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THE GREAT PYRAMID STONE COURSES

Part One

Introductory Remarks

When Flinders Petrie performed his superb survey of the Giza plateau in two seasons from December, 1880 to May, 1881, and from October, 1881 to April, 1882, he included exquisite measurements of the stone courses of Giza 1. As he stated in The Pyramids and Temple of Gizeh, page 41:

23. For ascertaining the height of the Pyramid, we have accurate levels of the courses up the N.E. and S.W. corners; and also hand measurements up all four corners. The levels were all read to 1/100 inch, to avoid cumulative errors; but in stating them in Pl. viii., I have not entered more than tenths of an inch, having due regard to the irregularities of the surfaces. The discrepancy of .2 inch in the chain of levels (carried from the N.E. to S.E., to S.W., on the ground, thence to the top, across top, and down to N.E. again), I have put all together at the junction of leveling at the 2nd course of the S.W., as I considered that the least certain point. It may very likely, however, be distributed throughout the whole chain, as it only amounts to 1.8" on the whole run.

Petrie’s Plate VIII was drawn to scale, with 1/ 200 for the horizontal and 1/10 for the vertical. Each course was drawn as a bar with a width corresponding to the thickness. Then each of the hand measurements were shown as lines at the top of the bars. Where the thickness varied considerably at the four corners these lines had corresponding separation according to the individual heights. Where the differences were small these lines crowded one another, sometimes not distinguished on his drawing. Because of the scaling one cannot determine the heights from the graph with precision. Since the graphical methods for showing the heights are subject to interpretation by conjectural reading, and hence not accurate, I do not include them in the following analysis. I use only the precise levels Petrie listed for the NE and SW corners.

In addition to his book, publication of Petrie’s numbers, and interpretation of the graphical values, may be found at

http://members.optushome.com.au/fmetrol/petrie/index.htm

where Graham Oaten provides the complete text of the book. This site is an invaluable resource for those who wish to directly obtain Petrie’s information.

Petrie’s list of levels was also published by D. Davidson and H. Aldersmith, The Great Pyramid: Its Divine Message, William Rider and Sons, London, 1924.

His measurements of the levels are shown in Table One, below.

Since early historical times the differences in thickness of the stone courses of Giza 1 have been a puzzle. Many proposals have been offered through the millennia to explain the differences, none adequate to the data.

Two major problems confront us:

a) design purpose, and

b) logistical control.

Why did the designer include differences among the course thicknesses?

How did the builders keep order among the stones of different dimensions?

How much easier it would have been if every stone had been the same size.

Petrie was acutely aware of the design problem. In a suggestion from J. Tarrell, published by Petrie, they proposed that the purpose of the different course thicknesses was to illustrate different values for ancient measures, especially different cubit values. However, this suggestion seemed immature to such a vast enterprise. Why would the stone courses be used to show such information, when it could have been shown far easier in the dimensions of cubit rods and architectural design? The proposal seemed grossly inadequate to what seemed to be complex data. It demonstrates how wildly surveyors and students have reached to understand the puzzle.

I shall now provide the explanation. The stone courses were part of a display of mathematical knowledge built into the pyramids, but never before recognized. Persons uneducated to the necessary technical level sought mythical, mystical, superstitious, and imaginary answers, often wilder in their speculation than Petrie’s attempt at a technical answer. Those who are educated to the degree demanded by the data do not accept that the ancient Egyptians could have had such knowledge. Therefore, they never bothered honest or solid examination. The general attitude of modern conservative and traditional minds is one of cult expression on all aspects of the pyramids.

Quarry and Placement of the Stones

Petrie attempted to explain the severe problem of sorting and arranging the stones to maintain constant course thickness at each height, with different thicknesses laid one upon another. As he stated:

It would require far longer labour to cut out and trim the stones to gauge, than to transport and fix them. Hence the cutting must have been started when the preliminary work was laid out, and must have continued during the ten years of preparations, before the twenty years of actual construction, according to the very likely statement at the time of Herodotus. There must have been, therefore, about a third of the whole material, or nearly a million blocks, lying out at the quarries, covering hundreds of acres, before any building began. This would give opportunity to sort over the thicknesses, so as to avoid needless waste in trimming and then the blocks which were nearly of the same height would be fairly dressed to correspond for one course.

The impossible nature of Petrie’s proposal may be easily calculated. (I am surprised he did not make the calculations.) Useable area on the Giza plateau is less than one square mile, perhaps 500 acres. Refer to the study by Mark Lehner, The Development of the Giza Necropolis: The Khufu Project, Mitteilungen des Deutschen Archäologischen Instituts Kairo, 1985.

While this space was available for Giza II, by the time of Giza 1 a significant portion of space had already been consumed in the Giza II pyramid and causeway constructions. Additional space was lost by the local quarries, ramps, and runways.

If Giza 1 contained four million stones, each with an average dimension of three feet, and the stones were laid out in double rows, with aisles between to move them past one another, more than two square miles would be required for the entire lot.

It would have been impossible to assemble such a vast array of stones in the available space.

Since this proposal invokes random cutting at the quarry, they would be laid out randomly on the plateau. This means that men would be walking up and down two square miles of terrain measuring to find the correct sizes for the next course.

Further, the great difference in thickness from course to course would suggest that there was no control at the quarry. Why would the stone cutters be permitted to so many different sizes of such great random variation?

Then the stones would have to be moved to the pyramid, a doubling of the work to move from the quarry to the site.

All in all, the scheme was a brave attempt to put some reason into the mechanisms of stone cutting and handling to maintain different course levels, but was really fantastic.

We then are left with two choices: The stones were cut to size at the quarry, or they were rough cut at the quarry and dressed to size on site. But we are still faced with the problem of the great variation among courses.

If the stones were cut randomly in the quarry, measuring from one to three cubits in thickness, and these transported to site, the site supervisor would have the task of sorting them. There would be random size stones, randomly arriving. He would need space to locate odd stones while he selected correct sizes for the course he was working. The unneeded stones would back up onto the plateau, and the scheme would collapse to a form of Petrie’s proposal. We can see the impossible task with such methods.

This logic shows that stones could not have been cut to random sizes under any scheme. The construction manager would be thrown back to the same problem.

Therefore, we are left with the fact that stones had to be cut close to desired thickness at the quarry, with the possibility of being fair dressed on site. The problem then reduces merely to the location of the fair dressing.

The other part of the problem is the sequence of course placement. Either one course was laid completely before another was started, or several near courses were in progress at the same.

The advantage to one course at a time is that all stones were cut to that size. There would be no confusion among work gangs as to which thickness to cut, and no problem in logistics to sort different thicknesses. Then the pyramid progressed course by course.

If they were laid in several layers simultaneously, with some areas of the pyramid receiving smaller or larger stone than others, how did the site supervisor control the flow of the stones to maintain order in the work? Again, this would have required advance planning, with markings of the blocks to indicate the course number, and a sufficient number of stones used in previous courses to provide a base for the layering of higher courses. This sequencing would have to be transferred back to the quarry master and his supervision of dozens of crews to ensure that the proper sequence was maintained. Clearly this scheme would have created a maze of logistics, merely to cut and move in the proper sequence.

Of course, when one layer was substantially along, it would have been possible to begin the next, but this had to be carefully scheduled and controlled.

We should remember that the courses were held to within tenths of inches. As I shall show in the analysis below, the courses were actually held in thickness to within +/- 0.02 cubits standard deviation, or less than one-half inch. While fair dressing could have been done on site, it again would have been far easier to cut to the desired thickness, with desired control, at the quarry — if the quarry personnel knew beforehand what thickness was requested by the site supervisor.

All in all, there seems to be no advantage to laying more than one course at a time. If the multiple-course scheme had been used a problem in logistics would be created of which stones would move to which locations around the site. There would be no greater work for the more simple scheme, and the logistics problems would be limited to quarry rates and rate of movement to the site.

The Layering Design

Previous analytical attempts to study the course variations were inadequate to informative graphical display, and failed to comprehend the high mathematical nature of the construction.

Figure One includes three different graphs. The first is the stone course thickness plotted against the pyramid height. The second shows the difference in course thickness from the SW to the NE corners. The third shows the difference in course level from SW to NE.

Note that I did not attempt to hold to course thickness along the horizontal axis, as did Petrie in his Plate VIII. I merely plotted the data as equidistant points. I viewed the data the same way I would in any other graphical plot.

We are immediately struck by the fact that the course thickness sequences are not random, but follow mathematical forms. I had seen similar displays on electronic oscilloscopes in my engineering experience. The individuals groups imitate electronic charge decay, triggered by random noise bursts, finally fading to a quiescent state.

I personally do not believe any human being on this planet could have devised a scheme with heavy stones that would so closely simulate the behavior of a modern electronic circuit.

It is simply incredible.

Figures One B and One C again illustrate displays observed on modern oscilloscopes, here representing noise bursts in electrical circuits.

Indeed, these are amazing, tantalizing, and intriguing displays.

In all calculations of stone course thicknesses and levels (heights) I use the average of the SW and NE corners, unless I indicate otherwise.

I make the following general observations:

Course Thickness versus Height

  1. A line drawn from ground level with a thickness intercept of 2.8 on the Y (thickness) axis, sloping down to one cubit thickness at 280 cubits height, just touches the first course thickness starting the first decay curve, and subsequent similar points at courses #35, #98 and #99, #118, #144, #180, and #196. A rigorous calculation of the regression line for those points yields a Y intercept at 2.81. The X intercept at 280 cubits is 0.998. This shows that the original height of the Great Pyramid was intended to be 280 cubits, with ideal intercepts at 2.80 for the Y (thickness) axis and 1.00 cubits for the X (height) axis. Note that the Y thickness intercept is exactly 1/100 of the X height intercept.
  2. Except for quarry control of cutting, all stone thicknesses lie within the band defined by this sloping delimiter line and one cubit.
  3. Generally the decay curves all seem to tail off to approximately one cubit thickness. However, some of the curves at the lower part of the pyramid seem to decay to a level somewhat above this value, while the curves near the top of the pyramid seem to rest on or near that value.
  4. At casual glance some of the individual curves seem to be linear, while some seem to be exponential. Differences among the slopes of regression lines drawn through the individual curves suggest different purposes in the display.
  5. Other scatter in the points is evident but without obvious reason.

Differences in Course Thickness

  1. The "noise" scatter in the plot of the difference in course thickness from SW to NE corners shown in Figure One B is also highly intriguing. From a "wild" burst, or loose control, over the height of the first decay group, the differences were closely controlled through the next two decay groups, to then settled to some general dispersion. Clearly, the construction engineers could tightly control the thickness of the courses if they so desired, but decided to not hold them so tight for most of the pyramid height.
  2. The difference in control between the first group and the next two groups shows that the builders intentionally created the dispersion difference. They were attempting to communicate further knowledge through this method. Today we would call that difference a result of production control, with evaluation of the production through Statistical Quality Control techniques. More on this later.
  3. More exact display of the data between the SW and NE corners shows that the difference were again not accidental, but intentional. While data for the NW and SE corners, or along the entire length of the courses, are missing, we can see how informative merely these two points become.
  4. Sequences occur in which the values see-saw back and forth, illustrated by the courses around 100 cubits in height, (Set #4), and then again at 160 cubits height (Set #6). There is a pattern also above 200 cubits height. These are not random variations; as I shall show, they were intentional.
  5. Similar intentional large reversal of thickness can be seen several places. The beginning of Group #4 at Courses #43 and #44, the middle of Group #4 at #57 and #58, at the beginning of Group #7, courses #97 and #98, and at the end of Group #9, courses #144 and #145, show this intent.
  6. These intentional reversals were not done to bring the corners back to equal levels. Rather they occur at the beginning of new (but not all) decay curves and serve as markers of those decay groups. This implies that the designer/builder was using methods to attract our attention.

Differences in Course Level

  1. For someone acquainted with mathematical analysis, the graph of the difference in course levels of Figure One C suggests almost a mathematical integration of the alternations found in Figure One B. The two are mathematically related because a change in course thickness inherently demands a change in course level. We can see this at the large alternations.
  2. The plot at the upper part of the pyramid after course 101 shows a consistent "drift" toward one direction with the SW thicker than the NE Again, this must be intentional, not accidental, and suggests that the designer/builder is drawing our attention to other display.

I shall now go on to discussion of the individual features of the course thicknesses.

Table One

 

 

Figure One Enlarged

 

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