The Great Pyramid of Khufu has been a popular subject of speculation for centuries. How was it constructed? Who built it? How old is it? What is the significance of its dimensions, its shape, its orientation with respect to the other Pyramids at Giza? What was its purpose? These topics and others are discussed in many books and on many websites. (Check here for a collection of links and references which vary from serious scholarship to rather absurd but imaginative fantasies.) In particular, one can find in this literature extensive data about the dimensions of the Great Pyramid and its various chambers and passageways as well as all the significant angles in that structure. Strangely, one finds far less speculation and also less data concerning the other major Pyramids in Egypt. A number of these other Pyramids are still standing: The Step Pyramid of Djoser at Saqqara, the Bent Pyramid and the Red Pyramid of Sneferu at Dahshur, and the Pyramids of Khafre and Menkaure at Giza. Excluding the Step Pyramid of Djoser, these Pyramids as well as the Pyramid of Khufu were all built during the 4th Dynasty over a period of just one century. They all have the shape of a true pyramid, but the details of their construction are somewhat different. Even the slopes of their faces differ. And that is the topic of this page.

I first became aware of the questions and speculations about the shape
of the Great Pyramid of Khufu while reading A. Pochan's book __The Mysteries
of the Great Pyramid__. There one finds a long discussion of the relationship
of the slope of the Pyramid's faces to the number ** &pi** and another
famous relationship involving the number

When I first read Pochan's discussion of the various relationships, the one involving 9/10 (as the slope of the edges) seemed to me to be the most compelling for the following reason. One of the many challenges for the builders of the Pyramids would have been to make a structure that rises beautifully and accurately to its apex. The silhouette of a pyramid is dominated by its edges. It must have been of the utmost importance to the builders that these edges be constructed as straightly as possible. Thus, in placing the outer blocks of each course, it would seem logical to carefully position the corner blocks first and then to fill in the remaining outer blocks of that course. If the slope (or the inverse-slope) of the edges is chosen to be a simple rational number, then it would be easier to find the correct position for those corner blocks.

However, Pochan also discusses the exercises in the Rhind Papyrus which concern the computations involving the "seked" of a pyramid. There one learns that the Egyptians represented the seked (which is just the inverse of the slope of each face) as a certain number of palms and fingers of horizontal distance per cubit of vertical distance. This documentation has to be considered as a significant clue. It would seem to make the relationship involving 14/11 (as the slope of the faces) quite compelling since that is equivalent to a rather simple value for the seked: five palms, two fingers per cubit. In addition, as Pochan points out, the faces of the finished pyramid would have been smooth (such as one can still see in the top portion of the Pyramid of Khafre). It would have been necessary for the artisans to cut the many outer blocks in a precise way to achieve this effect. Those blocks would have to be cut to have the same slant-angle as the faces.

Most scholars who have looked at this issue seem to believe in the theory that is suggested by the exercises in the Rhind Papyrus. That is, the architects who designed the pyramids chose the slant-angle of the faces just by specifying the seked to be five palms + 0, 1, 2, or 3 fingers per cubit. For example, the article *Mathematical Bases of Ancient Egyptian Architecture and Graphic Art* by G. Robins and C. Shute (Historia Mathematica, May, 1985) devotes several pages to arguing in favor of this theory. It is strongly supported by the measurements of many of the pyramids which are still standing, but not all of them. When I examined the slopes of the edges of the various pyramids, I was led to another theory: The architects designing the pyramids may have chosen the slope of each edge to be a simple rational number. It turns out that I was not the first person to suggest this. Later I learned
that a similar idea had also been proposed earlier
(around 1960 or so) by the Egyptologist Jean-Phillipe Lauer.

To be more precise, the theory that I proposed (to myself) was that the architects designed their pyramid projects so that the inverse-slope of the edges would be a rational number of the form 1+1/n for some value of n. As I will explain below, such numbers would seem to be a natural choice for two reasons: (1) They can be very simply expressed in the Egyptian notation for numbers. (2) The lesson learned from building the Bent Pyramid may have led the architects to choose a number larger than 1 for the inverse-slope of each edge. My theory is also strikingly supported by the measurements of the standing pyramids.

I have come to believe that both of these parallel theories are partially correct. They represent two approaches to choosing the shape of the pyramids and both may have been used. For each pyramid project, the architects may have specified the inverse-slope of either the faces or the edges, and then computed what the other inverse-slope must be. Since the two approaches involve different ways of specifying an inverse-slope, they may have played a different role in the construction. For example, one could imagine the builders using the inverse-slope of the edges to build up the corner sections of the pyramid (perhaps many courses at a time) with precision. It would also be necessary to have the diagonal lines for the pyramid precisely determined to do this. The remaining blocks for those courses could then be filled in. The inverse-slope of the faces (i.e., the seked) could then be used for that purpose, and also for the final smoothing out of the outer surface. That final step would involve cutting the casing blocks with precision.

I will discuss each theory based on data from __The
Complete Pyramids__ by Mark Lehner, concentrating on the pyramids built during the 4-th dynasty. Many aspects of the construction of these pyramids differ, and this seems to be true for how the slopes
were chosen too. At least one of the two theories matches the data extremely well for each of these pyramids, and this may reflect the intentions of their architects. For the Great Pyramid of Khufu, both theories match the data. That is, the seked of that pyramid is close to five palms, two fingers *and* the inverse-slope of each edge is close to 1+1/9, both with very impressive accuracy. It is tempting to believe that the architect of that pyramid was fully aware that either one of those relationships would lead to the other. Indeed, this may have made the choice of five palms, two fingers for the seked of the Great Pyramid particularly attractive. However, I see no reason to believe that the architect was aware of the relationships involving ** &pi** and

Lehner's book gives the slant-angle for the faces of various pyramids as well as the height and the lengths of the sides. It is then rather easy to calculate the
slant-angle of the edges and the corresponding slope, or inverse-slope. The following table summarizes this data. There are two entries for the Bent Pyramid corresponding to the lower portion and the upper portion. The double entries for the Pyramid of Menkaure reflect the fact that the base is not square.
43 1.058703 33 1.497233 51 .804615

**PYRAMID****ANGLE OF FACE ****INVERSE SLOPE OF FACE****ANGLE OF EDGE****INVERSE SLOPE OF EDGE**
BENT PYRAMID
54
^{o}27'44"
^{o}22'00".714288
44
^{o}42'36"
^{o}44'20"1.010156
RED PYRAMID
43
^{o}22'00"1.058703
33
^{o}44'20"1.497233
KHUFU
51
^{o}50'40".785667
41
^{o}59'15"1.111101
KHAFRE
53
^{o}10'00".749003
43
^{o}21'07"1.059250
MENKAURE
51
^{o}49'38"
^{o}10'46".786154
41
^{o}38'08"1.124919

In the extant Egyptian mathematical papyri (such as the Rhind Papyrus), one learns a number of things about Egyptian mathematics. One striking thing is that their notation for numbers is quite different than the modern notation. The only numbers that one finds are rational numbers and they are almost always expressed as a whole number plus a sum of unit fractions with distinct denominators (i.e., fractions with a numerator equal to 1). For example, the rational number 3/5 would be expressed as 1/3 + 1/5 + 1/15. In the Rhind papyrus, one finds a table for dividing 2 by all odd numbers from 3 to 101. The answers are always given in terms of unit fractions. For example, the answer for 2 divided by 5 is given as 1/3 + 1/15 and 2 divided by 13 is given as 1/8 + 1/52 + 1/104. (Of course, these expressions would be written in Egyptian characters rather than Arabic numerals as here.) A number like 10/9 (which is a very good approximation to the inverse-slope of the edges of the Great Pyramid) would be expressed quite simply as 1 + 1/9. However, 11/9 would have a more complicated expression: 1 + 1/6 + 1/18. The number 11/14 (which is the approximate inverse-slope of the faces of the Great Pyramid) would be written as 1/2 + 1/4 + 1/28. If the architect wanted to choose a number greater than 1, but still close to 1, a number of the form 1+1/n would be quite natural.

Now to the various pyramids.

THE BENT PYRAMID: The faces of the lower portion of this pyramid make an angle of 54^{o}27'44" from the horizontal. But the faces of the upper portion make a much smaller angle of 43^{o}22'. The inverse-slope of the faces for the lower portion is .714288. For the upper portion the inverse-slope jumps to
1.058703. When one computes the inverse slope for the edges, one finds 1.010156 for the lower portion, 1.497233 for the upper portion. These numbers might
suggest that the architect's intention was the following: The inverse-slope of each edge of the lower portion was intended to be 1. This corresponds to a 45^{o}
angle for each edge. What could be a simpler choice? But then, for some reason, the inverse-slope of the edges of the upper portion was increased to 3/2 (that is, 1+1/2).

I
was actually quite
excited when I noticed that the edges of the Bent Pyramid had such simple
slopes (an observation which J-P. Lauer had already made some 40 years earlier).
It would
be worthwhile to do more accurate measurements of the Bent Pyramid using
modern technology. I have relied on the data from Mark Lehner's book.
Based on that data, the inverse-slope of the edge of the lower portion is
slightly greater than 1, but is off by about 1%. But other references
that I've consulted give a slightly
different angle for the faces of the lower portion and, therefore, for the
edges. Based on one
source,the angle for each edge turns out to be even closer to 45^{o}, but still slightly
less.
Based on another source, this angle turns out to be slightly greater that 45^{o}. Also,
these
measurements only give an average slope. Undoubtedly, the slope of the
edges is not
really constant and so measuring the variation might provide useful data.
Unfortunately, the condition of this pyramid would make it difficult to do such measurements.

Another obvious problem is that it is not clear what level of accuracy one
would
expect the Egyptian builders to achieve. This level of accuracy would have
undoubtedly improved over the century of pyramid building. One would assume
that the builders would make some simple device in order to uniformly
achieve the
desired slope either for the edges or for the face. Such a device is
suggested
by Peter Hodges in his book __How the Pyramids Were Built__. The device
is a
moderately large right triangle constructed out of wood - something that
could be held by one or two workmen. The hypotenuse should have the
desired
slope with respect to the horizontal if one of the two legs of the right
triangle is held perfectly vertical. The device has a plumb line attached
to that leg so that the workmen could tell when it is vertical. Hodges suggestion seems quite credible since it is known that the Egyptians did use plumb lines. Such a device would make it possible to maintain a fixed slope even though the heights of the blocks used for different courses might vary considerably. (For the Great Pyramid, there is considerable variation from course to course.)

To get the seked for the lower portion of the Bent Pyramid, one just multiplies the inverse-slope found in the above table by 7. The result is almost exactly 5. That is, according to Lehner's data, the lower portion has a seked of five palms (per cubit). This is indeed very strong support for the conventional theory. It suggests that the fact that each edge has a slope (or inverse-slope) approximately equal to 1 is just an accident, nice as it seems. However, the upper portion is not consistent with the conventional theory. The seked of the upper portion cannot be expressed simply in terms of palms and fingers. And so, one could imagine that the architect switched from choosing 5/7 (five palms/cubit) for the inverse-slope of each face for the lower portion to the alternative method of choosing a simple inverse-slope for the edges of the upper portion, namely 1+1/2. Based on the data from Lehner's book, this turns out to be fairly accurate. The inverse-slope differs from 1+1/2 by about .2%.

There are also theories about why the Bent Pyramid has its peculiar shape.
One credible theory was proposed
by Kurt Mendelssohn in his excellent book
__The Riddle of the Pyramids__. He suggests that at the
time that the Bent Pyramid was in the process of being built, another
pyramid -the one at
Meidum,
which may have also had a
rather steep slope - had collapsed. For this reason, or possibly one of
several other reasons that have been proposed, the architect made the
decision to reduce the slope of the pyramid drastically. Increasing the
inverse-slope of each edge from 1 to 1+1/2 would accomplish that. All of
the Pyramids of the 4th dynasty which were built later are less steep then
the bottom portion of the Bent Pyramid. Accordingly, the edges have an
inverse-slope which is greater than 1.

THE RED PYRAMID: The slant-angle of the faces is just 43^{o}22'. That corresponds to setting the inverse-slope of the edges as 1.497253, which is the
same as the upper portion of the Bent Pyramid. The inverse-slope of each edge would then be quite close to 1+1/2. The kind of triangular device suggested by Hodges could easily be constructed with a high degree of precision for that value of the inverse-slope. As noted above for the upper portion of the Bent Pyramid, the seked for this pyramid does not have a simple value in terms of palms and fingers (per cubit).

THE GREAT PYRAMID OF KHUFU: As already mentioned, the inverse-slope of each edge is extremely close to 1+1/9. But the seked of this pyramid is also quite simple. It is five palms, two fingers/cubit. That is, the inverse-slope of each face is extremely close to 5.5/7=11/14. The two theories both seem to be valid for the Great Pyramid. I suspect that the architect was fully aware of this and chose the seked to be five palms, two fingers knowing that the inverse-slope of the edges would also turn out to be quite simple. (Or perhaps, it was the reverse.)

THE PYRAMID OF KHAFRE: The angle that each face of this pyramid makes with its base is 53^{o}10'. The inverse slope of each edge turns out to be
1.059250. This number is approximately 1+1/17 (accurate to about .04%). The inverse-slope of each face of this pyramid is approximately 3/4. The seked is 5 palms, one finger/cubit. But, based on Lehner's data, the accuracy is not nearly as good as for the Great Pyramid. (It's accurate to about .13%.)

THE PYRAMID OF MENKAURE: This is the last and smallest of the three great pyramids at Giza. It is interesting to note that, according to Lehner's book,
the base of this pyramid is not a square. It is rectangular and there is more than a two-meter difference in the lengths of the longest and shortest sides. At first,
when I computed the inverse-slope of each edge, it turned out to be approximately equal to 1+1/8 , but not with a high degree of accuracy. However, I had assumed
that the base was a square. When I learned that the base was not a square, I again computed the inverse-slope of each edge and found that it was indeed very
close to 1+1/8. Thus, this pyramid provides encouraging support for my theory. Because the base is not a square, the slant-angles of the faces are not equal. One pair of faces has an inverse-slope very close to that of the faces for the Great Pyramid of Khufu. That is, the seked would be close to five palms, two fingers/cubit. But the other pair of faces has a seked which cannot be expressed simply in terms of palms and fingers.

COPYRIGHT © 2000 RALPH GREENBERG

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