Copyright, Ernest P. Moyer, Author
Revised January, 2003
The Great Pyramid at Giza (Giza I) is a gem in pyramid building, and receives virtually all the attention of the world. But it was not built in isolation. It was part of a construction project that included three other large pyramids, and two minor ones. The next important member of the project is the Giza companion, Giza II. The remaining two are located at Dahshur, usually denominated the Flat and the Bent. Together, those four constitute 67% of the total Egyptian pyramid volume.
The Step Pyramid at Saqqara, and the Meidum pyramid appear to be evolutionary projects approaching the size of the large four, but all other pyramids are puny in comparison. As I shall show, the Meidum and the Menkaure pyramid at Giza served as the terminus a quo and the terminus ad quem respectively of the Great Construction Project. The last two add to the four large pyramids to make the 75% of all Egyptian pyramid volume, all within the Fourth Egyptian Dynasty.
Examination of the size and the slope of the exteriors of the four great pyramids show them related to one another mathematically.
I shall review the technical data of each in turn, in the sequence of the Flat, the Bent, and Giza II, while reserving most attention for Giza I. I shall then show how they are related.
My sources are:
Maragioglio and Rinaldi summarized the data collected by other previous workers. Some of their data was reported from sources not cited above. Also, their work was not always the most accurate. So we must be careful. Hereafter I refer to this work only as Maragioglio.
The North pyramid at Dahshur is the Flat; the South pyramid is the Bent. They are a companion pair, the same as the companion pair at Giza.
|Perring via Maragioglio #1||219.42 m||104.49 m||43o 36' 11"|
|418.74 cubits||199.41 cubits|
|Perring via Maragioglio #2||219.28 m||104.42 m||43o 36' 11"|
|418.47 cubits||199.27 cubits|
|Reisner via Maragioglio||218.5 m||104.4 m||43o 31' 11"|
|416.98 cubits||199.23 cubits|
|Calculated Mean||418.06||199.30||43o 38' 6"|
Mark Lehner rounded the height to 105 meters (200+ cubits), but he probably obtained his values from one of these sources. I do not give the values from Fakhry because he reported them merely as round numbers.
The three height measures suggest that the pyramid was intended to be 200 cubits, with a mean of 199.30. The difference is 0.15%.
The three base measures suggest that the pyramid was intended to be 420 cubits, with a mean of 418.06. The difference is 0.46%.
Petrie recognized this design. On page 162, in Chapter XVIII, entitled Architectural Ideas of the Pyramid Builders, he mentions that the angle of 43º 36' 11" from Perring fit an exact theoretical rise of 20 on 21. He then goes on to remark:
p 163. The angles determined by Perring for Colonel Vyse cannot be considered very satisfactory for comparison with theories, as they seem in one case to be distinctly in error (in the Second Pyramid angle) ; and some of the observations are so extremely near to theoretical angles, that they seem to have been modified by the observer. But taking those Pyramids of which I measured the angles repeatedly in many ways, the variations from the slopes which would result from integral amounts, is usually about half the probable error; and the variation only equals the probable error in the Third Pyramid, which is least accurately built. From these close coincidences it seems clear that the rule for slopes in designing, was to set back the face an integral number of cubits, on a height of an integral number. The use of angles of 4 on 3 (which has hypothenuse 5), and 20 on 21 (which has hypothenuse 29), seems to suggest that the square of the hypothenuse being equal to the squares of the two sides may have been known; particularly as we shall see that the use of squared quantities is strongly indicated in the Great Pyramid.
We must agree. Perring's values are much too close to the theoretical. They probably were adjusted to meet that ideal. Nevertheless, the difference of -5' angle from Reisner is a mere 0.2% from the ideal.
The measured values could be evaluated various ways. For example, Riesner's numbers of 199.23 height and 208.49 half-base length gives an angle of 43º 41' 56", 5' higher than the ideal, rather than the 5' lower he reports. Apparently he measured the slope, the base, and the height all independently of one another.
The following Table shows the results from various combination of the numbers.
I use the mean of the height (199.30) and base length (418.06) from the three reports above.
H = 199.30
B = 418.06
H = 200
B = 420
|1||20/21||43o 36' 10"||0.9524||
|43o 37' 30"||0.9531||
|43o 41' 56"||0.9556||
|4||Reisner Measured||43o 31' 11"||0.9496||
The Hypothetical from Measured Means columns contain the base and height calculations from the respective means of the height and base. The Hypothetical from Ideal columns show the same assuming the ideal height and base dimensions.
The angle calculated from the mean values of the height and base measurements is only 1' 20" higher than the ideal.
The midpoint between the angle calculated from Reisner's base and height dimension, and his measured slope is equal to 43o 36' 33". This is only 20" from the ideal.
Undoubtedly, the builders attempted a pyramid with the 20-21-29 proportions, within their ability to control, and our ability to measure. In fact, our measurements today are not equal to their design ratio. Unless we make better measurements, with great care, we will not be able to actually "prove" the theoretical model.
From this evidence we can conclude that:
The height of the pyramid was intended to be 200 cubits.
The base of the pyramid was intended to be 420 cubits.
The Flat pyramid at Dahshur was designed as a 20-21-29 integral Pythagorean triangle.
This pyramid is composed of a frustum, topped by a true pyramid shape.
Maragioglio reporting Petrie gave the following values:
|Lower Slope||54o 31' 13"|
|Upper Slope||43o 21'|
Again, I suspect the report from Fakhry. His values are all identical to Petrie except the upper height. His values for that portion are exactly 4.0 meters less than Petrie's, again raising suspicion about his integrity. Unfortunately, we do not have comparison measurements from other investigators. Hence we cannot calculate mean values, but must base our judgment strictly on Petrie's work.
Maragioglio personally observed great cracking of the stones comprising the structure. There was settling, not due so much to earthquakes, but to compression of the ground. Therefore, he commented that dimensions and slopes may not exactly reflect the original construction. However, as we shall see, this does not seem to seriously affect our interpretation of the structure.
First observation shows that the height of the two Dahshur pyramids are nearly equal, 199.30 for the mean of the Flat, and 200.51 from Petrie for the Bent. This difference is 1.2 cubits, about two feet out of 344, or 0.6%.
Petrie's base length measure differs from an ideal of 360 cubits by 0.08, or 0.1%.
Remember, Petrie was an objective observer who quelled earlier mystical notions about the Great Pyramid dimensions recounting human history. His comment about Perring shows his desire to obtain reliable numbers. Therefore, he did not adjust his measurements to some ideal.
The base length seems related to the Flat in a neat ratio of 420/360.
In order to aid analysis I assumed the two were designed to be identical in height with the bases related by some easy mathematical ratio. I determined that I could more easily compare the two if I divided one half of the base of the Flat pyramid into 63 parts. 210/63 makes each part 3 1/3 cubits.
Maintaining the ratio of 20/21 the height of the Flat would be divided into 60 parts. 200/60 again gives 3 1/3 cubits per part.
This permitted division of the Bent half-base length into comparable units of 54 parts. 180/54 also makes each part 3 1/3 cubits.
Then the task was to examine how each aspect of the structures related between the Flat and the Bent.
In order to show the design parameters I sketched an outline of the Bent pyramid compared against the Flat. The Figure shows the design ratios of the various segments, based on the analysis divisions I propose.
From this analysis I discovered that the position of the bend fits within a simple geometric framework.
The ideal height of the bend would be 28/60 X 200 = 93 1/3 cubits. Petrie's measured value is 93.64. The difference is 0.31 cubits, 6.4 inches, 0.3% from the proposed design model.
The horizontal location of the bend from the theoretical model would ideally be 34/63 X 210 = 113 1/3 cubits from the center of the pyramid. We do not have a direct horizontal measure; we must depend upon calculation from the height and slope. If we take Petrie's height of 93.64, and his angle of 54o 31' 13" the slope would be 1.403. (This is an ideal of 7/5.) The calculated horizontal distance from the point of the bend to the outer edge of the pyramid is 66.57 cubits. This compares to 66 2/3 cubits from the proposed model. The difference is 0.097 cubits, 2.0 inches, or 0.14%.
The slope of the bottom portion according to the model is 93 1/3 divided by 66 2/3 = 7/5 = 1.4 for an angle of 54.4623 degrees, or 54o 27' 44". Petrie measured 54o 31' 13", a difference of 3' 29" or 0.1%.
The model slope of the upper section is 106 2/3 divided by 113 1/3 = 0.9412. This is a slope of 16/17 and provides an angle of 43o 15' 51". This compares to the ideal 20-21-29 angle of 43o 36' 10". Petrie's measured angle of 43o 21' falls between the theoretical value for the Flat pyramid and the proposed model for the Bent.
The height of the top portion from the model would be (32/60 X 200) = 106 2/3 cubits. The measured value is 106.87 cubits. The difference is 0.27 cubits, 5.5 inches, 0.2%.
We can see how well the proposed model fits from the proximity of Petrie's measurements to the ideal, within a few inches and minutes of angle of one another. This strong agreement supports a conclusion that the proposed model was the basis of the pyramid design.
Petrie said nothing about this design model. He proposed none other, as far as I can determine. Therefore, the proximity of his measurements to the model make the evidence more graphic.
Therefore, we can conclude that:
The height of the Bent was designed the same as the Flat at 200 cubits.
The base of the Bent was designed at 360 cubits, in proportion to the Flat.
We can see from the diagram that the bent portion protrudes slightly above the slope of the Flat pyramid, if we assume these design criteria. The calculated amount is about 1.6 cubits, 33 inches, or about one yard.
Ideally, we would desire the bend of the Bent pyramid to fall directly on that slope line. The interested reader can perform the calculations to determine how much the base and height values should change to achieve that result. Of course, such exercise will force a more complex solution to the design.
Within design criteria the Bent pyramid outline was intended to fall within the outline of the Flat.
The Flat and the Bent were designed as companion structures.
From the evidence thus far presented the design criteria show that the pyramid builders used simple arithmetic ratios to achieve their measures.
From Petrie's measurements:
|Slope = 53o 10' +/- 4' as the best statement.|
The crude state of the exterior prevented more precise measurements.
The ideal slope of a 3-4-5 Pythagorean triangle is 53o 7' 48". Petrie's measure differs from this by +2' 12", or 0.06%.
Petrie calculated the height at 5,664 +/- 13 inches = 274.55 +/- 0.63 cubits, for an ideal of 275 cubits. From Petrie's angle and base numbers I calculate 274.21 cubits. The difference from the ideal is 0.16% and 0.29% respectively.
Within measurement error:
Giza II was designed as a 3-4-5 integral Pythagorean triangle.
Petrie recognized this design criteria. He listed this pyramid in his Table on page 162 as having a slope of 4-on-3, with a difference of 2' +/- 4' error. Refer to his remarks above.
Note that the measured mean base length of Giza II of 410.80 cubits is less than the measured Flat base length of 419.63 cubits by about nine cubits. I shall return to further discussion.
Later I shall discuss insights offered if we take the ideal height at 274 cubits. This value falls within the range of Petrie's error.
From Petrie's measurements:
|Slope = 51o 50' 40" "weighted mean up north face"|
Petrie noted that "the South face should not be included with the North, in taking the mean, as we have no guarantee that the Pyramid was equiangular, and vertical in its axis."
Petrie gives calculated height at 5,776 +/- 6 inches. This would be 279.98 +/- 0.29 cubits, for an ideal of 280 cubits.
From Coles measurements:
|North||(9065.1) 230.253||439.41||-2' 28"|
|East||(9068.5) 230.341||439.58||-5' 30"|
|South||(9073.0) 230.454||439.80||-1' 57"|
|West||(9069.2) 230.357||439.61||-2' 30"|
|Mean||(9068.9) 230.351||439.60||-3' 06"|
Thus we can see that Cole differed from Petrie on the mean around the four sides by no more than +0.01 cubits. Their differences overall appear to be no more than error of measurement. However, see analysis from Geographical Orientations in a companion paper.
The difficulty in determining the design criteria for the Great Pyramid is its proximity to three different but unique geometric solutions. The pyramid literature is rife with debates over the correct solution.
The height of the pyramid represents the radius of a circle, while the base perimeter represents the circumference of the same circle.
This is expressed mathematically as:
2 Pi X H = 4 S
where S is one side.
The square of the height is equal to the face area of one side.
This is the famous Golden Ratio solution, or Golden Section, also known as the Divine Ratio.
This ratio was used as a design guide in many ancient structures, has been discussed extensively in mathematical and architectural literature, and is well known in the field of art.
It is expressed mathematically as:
Equation 1: a/b = b/(a + b)
The ratio is mathematically unique. Numerical solutions from the above equation are 0.618034 and 1.618034.
For the pyramid design it is expressed as:
Equation 2: H2 = B X [H2 + B2]1/2
where B represents one-half the base length.
The face area of one side is one-half the base length times the apothem slope distance. The apothem slope distance is the square root of (the sum of one half the base squared plus the height squared), according to the Pythagorean theorem.
Equation 2 has the same mathematical solution as Equation 1. It is expressed as:
Equation 3: H2 /B2 = 0.618034 and 1.618034.
The area of a circle is to the square of its diameter as 11 to 14.
This ratio was given by Archimedes (c. 287-212 BC) in his treatise, Measurements of a Circle, Proposition 2.
It is expressed as
Pi R2 /(2R)2 = 11/14
This is the famous Pi approximation of 3 1/7 = 22/7. (When multiplied out Pi = 44/14 = 22/7.)
The actual ratios are:
11/14 = 0.7857 (Inverse = 1.2728)
Pi R2 /(2R)2 = Pi/4 = 0.7854 (Inverse = 1.2732)
From these calculations one can readily see the small difference between the two ratios.
We can now compare the pyramid dimensions subject to these three different criteria. Petrie's calculated value of the height was 280 cubits within error. To make the analysis easier I shall assume that value in all of the following. I first offer the simple solutions; later I discuss the interaction among the measurements.
From Proposal #1:
If the radius of the hypothetical circle is the height at 280 cubits, then the circumference is 2 Pi X H = 1,759.29 cubits.
The sum of the Petrie values is 1758.37. This differs from the theoretical model by 0.92 cubits, or 0.05%.
Certainly this hypothesis seems correct, within measurement error, with difference between the theoretical and the actual even better than the other pyramids discussed above.
Hence, we could conclude that the pyramid was built according to this design criteria.
Author: Ernest Moyer
Date: Aug-16-02 04:23
(Note: These comments were in response to an article posted on the Hall of Maat website by George M. Hollenback.)
Although you may fault Lumpkin for false reasoning the fact of Petrie's measurements still stand.
His values for the true base have a mean measure of 9068.8 +/_ 0.5 inches See page 39 of The Pyramids and Temples of Gizeh.
From the mean of his slope measurements he calculated the height at 5776.0 +/- 7.0 inches. See page 43 of The Pyramids and Temples.
Half the perimeter is double the mean side measure. 18,137.6 divided by his calculated height yields a ratio of 3.1402.
The extremes from Petrie would yield 3.1439 and 3.1364.
(I round off to four figures because of the uncertainties.)
From Proposal #2:
Solutions based on the Golden Ratio are found as follows:
From the measurements of Petrie and Cole, H = 280, B = 219.8. The square of these values in ratio are 2802 / 219.82 to give 1.6228 and the inverse of 0.6162.
These values calculated from measurements are in error by 0.3% from the mathematical ideal.
An ideal half-base length of 220 cubits would give 0.6173. This is in error by 0.17%.
We can calculate the error from the ideal Golden Ratio another way. The base length necessary to give an ideal of 1.618034 is 2802 / 220.122. The measured base length is different by 0.14%.
Based on this amount of error we could conclude that the pyramid is designed according to the Golden Ratio.
From Proposal #3:
The slope of 11/14 is 0.7857. The inverse slope of 14/11 is 1.2727.
The slope from Petrie's calculated height and the base measurements of both he and Cole is 280/219.8 = 1.2739.
The error from the ideal of 1.2727 is 0.09%.
Hence we could conclude that the simple Archimedes ratio of 11/14 was the criteria for the construction of the Great Pyramid. In fact, this criteria presents the least error of the three proposals from the ideal model.
This proposal would agree with our earlier assessment that the builders used simple arithmetic ratios for their design.
The uncomfortable and uncanny aspect of the three design criteria is that the solutions all fall within construction tolerances, or most likely, our measurement errors, of the pyramid. Even more, the errors on this Great Pyramid for all three solutions are less than the errors on the proposed designs of the other three great structures.
Someone certainly knew his mathematics.
Consider the slopes and angles in comparison:
|Pi||1.27325||51o 51' 15"|
|Golden Ratio||1.27203||51o 49' 39"|
|11/14||1.27273||51o 50' 34"|
|Petrie's Measured||1.27280||51o 50' 40"|
(Petrie's slope calculated from his base measured values and 280 cubits height is 1.2739.)
Hence Petrie's weighted mean falls in the mid-range of the theoretical ideals and, within calculation error, exactly on the 11/14 ratio.
Petrie noted differences in slope among the four sides of the pyramid. Refer to his remark above about the slope of the south face. A concavity also exists on each of the sides, and makes more exact determination of construction intent highly difficult, even if the pyramid were in perfectly preserved condition.
Hence we shall never be able to conclude from the present state of the structure, one way or the other, which of these three criteria were used to design the Great Pyramid.
The 11/14 ratio seems most likely, and would follow the design of the other Great Pyramids, based strictly on slopes.
Other exterior design elements tie these Great Pyramids to one another. We can see that they form a geometric set in that each is uniquely different in its display of elementary mathematics. Two are based on simple Pythagorean ratios: 3-4-5 and 20-21-29, one at Dahshur and one at Giza. For the pair companions one illustrates a frustum, and one illustrates mathematical solutions that could be Pi, the Golden Ratio, or Archimedes Ratio. One is at Dahshur, one at Giza.
Clearly, all four had to be planned before any one was built. It stretches human credulity to propose that they were built according to random whim by the Pharaohs.
All four integral heights seem to be the starting point for calculating the base lengths. These are 200 cubits for the two pyramids at Dahshur, 275 cubits for Giza II, and 280 cubits for Giza I.
If Giza II were increased in height to that of its companion, 280 cubits from 275, as we see for the two pyramids at Dahshur, the length of the base would increase from 410.8 to 421.2 cubits. Then the base lengths of the Flat at 418.5 cubits, Giza I at 419.8, and Giza II at 421.2 would all be in close proximity to one another.
If we were to sketch the outlines of the four pyramids to scale for this idealized scheme we would discover that the two Giza pyramids would touch at the top, the two Dahshur pyramids would touch at the top, the Bent falls within the outline of the Flat, and the Flat and Giza II very nearly coincide at the base, at 2.7 cubits difference.
The following table summarizes these design differences. Where possible I take the mean of the reported values.
|Flat||Bent||Giza II||Giza I|
One can immediately recognize how the Comparison Ideal provides nice round numbers for all four designs.
Refer to the two sketches, one showing the designs as they were built, and the other according to this idealized scheme.
Given the refinement of the great pyramid designs, and their clear connection to one another, why did the designer not make the Giza II height equal to Giza I to provide the nice round numbers?
At this point we should recognize the finesse by which he created his designs.
Earlier I mentioned the possibility of Giza II design based on a height of 274 cubits rather than 275. Exploration of this possibility offers other insight.
Consider suggested ideal cubit dimensions for Giza II of 205.5, 274, and 342.5 for one-half the base length, the height, and the slope distance. These are in easy ratio of 3/2 X 137, 4/2 X 137, and 5/2 X 137. 137 is a prime number. Prime numbers are used elsewhere in pyramid construction. Thus it is possible that the designer chose this display of mathematical knowledge, rather than simply equating the two Giza pyramid heights.
(The earlier proposed number are 206.25, 275, and 343.75 respectively.)
Note that Giza II with a height of 280 cubits would have had multiples of 70 to obtain 3, 4, and 5 X 70, instead of multiples of a prime number.
Such simple design ratios are in the other Great Pyramids.
The Flat uses multiple of 10 to obtain the cubit measures: 20 X 10 for the height, 21 X 10 for the base, and 29 X 10 for the apothem.
If the Great Pyramid was designed according to the Archimedes ratio it uses multiples of 20 to obtain the cubit measures: 14 X 20 for the height, and 11 X 20 for the base. (The ratio of 14/11 does not provide an easy slope distance for the apothem.)
Such examples of simple round-number multipliers for pyramids are found in the Rhind Mathematical Papyrus. See detailed analysis by Marshall Clagett, Volume Three, Ancient Egyptian Mathematics, American Philosophical Society, Philadelphia, 1999.
The Bent pyramid uses multiples of 40/3 and 20/3 to obtain the simple trigonometric ratios.
The ratio of the height of the frustum to the base of the frustum is 93 1/3 to 66 2/3, or 7/5. The multiplication factor to obtain the cubit lengths is 40/3. Stated otherwise:
(93 1/3)/7 = (66 2/3)/5 = 40/3
The top of the Bent has a multiplication factor of 20/3:
(106 2/3)/16 = (113 1/3)/17 = 20/3.
The ratio of the two multiplication factors is 2:1.
These factors become more interesting when it is recognized that the frustum of the Bent contains numbers related to Giza I and the Flat.
The height of the frustum is 93 1/3 = 280/3 cubits. The width of the frustum is 66 2/3 = 200/3 cubits. Therefore the height of the Bent frustum is exactly 1/3 of the height of Giza I, while the base of the frustum is exactly 1/3 the height of the Flat.
Or, to put it another way, the ratio of the height of Giza I to the height of the Flat is 7/5, and this is the slope of the Bent frustum.
Other factors should be noted:
Decimal multiples are used in Giza I (20) and the Flat (10). One is at Giza; one is at Dahshur.
Fractional multiples (1/3) are used in the Bent. With a height of 274 a prime number is used as a multiple in Giza II. The first is at Dahshur; the second is at Giza.
Giza II and the Flat use integral Pythagorean relationships, 3-4-5 and 2B21-29. One is at Giza; one is at Dahshur.
Giza I and the Bent do not use integral Pythagorean relationships. One is at Giza; one is at Dahshur.
The decimal multiples and the integral Pythagorean numbers are not in the same structures at Giza and Dahshur.
This mix of design elements can be seen more clearly in the following table:
This arrangement of elements can hardly be accidental, and shows intent by the designer to tie all four pyramids together in one grand architectural scheme.
The silliness of previous speculations can be noted from this remark by I. E. S. Edwards, who once held a wide reputation for his knowledge of the pyramids:
"The temptation to regard the true Pyramid as a material representation of the sun's rays and consequently as a means whereby the dead King could ascend to heaven seems irresistible."
Such primitive notions would have been a great disappointment to the greatest architectural genius of all historic time.
These pyramids show an enormity of project beyond most humans to conceive. My proposed solutions are beyond the belief structure of most modern professional people. The height and base lengths were not some random assemblage of stone, spurred on by imminent death of the Pharaoh. The immensity of the projects took them far beyond individual human mortality.
Furthermore, it took a single mind, a genius to devise this scheme. It was not done by committee, nor by some random social evolutionary unfolding. If the Great Pyramid Project required three or more generations to build, according to Egyptian traditions, more than one generation of engineers was required to see the final result. We can see from the structures that each was completed. They did not depend on the lives of the Pharaohs to reach those results. The immensity of the projects must have required a social commitment that went beyond Pharaonic whim. But such view demands that the scheme be passed on from one person to another, and one administrative dedication to another, with sufficient successive genius and economic control to reach the final result.
The Great Pyramid Project is an astounding social phenomenon that has never been equaled.
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