Copyright 2001, by Ernest P. Moyer
Revised February, 2003
We are now well into the mathematical genius of the Great Pyramids designer and builders. Many professional people of today, including college professors and archaeologists, are not trained to this level of mathematics. Only persons trained in the physical sciences, geology, astronomy, meteorology, engineering, and so on, have such background. Therefore, many persons will not be able to follow the presentation. But I feel an obligation to take the evidence where it leads, even to the highest intellectual levels within the comprehension of man. For those who may be at a loss, I suggest you may still obtain some sense of the pyramid designs from perusal of the graphical figures, and the discussion.
The linear regression lines on the natural logarithm plots of both the compass orientations and the mean side deviations of the Egyptian pyramids, when plotted against the pyramid numbers, show that they are mathematically, (not necessary physically), related to one another. This relationship is expressed by the equation
ln (Dev) = -1.27 + 1.17 ln (Or)
where I substituted the numerical values for the intercepts and slopes.
With explicit statement of all elements the expressions are as follows:
where A is the antilogarithm of the expression in the brackets, and B is the factor 10/gB.
Note that if B = 1 the expression devolves to Dev = A(Or). This is extremely interesting for it means that the side deviations were designed to be linearly related to the orientations when B = 1. If B is slightly different from 1, the relationship over small numbers might appear as linear. B is actually 1.171. Hence, the designer could approximate the relationship as 1:1.
When I first examined the relationship between the two I did not derive the rigorous mathematical forms above. I simply plotted the numbers from Table II against one another.
Figure Five shows this plot with the mean of the absolute side deviations (folding the negative side of Figure Two over to the positive side) in parts/10,000 against the mean orientation in minutes of arc west of north for each structure, This is a linear plot, not a logarithmic plot. (We may use a linear graph because B = 1.17.)
This graph shows the strength of correlation for five structures with measurements all from Petrie. No structure departs more than 5 parts/100,000 on average difference in base length nor more than one minute of arc in orientation from the calculated regression line. The calculated slope is 0.49 parts/10,000 per minute of arc. The calculated intercept on the abscissa is 2.2 minutes. This is the asymptote for Figure 1.
Figure Five raises crucial questions. Its meaning must be clearly understood.
If the side deviations and the mean orientations are physically independent of one another the strong linear correlation of Figure Five must be due to conscious human control.
When I first plotted the points, without benefit of the derivation, I was surprised at the strength of the correlation. The correlation certainly could not be due to accident. Some might feel that the three Giza structures, as a group, are not distributed sufficiently to demonstrate correlation. Then there would be only three data points showing correlation, not five. This view assumes unrelated randomness and normal distribution in the data. However, the calculated correlation coefficient for all five data points is 0.997. If we group the three Giza structures together and take their mean value, the calculated coefficient for their correlation with the Bent wall and Meydum is 0.998. Even more, the calculated correlation coefficient for the three Giza structures as a group alone is 0.905. Regardless of how we group the data a high degree of correlation exists among them. The mathematical equations above show that the strong correlation is not accidental.
From the mathematical derivation we can recognize the degree of intelligence and knowledge necessary to create the correlation.
Some might still argue that the correlation may be due to an inherent relationship in control that caused the sides length deviations to become smaller as the compass orientations improved. But as I described above, there is no organic cause why this should be so. For example, the builders might have constructed a pyramid misaligned 45% from the north with no side deviations. On the other hand, a structure with large side deviations may confuse the determination of the intended geographical orientation. But, as we can see, the small relative size of the deviations did not do so. The designer made careful choices to prevent such confusion. Clearly there is a strong, and clever, intelligence on display.
The strong linear correlation cannot be due to anything but design intent. A predetermined goal of maintaining construction control within exact equivalent limits of side deviation and compass orientation was executed by the builders. The correlation could only be by conscious planning and not merely uncontrolled improvements in measurement techniques. How could the builders of Meydum independently have caused the side lengths to deviate from the mean length in a manner related to mean orientation that would later correlate with the Bent enclosure wall, Giza 2 and Giza 1 strictly by accident?
The calculated regression line is very near an analytic ideal. The dashed line on the graph shows the relationship of Y = -1 + 1/2 X, where Y is side deviation in parts per 10,000 and X is minutes of arc. Of course, there can be no negative intercept for absolute values. I am merely extrapolating to the ordinate to determine the equation of the line. This equation requires that the designer calculated, and the construction engineers built, deviations in parts/10,000 and minutes of arc in angle.
The designer could "pull" the Giza values slightly to reinforce the linear regression of Figure Five, while not upsetting the slopes of Figures Three and Four.
One can see how the designer must have played with the numbers, seeking both intercepts on Figures Three and Four, and slopes, to provide numbers that would give the simple equation on Figure Five, while still making the numbers based on π and ε apparent to later investigators.
The proposed theoretical regression line is only 2% from the line calculated from the measured values; the intercept is in similar proximity. The difference may be due to our measurement limitations and not due to the construction limitations of the builders. In other words, the builders are testing the limits of our modern measurement methods.
If we accept that the pole position has moved 0.2' or 0.3' since construction of the pyramids 4500 years ago the calculated regression line falls even closer to this analytic ideal.
The simplicity of -1 for the ordinate and a slope of ½ certainly are not natural or accidental numbers. But this has dramatic impact on our understanding of the intelligence and mathematical ability of the designer. We would not be able to discover this relationship except by plotting it on a graph. Statistical or other mathematical methods would not make it obvious to us. Therefore, the designer had to use similar graphical design methods, (or a more sophisticated mathematical equivalent we have not yet discovered). In turn, this must mean he sat down with ruler and pencil (or cubit stick and stylus) and consciously laid out such scheme. My graph is merely a reflection of something he did nearly five thousand years ago.
But even this graph would not be possible if he did not fully understand exponential and logarithmic relationships. He had to know that a value of B near one would provide a regression line that appears linear. This means that he must have attempted many trial plots, with use of π and ε to provide numbers we could easily recognize. He had to have great finesse in higher mathematical relationships. He truly was a genius.
Highly disturbing is the fact that we have not progressed in our modern thinking to the point that we can arise above intellectual attitudes about ancient Egyptian cults, nor have we possessed the equivalent intellectual finesse to see through his design. How many "experts" are willing to accept that the pyramids were all tied together in one grand design, or that such ancient knowledge could have existed.
Once again we can see that Cole's data causes the Giza 1 case to fall considerably off the line and again places doubt on his measurements. If they are more correct than Petrie's they would refute the correlation for that data point. (I do not show the data for the Giza 1 sockets, which were merely construction control points.)
As we unfold the mathematical relationships we can more readily recognize why the placement of Giza 2 before Giza 1 in chronology is forced upon us, even though it violates the accepted chronological order obtained from historical and archeological data. However, this would not refute the correlation point on Figure Five since that point is independent of chronology.
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