders appear to have chosen the slopes of their passages to match those of the "stairways" of primes through those pyramid-shaped number arrays.

2.2. The angles to compare

Click on the link to the two Tables here, and you can evaluate from the slopes they list how closely the number diagonals resemble those passages.

Table 1 shows, as we saw earlier, the angles of the major prime- solid strings and prime- rich edge- parallels, together with their repetitions in the first two number pyramids. Those that are shaded have close parallels in stone.

Table 2 - 1 presents all the stone pyramid passages for which I found angles cited in my sampling of books and journals on ancient Egypt. Some of these angles vary from author to author, so I listed them side by side.

The first column of passage angles gives those from Badawy’s compilation; the second one is based on the 1993 edition of I.E.S. Edwards’ regularly updated perennial "The Pyramids of Egypt"3. The third column groups under the "Other" heading groups all the mentions of such angles that I encountered in a few additional books and articles4.  I also added the entrance trench into a royal mastaba from the pyramid time.

The main list of those passage angles follows the customary chronological order from about 2575 BCE, for the start of construction at Meidum, to about 2246 BCE when Pepi II was crowned.

However, the precision of the dates you see here is only apparent. The absolute years for Old Kingdom events in this book are the commonly accepted version, based mostly on Baines and Malek5.  Still, they are not as well anchored as their exact numbers seem to imply.  For instance, the historian of science Marshall Clagett warns: "There is still considerable uncertainty about precise dates."

As a group, the dates listed here could easily be off by a century or even two, but their relative sequence and the lengths of most reigns are fairly well established.

Omitting for now the much narrower "air shafts" in Khufu’s "Great Pyramid", the second of these Tables shows that eleven of the eighteen main stone passage angles match the apparently preferred 1 : 2 slope within two thirds of a degree or better.

In addition, two other passages coincide closely with comparably conspicuous number alignments from the first polygonal- number pyramid:

the 21.6670 for the lower entrance into the pyramid of king Khafre is only 8 arc minutes off from the 21.800 edge- repetion of the triangular- number pyramid; and

the 18.50 passage for king Pepi II’s queen Udjebten differs by only 4 arc minutes from the 18.4350 slope of prime- solid strings in that same triangular- number array as well as of an edge-repetition in the square-to-square version.

That even these apparent exceptions turn out to follow the number slopes lends further support to the idea that the pyramid architects may have borrowed some of their design features from the number pyramids.

But before we discuss the additional evidence for this, as well as their probable motives, let us briefly examine the factors that could have affected the stone slopes we observe.

*

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