robin cook Wrote:
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> Moving on to the Emhotep site they give 39.6
> degrees for QC south but 36.7 (?) for QC north.
> They state a length of 63.6 m (~ 121 cubits) from
> the start of the shaft (presumably just the
> angled portion) to the blocking stone - this would
> make it probable that the hieratic glyphs read 121.

Their value for QCN was an average slope for the whole of the shaft, including the west-pointing portion. This appears to me to be a 3:4:5 design.

Regarding the QCS shaft, the Project Djedi team analysed the video footage of the walls of the entire shaft and discovered a second set of marks just 3.7 metres from the blocking slab. This is about 7 cubits. It is likely then that these marks record a distance of 114c from the inlet in the QC.

Curiously, the elevation of this point is about 114 cubits. If we take the horizontal portion at the bottom of the shaft to be 4c, then this makes the sloping length of the shaft to this point 110c - yes, another example of the 110c element, found three times in relation to the KCN shaft - while the height of the QCS shaft to this point is approx 114c - 44c = 70c, the same as the height of both the KCS and KCN shafts in Gantenbrink's model. Thus we can say that the basic model for the QCS shaft, as defined by the second set of marks in the shaft, is 70-85-110.

We can see that QCS expresses the two elements of 70c and 110c as are expressed in the dimensions of the KCN shaft. One of these shafts is derived from the other. Which comes first - KCN or QCS? I'm not sure.

> So we come back to Gantenbrink, who gives this information -
> KC south horizontally 1.72m. Shaft short sections
> 39.20° and 50.54°. Then 45°.

The bottom section, 39.2°, is a seked of 8½ palms, the same as the entire sloping portion of QCS. I think Bauval is right that QCS points to Sirius, given Sirius' importance in Egyptian cosmology. Might not the bottom segment of KCS point to Sirius also?

> He remarks - "despite
> the extreme fluctuations in the initial section,
> the shaft seem to proceed with great exactitude
> and constancy".
> KC north horizontally 2.63m. Shaft 32.60°.

Gantenbrink is misleading us here. He has declared himself opposed to an astronomical interpretation of the shafts, and he is using the idea of an average slope for KCN which is around 11 palms and run with that. At the same time, he has found that the final portion is 31.2°, and left this out of the equation. Gantenbrink published his results in 1999 at cheops.org, and hasn't changed them since. John Wall piled on to this anti-astronomical interpretation and published an article on this citing Gantenbrink's seked model. (Gantenbrink's opposition to an astronomical interpretation of the shafts may lie behind his premature revelation of the results of the DAI survey and his falling out with Stadelmann and Hawass.)

More recently, Giulio Magli, using Stellarium, has found that the final angle of KCN points accurately towards Thuban's culmination in 2550 BCE.

> QC south horizontally 1.96m. Shaft average 39.6078°
> QC north horizontally 1.93m. Based on 14
> measurements made over a distance of 17 meters,
> the shaft's angle of ascent fluctuates between
> 33.3° and 40.1°.

The QCN shaft has two distinct slopes at its lower end that do not appear at first glance to have either an astronomical or geometric order behind them. Bauval reckoned that the shaft pointed to Kochab. That's as good an astronomical determination as any.

As for QCN's geometric model, I would guess that the overall slope is 3:4, and if QCN sports a set of marks 7 cubits below its blocking slab (which has not been penetrated) similar to the marks found found near the end of the QCS shaft, then I would guess that the model the theorized QCN marks convey is a 3:4:5 triangle with specific dimensions 66-88-110.

> The horizontal lengths all being different suggest
> that the virtual 'focal points' of the shafts must
> lie below the chambers and Gantenbrink gives his
> reasoning for the origin of the KC shafts 22 south
> of KC centre. But where does he record "that the
> final 16% of this shaft is set at a consistent
> angle that is slightly flatter than the 7:11 slope"?

Please reread what I wrote where I cite Gantenbrink on the 'Mankiller' tunnel. He gives this segment as being 11 metres long.

> I dragged up a drawing I made many years ago based
> on Gantenbrink to ease the present discussion -
>
>

I would leave the QC shafts out of it. Just go with the KC shafts for now. The exit points of the upper shafts define the top of a giant 198c x 198c square, with the line of the KCS shaft defining the diagonal of the square. We can see that this giant square is the same dimensions of the square at the 154c level from which it is dropped. We know that the diagonal of this level is 280c, so we can also see that the diagonal of this giant square is 280c. That is, KCS, in the Gantenbrink model, expresses the 99:70 value for rt2 while the line of KCS as the diagonal of the 198 x 198 square expresses the 140:99 value for rt2.

Also, taking the two diagonals of the giant square, we see that they cross in the gabling of the Queen's Chamber. That is, the gabling of the QC is
placed at the centre of the giant square.

> - KC south implies a square of 198 cubits, hence
> 44 below pyramid base - in turn the origin for QC
> south coinciding with the line of KC south on the
> height of the pyramid.

I am not so sure that the line of the QCS shaft pertains to this giant square. If I recall, Jeremy Potter has calculated the intersection of the KCS and QCS lines as being 55c below the base, not 44c below the base, but I stand to be corrected.

> Your scheme is indicated by blue lines. Now
> Gantenbrink, in the 1997 paper he presents on his
> website, stresses the importance of finding common
> reference points in the design of Khufu, saying
> that "The ceiling height of the QC passage
> constitutes a common quantity at the lower end of
> the Great Gallery" (with the ceiling height of QC
> north shaft exit), so the floor joint you mention
> to obtain your 110 vertical dimension may not be
> so important (although I would not be so strict -
> floors, ceilings, and centre axes should all be considered).

You are referring of course to his "common reference points" which are at the upper (S) and lower (N) ends of the Grand Gallery.

While I agree with Gantenbrink that the upper reference point has elevation 84c, I disagree with his analysis that the lower reference point - and the ceiling line of the long horizontal passage - has an elevation of 44c, though for a first draft of the design I would say this is correct. Gantenbrink himself admits it is about 5cm (half a palm) off this elevation.

The bottom of the N wall of the gallery is 18 palms above the joint in the floor which marks the joining of the ascending passage and the grand gallery. This joint has elevation 41c 2½p, which gives the elevation of the "common reference point" as 43c 6½p. This elevation affects only slightly the geometry of the QC shafts.

> On the other hand is Gantenbrink so meticulous
> as he presents himself - he states that Descending
> and passages have the same slope, which is not a fact.

Miatello makes the same mistake, giving the seked of the GG and AP as 14 palms. As many of us know, the geometry of the AP and GG is much more complicated.

Butler also makes a mistake regarding the line of the floors of the GG and AP, assuming them to be one consistent sloping line, of length 163 cubits; but as a first impression, not bad.

> And how come he and Legon, both of whom
> are familiar with Petrie's figures, nevertheless
> promote the 154 KC shaft exit height?

I think the 154c level represents the primary design level. It is identified in the outer course structure as the top of course 104. The diagonal at this level is 280c, the same as the height of the pyramid.

This level also defines the top of a giant square, whose diagonal, represented by the line of the KCS shaft (taken from its original design level at 154c elevation, not its adjusted level at elevation 152c 5p), is also 280c.

The 154c level is also strongly related to the level of the floor of the KC (82c elevation). There are a few ways to see this relationship. The diagonal at the 154c level is the same as the height of the pyramid while the diagonal at the 82c level is the same as the base of the pyramid (440c). Also, the base of the hypothetical pyramid at the 154c level is 198c, the same as the base of the hypothetical pyramid at the 82c level. The latter coincidence seems to be a secondary phenomenon, a result of these two levels deriving from different expressions of the 198:280 proportion.

> If the start of KC south is at 77 vertical and 22
> horizontal from centre; meets the horizontal shaft
> part 8 cubits from KC centre; is stated to be 45
> degrees; and its presumed target (the belt) is also
> 45 degrees, it must hit the casing at 154 vertically.

Correct.

> So an adjustment to 152/153 can have
> nothing to do with stars.

Correct. Yet the 45° angle of the majority of the length, and the alignment to Al Nilam, is maintained. The KCS shaft decants about 1 palm below the top of course 103, which is 10 palms below the 154c level. The best reason I can come up with for this shift is that the top of course 103 is 200c long. I regard this as an archaism, relating this pyramid to its predecessors, the Bent and Red Pyramids. I also think it draws attention to the length of the line at the 154c level, which is a whole number of cubits - 198c.

> And where does KC north bend?

I'm not sure, but I don't think it's all that relevant. Petrie has already shown that the bottom of the KCN shaft decants fairly precisely at the top of course 102. This is a 19 palms shift down from the 154c level. Our task as I see it is to account for this shift. I think the shift relates to the elevation of the junction of the floors of the GG and the AP.

> The putative adjustment (blue line) seems pretty large to me.

Yes, it creates two different levels, with different line lengths, one of 198c and the other of 200c.

> As to Dashur some interesting
> possibilities, but as Hamilton has said the survey
> data is a complete mess (and I'm an adherent of
> the 362 Bent base school and wouldn't want to clog
> up this thread going on about Dashur again).



> Maybe they should let the Russians come and measure the
> pyramids, they are usually pretty efficient ;+)
>
> I don't have an astronomy program at the moment
> but if anyone does (and has not yet become
> petrified by all this 'number crunching'), what
> are the altitudes of the suggested target stars in
> 2560 BC?

I use Stellarium. It's free to download. It takes into account the 1'-2' difference in the apparent altitude of a star created by atmospheric diffraction from its true altitude. Previous software used by Bauval and Legon did not do this, so their calculations were not as accurate as they could have been using Stellarium.

I have used 29.9791495*N; 31.1342005*E as the GPS coordinates of the centre of the GP, though I understand that (the late) Glen Dash has modified these figures slightly. Also, Stellarium requires an elevation above sea level. For the base of the GP, I used 60m.