Scholars seeking Ancient Egypt research less speculative (on both sides) than what is discussed below should check out D.Rawlins & K.Pickering at Nature 412:699 (2001/8/16). See also DIO 13.1  pp.2f.
Posted 2008 April:
In the 2007 Feb-Mar Ancient Egypt appeared an attack by John J. Wall upon an explicitly speculative DR theory (presented at the Greenwich centenary symposium, 1984) that the most important ancient Egyptian monuments may have been knowingly placed upon latitudes equal to unit-fractions of a circle. Wall's article classed this idea with “weird” science. He mailed the AE issue to DR anonymously — and never replied (non-anonymously) to DR letters to AE. (Wall belongs to a contentious “Hall of Ma'at”, which resembles and has contacts with another DR-embarrassed smearer, CSICOP. And he has been gift-published [2007 May] by yet another such dedicated creature, the DIO-loathing Journal for the History of Astronomy — of which more below.)
John-Wall's analysis was so contemptuous, even while amateurishly bungled in its science, that DR wrote a detailed letter on 2007/4/7 to AE (and thus to hiding goon Wall) wondering how such a mess could have occurred — including Wall's mis-rendering a latitude “pinched from DR” (p.4). The letter's footnote 6 noted indication of another's involvement in the AE attack.
AE replied on 5/5 without offering an explanation or owning to having erred in any respect. Wall has never replied directly.
While posting a mercifully softened version of the 4/7 letter, DR on 2007/5/30 wrote his 2nd and last letter to AE, asking whether AE could specify any expert who had refereed the article and suggesting that John-Wall name the party who couldn't even read angles' endings correctly. Despite follow-up DIO email inquiry, no (direct) response has occurred from either AE or Wall.
Shortly after his 2007 Feb-Mar Ancient Egypt attempt to portray DR as crank, Wall made his debut (2007 May) in the DR-hating Journal for the History of Astronomy, of M.Hoskin & O.Gingerich.
Posted 2009 December:
The above-cited Wall paper (JHA 38:199-206 [2007 May]) is
“The Star Alignment Hypothesis for the Great Pyramid Shafts”.
I concur with Wall that the ever-popular stellar explanations are false.
[The 45°-tilt shaft alone being sufficient disproof: were the engineers for Khufu's Great Pyramid supposed to sit about for decades, waiting for a major star to precess until it culminated at exactly 45° altitude?]
He then offers the plausible proposal that Egyptian measure might explain the angles (vs the horizontal) of the four Great Pyramid interior shafts. He cites ancient evidence for Egyptian definition of grade as the number of horizontal palms & fingers corresponding to a vertical cubit, where a cubit was 7 palms of 4 fingers each. Thus, his theory is that each of the shafts' angles A was intended to satisfy the equation:
where f is an integer (the number of horizontal fingers/vertical cubit, as just noted), there being 28 fingers in an ancient Egyptian cubit.
Wall cites the actual robot-aided measures of A by Rudolf Gantenbrink (Mitteilungen des Deutschen Archäologischen Instituts Abteilung Kairo 50:292-294  pp.293-294) for each of the four shafts: King North (KN) 32°36'08", King South (KS) 45°, Queen North (QN) 39°07'28", Queen South (QS) 39°36'28". To explain these data in accord with the above equation, he proposes ratios (angle-tangents) which are respectively: 28/44, 28/28, 28/34, 28/34.
Wall shows (p.200) that these predictions fit to within a mean error of merely 0.41 percent (using absolute-value residuals, which are never greater than rms). This agreement convinces him that “There is little doubt that the grades were intended as whole number[s] of finger:cubit” ratios.
It is unfortunate that the author of this paper is serving the interests of censorial establishments (e.g., see above) by acting as a slithering goon to assist in inhibiting public enlightenment as to the brilliance of archons. But even a goon has the right to have his views evaluated fairly, whether or not he is willing to grant others the same right.
Good Wall Paper?:
However, when DR finally got around to examining the John-Wall article, he found it curious that, if we invert the above equation's argument so that it reads
we get a better fit! — and this despite higher odds against the fit occurring by chance: Wall's residual-square sum S = 565 arcmin-squared (vs 828 for chance); while the DR inverse theory's S = 493 (vs 1840 for chance).
Which naturally suggested that the Wall fit was not so high-odds as the JHA presentation (just by percentages: Wall p.200) makes it appear.
Presuming there was JHA refereeing for this paper, it was evidently restricted to assuring such easy nits as that the celestial calculations agreed with conventional public programs' results (Wall nn.4&18) for the paper's attack on stellar-alignment theorists. But was anyone at JHA skeptical or capable enough to ask: what is the statistical significance of Wall's own cubit-finger theory's matches?
E.g., the 4th match, Queen South (QS), using
Wall's (p.200) horizontal 8 palms & 2 fingers
(34 fingers) is off the mark by Δ (O-minus-C) =
39°36'28" − arctan(28/34) = +8'.1.
But what if we had tried
adjacent 8 palms & 1 finger
(33 fingers) or 8 palms & 3 fingers (35 fingers)?
The fits would indeed have been worse but they raise the question:
how large is the target Wall is hitting?
For the QS case, the gap between the adjacent unit-fingers/cubit angles is
99'.3−, so that we may use 49'.6 as the mean distance between
potential solutions (via Wall's equation) for QS's angle.
Now, if the shafts' angles occurred purely at random, it is nonetheless
obviously impossible that angles in this vicinity can be more
than 1/2 this interval (24'.8) from a hit upon an exact Wall value
We call this half-interval H.
[The expected (chance) residual-rms S is 1/3 of the squares-sum of the four H; in the dimensionless unitary 4-cube discussed below, said S would equal 4/3, where maximum possible S is obviously 4.]
So, we will test the usual hypothesis that the matches are just chance. In the range under consideration, there is (near enough for our simple purposes) equal a priori probability for a random angle to be anywhere from 0' to H = 24'.8 distant from one of Wall's integral-ratio-grade solutions. If we re-scale this H to be unity, then the actual discrepancy (above +8'.1) is Δ/H = 8.1/24'.8 = 0.33 of dimensionless units.
Doing likewise for all four discrepancies and adding their squares yields a dimensionless sum S = 1.02.
Since there are four independent values of A, we are operating in a 4-cube of side unity, where the residuals are a 4-vector (components w, x, y, z), whose length is a measure of S (being its square-root):
To determine net probability P: since the probability density pd is virtually uniform throughout the 4-cube, we need only find what portion of it is contained within the hyper-spherical shell of radius r, centered at the (0,0,0,0) corner, each corner-bounded shell being a fragment equal to only 1/16 of a whole hyper-spherical shell. (For simplicity, our math development is examining only cases where the 1/16-shell is entirely within the 4-cube — that is, where r < 1. The 1% spill-over for r = 1.01 is too trivial to bother about.) So probability P is just the hyper-spherical portion's 4-volume, divided by the 4-cube's:
Thus, for Wall's fit, where r = 1.01 (square root of S = 1.02), we just round r to unity and find
which is not statistically significant.
Next, we examine the inverse proposal,
in which we base our definition of grade on a horizontal not vertical cubit.
The respective proportions that closely fit the Gantenbrink data are then:
4 palms 2 fingers (KN), 7 palms (KS), 5 palms 3 fingers (QN), 5 palms 3 fingers (QS). These figures are more simply expressed as: 18 fingers (KN), 28 fingers, 23 fingers, 23 fingers. Thus the respective ratios are: 18/28, 28/28, 23/28, 23/28. What is striking here is that these numerators obviously involve appreciably smaller integers than those of the denominators of Wall's fits, which creates correspondingly larger relative intervals, thus lowering the chance of a random hit. So, when the dimensionless residual-square sum for this approach is computed, it turns out to be S = 0.34; and r, the square root of S, is 0.58. The above equation for probability then finds:
Which is significant if not strongly so.
Another theory one might try (if for no other reason than to exhibit the variety of potentially fitting theories) is that the four shafts' grades are simply (ordered?) unit fractions of a circle. All three of the well-measured shafts are within 1% of such a fit (KN 1/11, KS 1/8, QS 1/9), and their H are much larger than those for the cubit:fingers theory, so that odds against chance will be significant.
Given the smallness of the samples adduced,
DR is not alleging that any of these theories (or his 1984 Greenwich one)
is certainly true or false.
[None are as near convincing as W.Petrie's finding that if the Egyptians believed π = 22/7, then the perimeter of the Great Pyramid's base equalled in length a circle of radius equal to the pyramid's height. The other two Giza pyramids seem to have grades of about 4/3 (Khafre's giant 3:4:5 demo?) and 5/4 (Menkure).]
The four shaft A data (some of which may see future improvement that could seriously alter odds computed hereabouts) are an infirm foundation for conclusions as things stand; e.g., even the better-fixed empirical data are too unsure: up to at least 0°.2 by G's estimate. But there are certain obvious ironies here with respect to Wall's attacks on DR's high-odds if speculative 1984 theory, coming from one who would inadvertently publish a blah-odds paper without he or his equally numerate, brave, & principled rewarders even realizing it.
Indeed, none of the main parties responsible for publishing this paper even understood (as in the case of the semi-tourist magazine 2007 Wall paper cited above) that Wall's theory is a statistical problem — much less thought to evaluate it on that basis. Considering the extent and dedication of Wall's labor on behalf of DIO's fearers&smearers and shunners&runners (whose mental shortcomings leave them few other options for countering DIO's all-too-accurate exposures of their pretensions), one might have expected them to have served him better — instead of just adding another notch to their already-overflowing embarrassment-score.
DIO thanks Keith Pickering and Johns Hopkins University's Hans Goedicke for advice.