In order to more explicitly illustrate the nature of the knowledge incorporated into the stone courses I show the first data set in the following Table and Figure.
Petrie’s Measurements From
The Pyramids and Temples of Giza
Cubits calculated at 20.632 in./cub.
The estimated decay curve shows how the data points generally follow a monotonic decrease with course height. Note that this curve is plotted according to height, and not according to course number.
I exclude two courses from the estimated curve. The first is #18, the course providing the floor of the entrance passage; the second is #19, part of the second data set.
If the facing stones had been cut and laid with the same height as the inner core stones, (which is all we have left after the rampages of the past centuries), and an investigator had discovered the mathematical arrangement, he would have been alerted to the fact of this interlacing of the first two data sets. This should have raised the question of purpose before the structure was violated. Of course, we shall never know because the pyramid was raped of its facing stones before a culture arrived on the scene who could interpret the mathematical design. Even today, more than a millennium after the first intrusion of the structure, we have not attained the cultural level to investigate with adequate competence. But an astute student should have been led to question the odd data points. He might have been led to wonder about course #18, and discovery of the entrance to the pyramid without blowing holes violently in the face of the pyramid.
Since the mathematical form of this first decay curve suggests a logarithmic decay I plotted the data points against a logarithmic vertical scale as shown in Figure Two. We can see how this brings to the eye a ready recognition of the mathematical form of the set. We immediately encounter the same design intent shown in the mathematical properties of the geographical orientations and side deviations of the Fourth Dynasty Structures.
This Figure shows how the course thickness varies with pyramid height for courses one through twenty-one. I show an estimated regression line which in linear on the logarithmic vertical scale.
The estimated intercept on the vertical axis is at ln (e) = ln (2.72) = 1.
The estimated intercept on the horizontal axis is at 40 cubits height. Hence the slope is 1/40 ln (cubits)/cubit height.
Note that this horizontal intercept is at 0.0 ln vertical scale. But 0.0 ln means 1.0 cubits in the real world. The designer brought all the data display in the stone courses into one cubit thickness base reference. We shall see this further in the following sequence of data sets.
The calculated actual regression line has a slope of 1/40.3 and an intercept on the vertical axis at 1.02 ln(cubits). The value of 1.02 on the vertical scale is hardly distinguishable on the graph. The calculated intercept on the X axis at one cubit thickness on the vertical scale, (0.0 ln), is at 41.2 cubits height, slightly above the estimated value. This calculated value differs from the estimated by about three percent.
I do not show the calculated mathematical regression line because it is so close to the estimated; it would merely have clouded the plot.
The interested reader can draw in the calculated values to determine the small difference from the estimated line.
The vertical intercept is nearly the same as the estimated delimiter line defining the maximum stone thickness with height, 2.72 vs 2.80. The delimiter line intercept could have been estimated at this point, rather than 2.8, but that would ignore the neat ratio of 1/100 for the delimiter line ratios. The calculated value of the delimiter line intercept at 2.81 strongly suggests the designer/builder intended 2.8 rather than 2.72.
The reader can see how Course #18 falls outside the three sigma band limit. Course #19 falls much farther away.
Refer to further discussion below.
The plot could be tested statistically for linearity. However, the neat solution to the regression correlation makes such statistical test superfluous. We should remember that we are analyzing the creation of a human mind, and not some unknown relationship from nature.
The scatter of data points due to the seemingly loose control exercised by the builders for this first decay curve is clear on the plot.
I calculated the three-sigma standard deviation band, or what production engineers would call "control limits," for the regression. This value was +/- 0.14 units on the logarithmic scale. For higher courses this means the designer/builder held the thickness differences to less than +/- 0.2 cubits, or within +/- 4 inches. For lower courses the thicknesses were held to within +/- 0.3 cubits, or +/- 6 inches. This shows that the first few lower courses were not held as tightly as subsequent higher courses, but this did not affect the data display for higher logarithmic values. The data points were still within the three-sigma band limits. The designer knew he did not need such tight control for the lower courses, while still maintaining the scatter on the logarithmic plot.
Note that the first course, which serves to define the delimiter line for course thicknesses with pyramid height, does fall within the upper three-sigma band limit.
We see two extreme points, each of a pair, that fall just on the limits of the three-sigma band. The first is a SW measure high at course #7; the other is a SW measure low at course #13. The partner measure in each case falls either on the estimated regression line, or close thereto. Course #12 with a similar large difference in measure between the two corners, and with the high value at the SW, straddles the estimated regression line.
Were these three pair intentionally caused to differ significantly more than the other pairs? Or is this merely part of the statistical distribution of builder control from corner to corner on the pyramid?
We may obtain a better idea by listing the differences for the respective pairs.
|0.00 - 0.01||0.01 - 0.02||0.02 - 0.03||0.03 - 0.04||0.04 - 0.05||0.05 - 0.06||0.06 - 0.07||0.07 - 0.08||0.08 - 0.09||0.09 - 0.10||0.10 - 0.11||0.11 - 0.12||0.12 - 0.13||0.13 - 0.14||0.14 - 0.15|
Given the relatively close control of most of the other data points we see that the three widely dispersed points are in a class by themselves. The distribution of control in thickness varies from 0.01 to 0.08 in log cubits, while the three other courses fall well outside this group. Hence, we can conclude that the dispersion of the differences of the logarithmic thickness values between the SW and NE corners for all nineteen courses does not follow a statistical normal distribution. The data strongly suggests that designer/builder controlled the thicknesses to deliberately create an artificial distribution.
This conclusion is suggested also by the placement of the three odd courses. We can see from the graph how they seem to be placed to fall on the three-sigma bands. This further suggests that the designer/builder knew statistical control of his product, (stone blocks), and could deliberately achieve such results. We might even conjecture that his knowledge was equivalent to our current knowledge in statistical control. He then held those course thickness differences from corner to corner to provide this information for us, as part of his communication with the future.
(We do not know how the other two corners, or measurement of other places along the courses, would modify our reasoning. It may be that the entire pyramid tilted slightly from South to North, or perhaps East to West. Regardless of this lack of data, we can draw our surprising conclusions.
We should also note that the distribution of samples from any parent distribution, rectangular, trapezoidal, or other, all follow a normal distribution. Thus the data can be treated as samples taken out of hundreds of stones in each course. This fact justifies our treatment of the data on statistical assumptions about normal distributions.)
From study of the pyramid geographical orientations and side deviations we know he was intimately familiar with graphical plots. He could achieve his results only by resorting to graphical displays. The data here cannot be understood by us except by using similar conceptual techniques. As I shall show on subsequence decay curves, the builder maintained control necessary to show extreme points on the plots.
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