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Control Finesse of Side Deviations

THE FOURTH DYNASTY - GREAT PYRAMIDS OF EGYPT

PART FIVE

Copyright 2001, by Ernest P. Moyer
Revised February, 2003

In order to address the problems created by the strong linear correlation of Figure Five I examined the data of Table II more rigorously.

I noted that the spread of individual side deviations from the respective pyramid mean seemed to follow certain patterns.  For example, I ranked the absolute values regardless of geographical side or sign.  That is, for Meydum they were 21.00, 12.00 8.30, and 1.40. The values for the Bent Wall were 11.00, 5.40, 3.70, and 2.00. Since Figure Five shows that the magnitude of the respective pyramid deviations was dependent on their sequence in the total project, I calculated the ratio of the individual deviations to the mean for each structure.  This normalized the deviations, or put them on equal footing with one another to permit more rigorous evaluation.  I show these normalized results in the following table.

TABLE III
Normalized Side Deviations in Parts/10,000

Pyramid  Mean
Deviation
N E S W
Meydum 10.70 -0.78 +1.96 -0.13 -1.12
Bent Wall 5.50 -0.67 -0.98 -0.36 +2.00
Giza 2 1.70 -2.06 +0.22 +1.41 +0.41
Giza 1 Core 1.20 +0.77 -1.92 +0.21 +0.92
Giza 1 Case (P) 0.72 +0.90 -1.67 +1.08 -0.31
Giza 1 Case (C) 2.40 -1.80 -0.18 +1.88 +0.26

(P) refers to Petrie; (C) refers to Cole.

If we now take these calculated values and rank them by numerical magnitude we obtain the following:

TABLE IV
Normalized Side Deviations Ranked by Magnitude

Pyramid Largest
Side
Deviation
2nd 3rd Smallest
Side
Deviation
Meydum 1.96 1.12 0.78 0.13
Bent Wall 2.00 0.98 0.67 0.36
Giza 2 2.06 1.41 0.41 0.22
Giza 1 Core 1.92 0.92 0.77 0.21
Giza 1 Case 1.67 1.08 0.90 0.31
Arithmetic
Average
1.92 1.10 0.71 0.25
Cole Case 1.88 1.80 0.26 0.18

I show Cole’s Case values separately because we have now established that his measurements do not possess the refinement necessary to become part of this study.

A brief look at the last table quickly reveals another startling result. The normalized values of the side deviations with respect to the mean, without regard to sign or side, show that the values for the individual pyramids are very close to one another in the respective columns. The largest normalized side deviations are spread from 1.67 to 2.06. The second largest deviations are spread from 0.98 to 1.41. The next two columns show a continuing monotonic decrease indicated by the mean values. This monotonic decrease is shown by the arithmetic average.

Clearly, the designer/builder predetermined how the side deviations of each structure would vary in magnitude, according to the size of the pyramid, but in ratio to make the normalized values nearly the same to one another. This fact reinforces our earlier conclusion that the Fourth Dynasty Great Pyramids were all part of one huge construction project. He appears to have done this (at this point in our analysis) without regard for side or sign. However, he was not so simple minded.

If we plot the mean values, while showing the range among the five structures, we obtain the following.

The open circles are the arithmetic means for the five structures. The vertical lines and bars show the spread in values. The closed diamond shows Cole’s values for Giza 1 case.

Once again we have an amazing arrangement of data, dramatically illustrated by graphical plot.

We can easily recognize that the variation by side, when plotted against the side number, shows a linear correlation relationship. The calculated regression line has an intercept at a normalized 2.35 parts/10,000 and a slope of -0.54 parts/10,000 per side.

To further enhance his mathematical finesse the designer used values which again show neat numbers.  When I calculated the regression line I found that the mean value of the side number was 2.5 and the mean value of the normalized side deviation was 1.0. 2.5 is the midrange between 1 and 4. 1.0 is the midrange between 0 and 2. You can see from Table IV that he apparently intended for the normalized side deviation to not exceed 2.0. The mean of the side numbers was inherently defined by 1 to 4 sides. (There could be no 0 sides, nor could there be 5 sides.) In fact, it appears that he was trying to reach 2.0 for the largest normalized side deviation, 1.0 for the second, 0.75 for the third, and 0.25 for the fourth. Note also that ln(10) = 2.30, very near to the calculated intercept value of 2.35.

The designer not only knew linear graphical relationships, including exponential and logarithmic forms — he also knew what it meant to normalize data, and to reduce them to a common reference.

In order to achieve such relationships the designer had to know in advance the methods of each structure.  He designed Meydum to be larger in deviation, and Giza 1 to be far closer. But he did so in such a way that the normalized values would fit neatly in relationship to one another as shown on Figure Six. In other words, he calculated the deviations to be related to the amount of geographical misalignment while relating them also to one another strictly on the basis of deviation.  On an absolute scale Meydum would seem to have values too large, not related to the other pyramids, while on a normalized scale readily shows that relationship.

Figures Five and Six now show explicitly that Meydum was intended to be part of the overall pyramid project. It does not sit in isolation, related merely by chance as one might claim from Figures One and Two.

We also can see that the side deviations of the Menkaure pyramid violates this scheme, and cannot be included on the last two Figures. It was not intended to be part of the geometric project.

We see further that Cole’s measurements come into increasing question. His values do not fit neatly on Figure Six. The second and third sides are respectively far too high and too low. This fact reinforces our respect for the superb measurements of Flinders Petrie, who stands out above other men.  Petrie was equal to the original designer in his measurements; Cole was not.

The remaining rubble of stones found at Meydum may have been originally intended to finish that structure, but  were not needed to tie it to the other 4th dynasty pyramids.  At some point, work on the outer casing was suspended because it was no longer necessary. The designer/builder may have intended that it remain an incomplete object to stand out through all the centuries as a marker to attract attention, and as an indicator of the terminus a quo for the project. This act may also have prevented confusion with the four much larger pyramids in their mathematical relationships.

The data of Figure Six might help us understand the range of errors, either in the construction, or in our measurements. I reviewed the maximum and minimum values for each side as shown in Table IV. I learned that of the eight limits on the Figure six were due to the two Giza pyramids. The median points also were populated by the Giza pyramids.  This merely tells us that the greatest errors took place where the requirements were the tightest.  Since the constructions test our modern measurement abilities we cannot say from these numbers if the spread in data is due to the former or the latter.

One is tempted to “force” the calculated means to some numerical clean value, such as 2.00 at side 1, 1.00 at side 2, and so on, with a slope of -5.0, to obtain neater regression points, but I could find no clear fit that would justify such manipulation of the numbers. The designer offered us striking information without resorting to such artificial values.

 

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