Part Five
When I first examined the Fourth Data Set I was completely baffled. I could not for the life of me figure out what the designer was about. I thought about it, pondered it, stared at it, turned it sideways and upside down. I broke it into pieces. I noted that a part of the group seemed to have a decay, courses #48 to #57, and that four courses, #59 to #62 formed a straight line on the data plot. See Figure Nine. But that didn’t help much. I felt that four data points were not sufficient to define another decay curve.
Refer to tabulation below.
I tried examination with both linear and logarithmic thickness plots. I tried both course height and course number. None of those ploys seemed to help.
I felt I could find a circle that would fit courses #48 to #57. I used courses #47 and #48 as pointers to the center of the circle and the widely split course #57 as another pointer at 90 degrees to the first. Indeed, it seemed that this was a reasonable solution.
I then thought that perhaps courses #59 to #62 pointed to the center of a second circle. I drew that circle and thought perhaps it also might fit to the data points. But then I was left with the odd courses from #43 to #46 and the strange arrangement from #63 to #66. I thought perhaps the designer used those points to "throw tangents." At that point I realized I was merely guessing.
I decided to back up. I took the plot from Data Set #3 and extended that graph to include this data set.
See Figure Ten.
I then began to see more reasonable patterns. If I drew a line between courses #43 and #46 (regression line B) I saw that it was parallel to the regression line of Data Set #3 (A). The spread in those two points was about the same as the calculated threesigma spread of Data Set #3 (shown by dotted lines). I found the same suggested visual arrangement for courses #44 and #45 extending downward to include #47, (on the threesigma band), and #49 through #53 (regression line C). Certainly, extension beyond #53 was not possible, but I was intrigued by the same slope and seeming threesigma dispersion of the three regression lines. Even the four odd course alignments from #59 to #62 were at the same slope, or approximately thereto (line D).
Was the designer leading me along a path of discovery?
I found I could include three data points each from regression lines A, B, and C, line (a). Then a parallel line included points on B and C, line (b). If I projected a line from courses #63 to #65 it passed through the data points at #61, the other widely split course #58, nearly through #50, near to #46, and to the upper point at #42 on Data Set #3 (c). Line (c) was then perpendicular to (a) and (b), or nearly thereto.
But this was all mere playing with lines and points. There had to be more sense than just casting lines about on graph paper.
However, I continued to be intrigued with the possibility of a curvature of some kind from #48 through #58. It then dawned on me that perhaps the designer was trying to show me an ellipse, rather than a circle. I plotted the ellipse I show on Figure Ten.
Of course I could adjust the center and size of the ellipse somewhat to fit the data points but the one I show seemed like a good fit.
Now I became far more interested. I noticed that the upper part of the ellipse just touched a horizontal line drawn from the mean of the two data points of the delimiter course #35. The right hand portion seemed also to just touch another delimiter line drawn vertically through the two points of course #66.
If the designer intended for course #35 to define an upper delimiter line to the ellipse I now understood why he had forced me back to Data Set #3. He "wasted" courses #43 to #46 to force my attention.
From this deduction I then removed the four previous data points making up lines (B) and (C), #43 to #46. If they were intended to lead me to recognize Data Set #4 they were not part of that design. This gave me Figure 11.
Ellipses can be defined by minor and major circles. The minor circle has a radius length equal to ½ the minor axis of the ellipse. I shall call that length "a." The major circle has a radius length equal to ½ the major axis of the ellipse. I shall call that length "b." I drew those into the graph. Mathematicians call these two circles the eccentric circles of the ellipse. The ratio of their radii defines the flatness of the ellipse. If the two radii are equal with b = a the ellipse collapses to a circle.
The area of a circle is equal to the radius squared times Pi. Area = π r^{2}. The area of an ellipse is equal to Pi time the product of the length of the two radii. Area (ellipse) = π a b.
I now understood why my initial guess at a solution to the data set invoked two different circles. I had the sense of them from my data plot, but not the solution. I was probing the two curves from the ellipse and its major circle.
In my first tentative evaluation of the ellipse and the two eccentric circles I probed the length of the radii. But they seemed such odd numbers I could not make sense of them.
Next I evaluated the areas of the respective circles from my graphical scale of inch units.
I then made a startling discovery. The area of the major circle was 36, Pi X 3.385^{2}. The area of the minor was 16, Pi X 2.257^{2}. Both were integral squares: 6^{2} =36; 4^{2} = 16. The area of the ellipse was Pi a b = Pi X 3.385 X 2.257 = 24. But 24 was the product of 6 X 4, the square roots of the areas of the major and minor circles.
How intriguing.
A man, five thousand years ago, designed the thickness of stone layers in the Great Pyramid in such manner that he would cause a mathematical interpretation that would yield neat numerical values. How did he know that I would use a scale in inches that would cause such a fascinating relationship? It was uncanny.
But my scale was merely a convenience for graphical presentation. It was not the real world. How would the ellipse and its major and minor circles relate to one another if I used the actual stone course cubit values?
The ratio was fivetoone; each inch on my graph was equal to five cubits in height.
I felt that I would come up with numbers that would not have such neat numerical relationships.
When I calculated with the real numbers I was stunned once again. I obtained another set of uncanny numerical relationships.
The area of the major circle was 900 square cubits. The area of the minor circle was 400. The area of the ellipse was 600.
900 is 30^{2}; 400 is 20^{2}; 600 = 30 X 20.
The man had designed the ellipse to obtain such neat numbers. These numbers showed that my assumption of the course #35 horizontal delimiter line, and the #66 vertical delimiter line, were correct.
I had now accounted for all data points from Set #3 through Set #4. There were no "wasted" data points, except as they were used to draw my attention to the solution.
I redrew the line through #59 to #62. I could make it fit within reasonable use as another "pointer" to the center of the ellipse and circles. This fit was as good as my previous guess of the same slope as the common regression lines. The minor circle passed through course #49. The major circle passed through course #63.
The increase of the square values from 36 to 900, and so on, was due to the fact that the scale multiplier was fivetoone. This number squared is 25. The multiplier 25 times the small scale area of 36 = 900, and so on.
The designer worked with areas, not radii. Would not attention have been drawn to the design if he had used simple radii numbers? Why would he make this choice?
He may have felt that our attempt to understand his methods would be elicited more with areas than radii. We would naturally ask why such odd radii, just as I did. This forced me to consider areas. When I had roughly calculated the areas I did not get exact neat values, but rounded off to what was an obvious design goal. I could check my error by reverse calculation: take the exact areas and divide by the square of the radii of the circles. 36/(3.385)^{2} = 36/11.458 = 3.1418. 16/(2.257)^{2} = 16/5.0941 = 3.1409. 24/(3.385 X 2.257) = 24/7.640 = 3.1414. How close I came to the value of Pi depended on the how close I had calculated the radii of the circles.
Given all this possible error, we can see how he was showing us that he knew the theoretical value for Pi.
If he had used neat radii numbers he would not have forced our attention to the ratios.
What led to his choice of value to display this ellipse? If he had the ratio of the major and minor circles too close to one another (such as 10/9) the stone courses would have appeared as a circle. We would not have detected an ellipse. Similarly, of they had been too far apart (10/1) the slope of the stones layer thicknesses on the graph would have displayed as nearly a vertical line. In order to attract our attention he had to stay within the limits of a reasonable display.
Note the ratio of the radii of the major and minor circles. For my scale drawing this was 3.385/2.257 = 3/2. For the stone design this was the same ratio since both numbers were multiplied by five. This ratio was chosen to make it easy to discern the elliptical plot. Other ratios could have been chosen: 4/3, 5/3, and so on. 3/2 seems the neatest to discover the design. If the ratios had drifted too far from simple values recognition would have been more difficult, in each case modifying the stone course plot curvature.
The ratio of 3/2 could have been done with radii in simple integers. If the numbers 3 and 2 were chosen the area of the ellipse in my scale would have been Pi X 3 X 2 = 18.849. (Using the multiplier of 5 for actual cubit values we obtain Pi X 15 X 10 = 471.124.) But this would not have been a neat area number to attract our attention.
Many other area values could have been chosen. But most would not provide neat round numbers. For example, if the designer wished to preserve his major circle at 900 and used a ratio of 5/3 the minor circle would be 18^{2} = 324, with an ellipse area of 540. These are not neat round numbers. And so on.
His choice was acute in bringing our attention.
(I discovered another curious coincidental fact. The product of the multiplication of 11.285 X 16.925 = 190.999. Or, in other words, the product of these two numbers is very nearly 191. The theoretical numbers are 16.9256875 and 11.2837917, to the nearest seventh digit. This yields a product of 190.9859. 191 is a prime number.)
In order to display these areas a fundamental assumption is made that the vertical scale of the graph is equal in units to the horizontal. I would not find the areas of the circles and ellipse to the exceptional values I described except through such equating. I could arrive at my conclusions only if I turned the graph into similar units on both scales.
But I used the logarithmic scale dictated by Data Set #3. I laid out Data Set #3 for convenience on the vertical scale. This gave a slope comfortable to the eyes. If I had expanded the scale the slope would calculate the same mathematically, but would appear steeper. Similarly if I had reduced the scale.
Such difference would have translated to Data Set #4, with the ellipse stretched or reduced accordingly on the vertical scale. My choice was fortuitous to provide a convenient graphical view. We know from the data that the thicknesses varied from one cubit to two (see Figure Nine), providing a logarithmic scale with a "base" starting at 0.0, and extending to 0.7.
One vertical scale inch is equal to the difference between successive ln units of 0.1. This may be expressed as
(ln T_{1}  ln T_{2}) = 1.0 inch.
From the rules of logarithms (ln T_{1}  ln T_{2)} = ln (T_{1} / T_{2}). Simple computation shows that this is true. It is true for any vertical thickness value when translated to logarithmic values. Hence ln (T_{1} / T_{2}) = 1.0 inch. Thus the operation of equating the vertical scale to the same scale dimensions as the horizontal is legitimate, with the translation of 0.1 ln units equal to one inch. This, in turn, can be translated to the number of cubits by the simple multiplier of 5.0.
This equality would not have been true if a linear vertical scale had been used, simply because the designer performed his calculations in logarithmic values. If he had performed them in linear values he could have used a linear vertical scale. See following Data Set.
Still another question is his general knowledge of mathematics. Did he know conic sections?
We can see from the design choices that our ancient genius was intimately familiar with ellipse mathematics. He could not have made those choices unless he had a good understanding. Of course we cannot extrapolate this display into general knowledge of conics. See following discussions.
SET NUMBER FOUR  

Course Number 
Cubits  Natural Logs 

N. E. Level 
N.E. Thick 
S.W. Level 
S.W. Thick 
Ave Level 
N.E. Thick 
S.W. Thick 

66  102.36  1.1487  102.4089  1.1778  102.385  0.1386  0.1636 
65  101.212  1.26987  101.2311  1.2359  101.221  0.2389  0.2118 
64  99.9418  1.24079  99.9952  1.2650  99.969  0.2157  0.2351 
63  98.701  1.26018  98.7301  1.2699  98.716  0.2313  0.2389 
62  97.4409  1.18263  97.4603  1.2117  97.451  0.1677  0.1920 
61  96.2582  1.26987  96.2485  1.2796  96.253  0.2389  0.2465 
60  94.9884  1.36196  94.9690  1.3426  94.979  0.3089  0.2946 
59  93.6264  1.39104  93.6264  1.4347  93.626  0.3301  0.3609 
58  92.2354  1.3862  92.1917  1.2408  92.214  0.3266  0.2157 
57  90.8492  1.09054  90.9510  1.2117  90.900  0.0867  0.1920 
56  89.7586  1.20202  89.7392  1.2020  89.749  0.1840  0.1840 
55  88.5566  1.22625  88.5372  1.2166  88.547  0.2040  0.1960 
54  87.3304  1.24564  87.3207  1.2263  87.326  0.2196  0.2040 
53  86.0847  1.24079  86.0944  1.2408  86.090  0.2157  0.2157 
52  84.8439  1.27472  84.8536  1.2844  84.849  0.2427  0.2503 
51  83.5692  1.28926  83.5692  1.2844  83.569  0.2541  0.2503 
50  82.28  1.35712  82.2848  1.3620  82.282  0.3054  0.3089 
49  80.9228  1.51221  80.9228  1.5219  80.923  0.4136  0.4200 
48  79.4106  1.72063  79.4009  1.7012  79.406  0.5427  0.5314 
47  77.69  1.72548  77.6997  1.6915  77.695  0.5455  0.5256 
46  75.9645  1.32319  76.0081  1.3668  75.986  0.2800  0.3125 
45  74.6413  1.83695  74.6413  1.8466  74.641  0.6081  0.6134 
44  72.8044  1.99205  72.7947  1.9921  72.800  0.6892  0.6892 
43  70.812  1.618  70.8030  1.5560  70.8100  0.4787  0.4420 