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Egypt
Origins

The Great Pyramid Stone Courses

Part Four

Statistical Quality Control

I learned the concepts and principles of Statistical Quality Control while I was employed by Bell Telephone Laboratories at the Western Electric Plant in Allentown, Pennsylvania from 1955 to 1962.

Consider a bolt in a rifle. If the bolt is too large in diameter it will not fit into the chamber. If it is too small is will not properly secure the charge, and will permit fly-back of gases that may harm the rifleman.

Hence, control of the diameter of the bolt must fit within certain values. But equipment to machine the bolt will produce variations. The idea behind quality control is too ensure that the bolt diameter will always fall within safe defined limits. Production engineers will measure a sample of bolts coming from the machinery to determine if the bolts fall within those prescribed limits. They will not measure all bolts, but will use statistical mathematics to estimate how many bolts may not qualify. Under a well-defined and properly controlled production run this estimate will permit escape of a very small number not meeting the requirement.

Rifle bolts are cut by machinery set to hold certain tolerances. The individual pieces are produced over and over again within those limits.

But the stone blocks of the pyramids were not subject to machined production. They were all hand cut. If the pyramid designer/builder were to hold his stone blocks to proper thickness to permit the mathematical display we have shown, he had to have control over the cutting and trimming.

I indicated earlier that Course numbers 29 through 31, and 32 through 34, were held to +/- 6 and +/- 4 millimeters respectively in thickness. I also indicated that it appeared the builder intentionally introduced variations to create not only a logarithmic regression line, but also to display three-sigma band limits. This last criteria would increase the care of stone cutting to a tighter tolerance necessary than what might be demanded merely to display a regression line.

On Figure One I show the variation in course thickness and level for all courses in Giza I. We can see that the first three decay curves have an abrupt increase in the tightness of production control from Data Set #1 to Sets #2 and #3. I show this in enlarged view in Figure Five. I calculated the standard deviation for each group. On the course thickness this improves by an order of 5:1 between the first data set and the next two, while on the course level this improves by an order of 10:1. The abrupt change shows that this improvement was not due merely to a learning process but by conscious intent. This graphical display clearly was intended to demonstrate the ability by the builder to hold the stone thickness and level to desired control. (For the moment I do not address the interaction between course thickness, or variation of the courses, and the levels.)

The standard deviation I show includes 95% of the courses falling within that value. For a course thickness of one cubit a control of +/- 0.023 cubits would mean approximately +/- 0.5 inches. The first data set are held to only +/- 2.6 inches. As I discussed earlier this wider spread was intentional to display the wider three-sigma bands shown in Figure Two. Again we should not conclude that this was a learning process.

If we knew the expected thickness for each course we could measure how well the stone cutting was held. In order to evaluate this ability I examined the regression line of Figure Three. If the builder intended for the last six data points to demonstrate his three-sigma band perhaps we could use those points to reach such expected value. I noticed that the three last data points were scattered around 0.24 ln values. The prior three data points were scattered around 0.30 ln values. I assumed those as the expected production goals and considered how well they were held, more rigorously than my previous estimates of +/- 6 and 4 millimeters. I plotted the differences from the expected values on Figure Six. We can see that my estimated nominal values must be close to the design intent of the builder, since the points seem to scatter randomly around them. (The second point for each course number is the SW value.) We can see that seven points fall well within +/- 3 millimeters, while the other five points fall within +/- 7 millimeters. If we had measurements from many points on the respective courses we would be able to build a more rigorous statistical distribution for each. This would show more firmly the expected design goal, and the true nature of the production control.

As I further examined Figure Three I saw that each of the courses seemed to fall on expected values that were within two decimal digits of the logarithmic values. I show these in the following Table.

Set #2 Calculations
Cubits Natural
Logs
Course
Number
N. E.
Level
N.E.
Thick
S.W.
Level
S.W.
Thick
Ave
Level
N.E.
Thick
S. W.
Thick
Nom.
Thick
NE
Diff.
from
Nom.
SW
Diff.
from
Nom.
34 55.1377 1.27 55.1425 1.275 55.140 0.2389 0.2427 0.24 0.0011 -0.0027
33 53.8678 1.275 53.8678 1.270 53.868 0.2427 0.2389 0.24 -0.0027 0.0011
32 52.5931 1.275 52.5979 1.260 52.595 0.2427 0.2313 0.24 -0.0027 0.0087
31 51.3183 1.347 51.3377 1.372 51.328 0.2982 0.3160 0.30 0.0018 -0.0160
30 49.9709 1.367 49.9661 1.347 49.969 0.3125 0.2982 0.30 -0.0125 0.0018
29 48.6041 1.362 48.6187 1.357 48.611 0.3089 0.3054 0.30 -0.0089 -0.0054
28 47.2421 1.415 47.2615 1.415 47.252 0.3473 0.3473 0.35 0.0027 0.0027
27 45.8269 1.44 45.8463 1.435 45.837 0.3643 0.3609 0.36 -0.0043 -0.0009
26 44.3874 1.493 44.4116 1.512 44.399 0.4007 0.4136 0.40 -0.0007 -0.0136
25 42.8945 1.566 42.8994 1.570 42.897 0.4482 0.4513 0.45 0.0018 -0.0013
24 41.329 1.57 41.3290 1.566 41.329 0.4513 0.4482 0.45 -0.0013 0.0018
23 39.7586 1.599 39.7635 1.604 39.761 0.4697 0.4727 0.47 0.0003 -0.0027
22 38.1592 1.672 38.1592 1.701 38.159 0.5141 0.5314 0.52 0.0059 -0.0114
19 34.1945 1.847 34.1993 1.895 34.197 0.6134 0.6393 0.63 0.0166 -0.0093

The Nominal thickness is the design expected value. I then plotted the difference from the expected value for the NE and SW corners in Figure Seven. If the expected value was held at two decimal digits we should expect to see the expected thickness for each course to center around those values. Indeed, Figure Seven confirms my estimates. If I were off by one place in the second digit the graph would show that course displaced by 0.01 ln. The only candidates for such possible false estimate would be Course #26, and the Set at 0.03 ln. But these differences might only be random variation in thickness control.

Note how the NE and SW points for Course #19 seem to be balanced around the 0.63 ln value, within random control error, suggesting that the designer intentionally was bracketing the upper three-sigma band limit. Course #22 also seems to be balanced around the regression line. The overall placement of the points determined both the regression line, and the three-sigma spread.

Hence we can deduce that the designer used values with two decimal digits to determine how he would design the regression line. This principle should apply to the other regression lines.

For each course the height would increase by a proportionate amount. This means the designer had to calculate the respective height and thickness values to adjust his regression line. He could not randomly hope to achieve the design goals by luck; he had to know each course before he started construction. The entire pyramid was calculated, course by course, before the first course was laid.

Next I examined Set #3.

Set #3 Calculations
Course
No.
NE
Thick
SW
Thick
Mean
Thick
Thick
Diff.
Diff. Diff. Nom.
Thick
NE
Thick
SW
Thick
NE
Diff.
SW
Diff.
All values in cubits Inches Mill. LN Values
42 1.3570 1.4250 1.3910 -0.0680 -1.403 -35.62 0.33 0.3053 0.3542 0.0247 -0.0242
41 1.4980 1.4880 1.4930 0.0100 0.206 5.24 0.40 0.4041 0.3974 -0.0033 0.0034
40 1.5460 1.5410 1.5435 0.0050 0.103 2.62 0.43 0.4357 0.4324 -0.0016 0.0016
39 1.6430 1.6380 1.6405 0.0050 0.103 2.62 0.50 0.4965 0.4935 -0.0015 0.0015
38 1.7590 1.7590 1.7590 0.0000 0.000 0.00 0.56 0.5647 0.5647 0.0000 0.0000
37 1.8510 1.8420 1.8465 0.0090 0.186 4.71 0.61 0.6157 0.6109 -0.0024 0.0024
36 1.9920 2.0260 2.0090 -0.0340 -0.701 -17.81 0.70 0.6891 0.7061 0.0085 -0.0084

We can see from Figure Four just how tightly the builder held control for this data set. I show the spread of the two corners in Figure Eight. When I first plotted this graph I thought perhaps I had made a mistake in my logic. The point pairs all fell evenly on both sides of the expected values. When I reexamined the plot of Figure Four I saw that my expected values were valid. The builder had achieved such good control that he met his design criteria with extreme exactness. His control was as good as that of the Second Data Set, or perhaps better. However, the last course in the set shows extreme spread, and suggests another design reason. It terminates the set. See Figure One. In fact, the two course with the widest spread, #36 and #42, are both terminator courses.

You can see from the above Table the differences in millimeters for course thickness from corner to corner. The tolerance is tightest in the center of the Set, and more loose at the ends. Again we must comment on the extremely tight control of stone cutting and trimming.

Pair Level Compensations

Another important aspect of the design was the compensation of the level when the designer used one level to create large differences from the north to south sides. This can be seen in Figure Two where those points with large spread have an adjacent spread that brings back the levels, Courses #7 and #8, #12 and #13, and #20 and #21. This is also seen in #30 and #31, Figure Six. #42 and #43 are in adjacent sets, Set 3 and Set 4, but they also compensate for one another. I tabulate the differences here.

Course
Number
NE
Thickness
SW
Thickness
Difference
in Thickness
Dimensions in Cubits
2.00 2.38 2.54 -0.16
3.00 2.34 2.20 0.14
7.00 1.90 2.14 -0.24
8.00 1.83 1.69 0.14
12.00 1.44 1.65 -0.21
13.00 1.44 1.25 0.19
20.00 1.14 1.17 -0.03
21.00 1.15 1.09 0.06
30.00 1.37 1.35 0.02
31.00 1.35 1.37 -0.02
42.00 1.36 1.43 -0.07
43.00 1.61 1.56 0.05

The compensation may also be seen in Figure Eight where the wide separation of Course #36 is followed by opposite compensation in Courses #37 through #41.

Only higher on the pyramid were the levels moved toward a consistently greater height on the south, above Courses #130, but then by less than tenths of a cubit. See Figure One C. I shall explain; there was another purpose.