robin cook Wrote:
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> So which design imperative carried most sway? I
> should say the 14/11 and 99/70 ratios.
>
> Strictly the level of the KC complex floor should,
> according to root 2 ratio, be 82 = 280 - 198, but
> is actually 82.1 by the 'Petrie cubit' (which also
> gives a base of 219.9 instead of 220). Is this
> result of some arcane plan, building error, or the
> consequence of teams charged with a certain job
> using a 'local cubit'?

The level of the floor is 82 cubits ½ palm above the base, consistent with the 12-palm height of the Great Step and the "whole cubit + 2½ palms" elevation of the four points in the sloping passage system's floor.

The system of elevations starts with the entrance, at the top of course 18, which can be designated as "Point A," which is 32c 2½p above the base.

The second point, "Point B," is the junction of the DP and the AP, marked by a special joint in the floor of the DP.

See Lepre on this joint at Point B and another joint further up the DP. Both joints in the floor, puzzingly, are not set perpendicularly to the line of the floor, unlike the rest of the joints in the floor of the DP.

In any event, the joint at Point B is situated 24 cubits below the entrance elevation at Point A, so Point B is 8c 2½p above the base.

The third point, Point C, is the junction of the AP and GG, marked by another joint in the floor. It is set directly under the north wall of the GG. Point C has elevation 41c 2½p, and so the height or rise of the AP is 41c 2½p - 8c 2½p = 33 cubits.

The fourth point, Point D, is the top of the floor of the GG, where it joins the Great Step, on the central axis. Petrie doesn't give the elevation of D, but he does give us the height of the Great Step, in inches, which works out to 12 whole palms, and he also gives us the elevation of the top of the Great Step, in inches, which works out to 82c 2½p elevation. From these data we can work out the elevation of Pt D as 80c 2½p. This gives the GG a height of 80c 2½p - 41c 2½p = 39 cubits.

> Nevertheless, it would seem that a fundamental
> geometrical design was intentionally refined.

Absolutely. However, the KC floor and the KC shafts deal with the geometry of sqrt 2 while the sloping passage system deals with the geometry of sqrt5. These are two different systems, separated vertically by a distance of 12 whole palms, exactly along the E-W central axis, embodied in the Great Step.

> The descending passage is justly famed for the
> precision of its execution, and laid out on the
> diagonal of the double square.

This is true for the part below the AP junction point, point B, down to the bottom of the DP, but the measures that Petrie gives for the upper part of the DP, i.e., the horizontal and vertical distances between Points B and A do not produce a precise slope of 1:2.

We will find that the vertical distance A-B is 24 cubits, but the horiz. distance A-B is 48c 2p. It appears that the DP around Point B was precisely 1:2 in slope, but that near the entrance the slope tapered off just slightly. The important datum established here was the whole cubit vert. distance of 24c. It is important because it is one-third of the vert. distance B-D of 72c.

The number 72 comes up in the geometry of the 1:2 slope, which we may presume forms the basic design of the AP-GG combined line. The right triangle with 1:2 slope and horiz/vert dimensions of 72/144 has a sloping length of precisely 161c. In other words, it produces an empirical triple of 72-144-161.

The exact sloping length, as shown by calculator, is 160.9969..., virtually 161 cubits.

If they knew Pythagoras' Theorem, they would know sqrt 5 as 161/72. This number is reckoned as the average of 9/4 and 20/9, the two simple conjugate values of the square root of 5 (in the same manner as 99/70 is reckoned as the average of the two simple conjugate values for sqrt 2 of 7/5 and 10/7).

This is the essence of Hero of Alexandria's technique for calculating any square root, and it is the algorithm used in modern calculators. Even though Hero lived in the first century CE, historians of mathematics recognize that it was known and used by the Babylonians ca. 1800 BCE.

The question we should be asking is: why didn't they use the 1:2 slope for both the AP and GG floors.

It seems they added two whole cubits onto 161c to give a combined length of 163 cubits (virtual). This effectively hides their knowledge of the Pythagorean Theorem and their precise reckoning of sqrt 5.

On top of this they have divided the line into two parts of 88c and 75c, both significant numbers in their own right, and miraculously have created whole cubit rises for the two new triangles of 39c and 33c, and a whole cubit run for the GG of 79c.

I am still staggered by their genius.