Figure One C shows a clear intent to separate course levels between the NE and SW corners at the upper part of Giza I. This is noticeable in the middle of Data Set #4 where a drift is first evident. Above Data Set #5 only two data points have the Northeast higher in level than the Southwest. Above Data Set #9 the difference is better than 0.1 cubits, 2.0 inches.
Why did the designer insert this obvious difference, with the pyramid sloping slightly from the South toward the North? We cannot argue that he was not able to measure levels; his ability to maintain them equal to one another is well evident from 40 to 80 cubits height, Data Sets 2, 3, and part of 4. While the widely separated courses #91 and #92 may have been used to create a pointer to the center of the ellipse shown in Figures 10 and 11, we also see them as indicating a demarcation in the trend of the levels.
The unusual arrangement of the stone courses is also shown in the nearly constant separation of the thickness above Set #8. See Figures 13, 17, 18, and 19. The number of data points where the SW and NE corners are very near each other in thickness is a small minority, certainly far less than to be expected from statistical randomness.
A notable aspect of these top courses is the proximity of courses #155 to #161, (above 212 cubits height), #167 to #179, and #185 to #195, to one cubit thickness.
In attempt to understand why the Designer placed the courses in this fashion I examined the data more rigorously. As part of this exercise I wondered how the data might look if we normalized all to one cubit width. Stated otherwise, what might the data display if each course were shown as different from one cubit, but taken from a standard reference point? I started out with one cubit width, and then assumed that each course was to be one cubit also, but different by an amount determined by the Designer. Since groups of courses were different from one cubit, (see Table One), this exercise would create an ever increasing difference from the one cubit reference width as we proceed through the courses.
For example, suppose the first course was one cubit in thickness, and the second course was 1.2 cubits. This would place it at 0.2 cubits difference from the assumed reference value of one cubit. If the third course were 1.3 cubits in thickness this process would place it at 0.5 cubits difference from the assumed reference value of one cubit.
The point at which we start (reference point) can be arbitrary. I chose the height of 262 cubits (course #201) because that was at the top of the extant structure. (The top courses of the pyramid also were all near one cubit.) This would avoid the much thicker courses at the bottom of the pyramid. I then worked down the pyramid, calculating how much each course was different in level from the assumed one cubit width.
Such a data plot is shown in Figure Twenty. This display is remarkable for the fact that instead of a wild scatter in points as appears on the preceding Figures we now have a highly regular pattern.
I had transformed the data to a different framework. Such transformation is known in formal mathematics from the Theory of Complex Variables. Once again the designer was showing us his wide grasp of higher mathematical concepts.
For a sequence of courses near one cubit thickness the plot is level. Then, as the Designer introduced thicker course to provide the decay curves in the higher Data Sets, there is a consistent shift to higher values of displacement from the original reference level. Three distinct groups are evident above course number 150. A possibility exists of another short group from 145 to 150. The graphical plot offers some suggestion that these groups are in round course numbers from 150 to 160, 160 to 180, and 180 to 200.
Now the "circle solutions" I show in Figure 19 take on an altogether different aspect. They merely differentiate between the groups. The flat set of tangent points extending upward from the bottom of the circles are those courses near one cubit thickness, and define the horizontal groups shown in Figure 20. This display leads us to deduce that the Designer never intended the decay curves to show some higher mathematical function; he used them to separate the upper groups. That is the reason I could not find supposed mathematical purpose.
Figure 21 shows the upper-most segments on an expanded scale. This helps us grasp the form of the design. (The reason the NE and SW corners are reversed from one another in apparent level is due to the use of difference in the calculation, and not merely the thickness.)
Figure 22 is expanded still further because I noticed that a plot of the points where they moved from one level of one cubit to another were not randomly scattered. In this case they appear to display a circular curve. The NE and SW corners seem somewhat difference in the respective radii.
I then took each of the segments near one cubit thickness and plotted them on expanded scales, Figures 23 through 26.
Now the amazing control by the builder, and his skill, becomes more apparent.
Figure 23 shows a difference of 0.14 cubits between the levels of the NE and SW corners. 0.14 cubits is less than three inches difference in course levels. However, the thicknesses decrease with course number. This difference in thickness from course to course provides the regression lines I show on the graphical plot. The slope of 1/40 means that the differences in thickness from course to course was 0.5 inches, or 12 millimeters.
We can see how incredibly well the builder could achieve such control.
The three-sigma band limits show an ability to control course thickness to +/- 9 millimeters of the design value 99% of the time. This control compares with the thickness control I discuss in Part Four.
Figure 24 shows a difference of 0.127 cubits between the levels of the NE and SW corners. This is a difference of about 2 ½ inches, not too different from level differences of the previous segment. Here the designer chose to increase the thicknesses with successive courses, rather than decreasing them, reinforcing our notice of his intentional design. The slope of 1/70 means that this difference in levels was only 0.3 inches between courses, or 7 millimeters.
The estimated three-sigma band limits put his control of thicknesses to +/- 10 millimeters 99% of the time. This is comparable to the control displayed in courses #153 to #161.
As the builder proceeded up the pyramid his control of course thickness and levels increased.
Figure 25 shows this for courses #185 to #195.
The slope of 1/120 means that he separated the course thicknesses by only 0.17 inches, or 4 millimeters.
Here his control of level is about +/- 12 millimeters 99% of the time.
We should not be confused between the average difference in course thickness shown by the regression lines, and his statistical control, shown by the three-sigma bands. Although the latter is wider than the difference in thickness, with individual courses sometimes falling out of sequence, the total array of data shows how well he did.
Figure 26 shows the same plot for the top extant courses, from #194 to #201. Here I have not shown the NE and SW corners as separate regression lines. This group was part of another circle formation, as is suggested on Figure 1, but the remaining stone courses do not permit us to determine the exact parameters. The high slope value suggests this. A similar sharp slope is true for the preceding segments.
We are now in a position to reconsider the sequence in which the courses were laid.
First, I assume that each course was held to thickness differences along the course from north to south as the data plots show, within cutting error. We could measure the pyramid at several points along each side to determine if this is so, but it would require the exquisite accuracy demonstrated by Petrie. Most modern surveyors would not be equal to such refinement.
Second, we will not know if the courses were held to the design thickness throughout the entire body of the pyramid, including the interior, unless we do excavations. But this point may be mute. What difference would it make to show how well the builder achieved his goals throughout an entire course when the control was so well displayed on the visible parts of the structure?
From the data we can immediately recognize how each course, and each stone in each course, was held to high precision. The construction engineers and site managers could not permit mixing of course stones.
Neither could they randomly lay courses. Each had to be held to the design requirements. Thus it seems highly unlikely that the dressing to precision took place on the pyramid. Such process would have required many crews dressing stones directly on site. What confusion there would have been!
We cannot visualize a realistic scenario other than fair dressing at the quarry. And we cannot visualize a realistic scenario other than transportation to site in the correct sequence for each course.
This evidence introduces another serious doubt. Ramps would have blocked view for site survey to holding the precise alignments of the pyramid construction. But now ramps would have blocked view for site survey to holding the precise thicknesses and levels. Ramps would block measurement of courses to ensure that the mathematical designs were held.
A solid reassessment of our ideas of pyramid construction must be achieved if we are to properly understand this most unusual Great Wonder of the world.
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