MJ wrote:

"However, I am labouring under the impression that yourself and others are arguing that not only did the 4th Dyn AEs not know the Pythagoras Theorem, but also that even if they did know it they would not have incorporated it in the design of a king’s burial chamber."

It does not necessarily follow that a knowledge of the 3-4-5 Pythagorean Triple, or any other Triple for that matter, means that the Egyptians knew of the Pythagorean Theorem, or that they knew that such a theorem might even exist.

The 3-4-5 Triple, and many other Pythagorean Triples and "near" Triples, can be easily discovered empirically. All that is needed is the ability to create an accurate right-angle, and the ability to accurately subdivide a linear measurement tool. The Old Kingdom Egyptians had both of these capabilities. They were adept at field surveying and in the accurate laying out, and construction of, sophisticated structures.

By entertaining the possibility that they were able to empirically discover many "true" Triples, and many more "near" or "apparent" Triples, it then becomes possible to arrive at explanations for many other design choices that one sees in Egyptian constructs. (I have been working on a draft of an initial paper on this subject dealing with OK pyramid design.)

Two examples of apparent Triples possibly known about at the time are the 49-50-70 Triple, and the 70-70-99 Triple. If one accepts both Petrie's finding for the slope of the core of two sides of the Red Pyramid, and also the current estimate of 45° for the slope of the casing of this pyramid, then one can see that it is not outside the realm of possibility there was an intent to incorporate both of these Triple relationships in the Red Pyramid's exterior design.

In this context, I also see it as being entirely possible for the remen and double-remen relationship to have been simply the practical application of an awareness of the above two apparent Triples. A square remen (i.e., 5 palms) was seen as having a diagonal of almost exactly 7 palms (a royal cubit). A square royal cubit (7 palms) was seen as having a diagonal of a double-remen (10 palms - with the 9.9 being fudged to 10. I suspect that in measuring large distances, or in situations where precision was desired, the more accurate 70-70-99 relationship would have been re-instated.)

All of this is, of course, theoretical. It may have been the way that they thought about things, and it may not. As there is much of value that can potentially be gained, I believe it certainly to be a line of inquiry worth pursuing.

Lee Cooper