Web Exclusive: Pyramid Schemes

December 1, 2009
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Figure 1. Goddess Nut pictured on the ceiling of the tomb of Ramses VI. The day and night are portrayed on the left and right side of the photos, respectively. Her head is at the far wall, and the circular discs are symbolizing the sun moving through her body.[1]


The orientation of significant buildings in ancient Egypt is a widely discussed and speculated upon topic. Measurements taken in modern times show an incredible similarity to the alignment of significant buildings, namely temples and pyramids, to true north. There is no description or explanation to confirm this intentional alignment, and there is no indication as to how it was achieved. However, given the fact that there is a striking pattern to the buildings, it is hard to imagine it was anything less than intentional.

A number of theories as to how and why the orientation was established have been put forward. From a surveying point of view, these will be looked into for their merits and weaknesses. The errors also must be investigated as, ultimately, they could be the biggest clue as to why and how orientation was established.

Methods of Orientation

Over the years, Egyptologists have proposed a number of methods of orientation. When evaluating the plausibility of each method, it is important to keep a few things in perspective. Firstly, the method must be simple, in line with the simplicity with which the Egyptians carried out all tasks. Secondly, the method must contain errors as no building was without error, though it could be possible that the error decreased with time--maybe due to increasing accuracy. This aspect is something that will be discussed later.

Discussion will for the most part refer to the orientation to true north. However, looking at the evidence from the time of the ancient Egyptians, it would appear that, in fact, the pyramids were aligned east-west.[2]

The significance of the sun rising and setting and the knowledge that the Egyptians had of astronomic movements are shown in the picture of the goddess Nut (Figure 1), taken from the royal tomb at Thebes. In the picture, Nut, goddess of the sky, with her elongated body, is shown to be ingesting the sun and then 12 hours later gives birth to it again, representing the new day. Horus, god of the sky, holds a vertical line bisecting the path of the sun. The Czech Egyptologist Zaba believes that this line is the meridian line, which would indicate that the Egyptians had knowledge of the meridian as the center of the sun’s path in the sky.[3]

Figure 2. Diagram of Edwards’ method showing the angle being bisected to find north.[2]

Rising and Setting of the Stars

Accepted by many people over the years, Edwards proposed a method of orientating the pyramids using the rising and setting of the stars.[4] Although his method is theoretically correct, it is questionable from a practical surveying point of view, especially considering what we know about the simplicity of life in ancient Egypt. Edwards’ method (figure 2) suggested (ibid) that a circular wall was built, which had to be perfectly horizontal, to create a false horizon.

A person standing in the center of the circular wall could then mark the positions of a rising and setting star on the wall when viewed from the center pole, and the angle between the two could be bisected to find true north. The wall was used, as atmospheric refraction would make it create the impression that the stars were rising higher on the horizon than their actual position, and this would in turn produce an error in the calculation of the meridian line.

Despite the skill of the Egyptians, it is doubtful that even the most skilled mason could build a wall as precisely as required; a both perfectly circular and perfectly level wall would perhaps be beyond their ability. Even if such a wall was built, and presuming that it would be possible to position the sighting pole correctly and accurately, there would still be a large human error affecting the observation. The observer would have to be immobilized for the length of time for even the slightest movement could significantly affect the observed angle, consequently achieving inaccurate results.[2]

Circumpolar Stars

Despite the severe flaws that using circumpolar stars have as a method of orientation, it still appears to be a popular and ever-developing theory. At the time of the pyramid building, there was no star fixed to the point of the pole that was visible.[2] Therefore, stars close to the pole were required to determine north. The closest star would have been Alpha Draconis at 1º40′ from the pole. In order to calculate true north, the most extreme eastern and western positions of the star must be visible and this angle bisected. For such points to be visible, 12 hours of darkness are required, something which may be only possible on equinoctial nights--limiting the scope for this method considerably.

Even assuming the stars were visible for the required time, the precise measurement required would be unattainable due to lack of instruments for such a purpose. Consequently, the resulting error would have disagreed with the errors that have been measured on the pyramids (see Errors in Elevation: Deviations From True North).

Recently, an even shakier version of using stars has emerged. Spence, a British Egyptologist, has proposed the use of two circumpolar stars that at the point of alignment, which could be ascertained through the use of the plumb bob, marked the point of true north.[5] Not only does this include the problems of the previous method, such as how they could have seen these stars at the appropriate time, but also it is hard to believe that the Egyptians would have the required knowledge to use this method.

Figure 3. Diagram of Neugebauer’s method.

Sun

The use of the sun to orientate buildings is, from a surveying perspective, the most feasible theory. The sun is overwhelmingly prominent in the Egyptian sky, and unlike the use of stars in orientation, it would have been generally visible and accessible.[2] The sun also had the advantage that--rather than the line of direction having to be transferred to the ground from the stars using the instruments used to define orientation--the sun produces the line on the ground in the form of a shadow. Evidence does not show whether the Egyptians had ability to determine the summer and winter solstices or the equinoxes. However, given the role of the sun in their lives, it is unlikely that the Egyptians failed to notice these phenomena.[6]

When a shadow was at its shortest, it always pointed to one direction: true north, the meridian. At the point of the shortest shadow of the year, it was the summer solstice. It was recognized that one year was the number of days between one summer solstice and the next.[3]

One such method of using the sun for orientation is proposed by Neugebauer.[7] Recognizing the serious difficulties in the astronomical methods, Neugebauer’s method requires the use of an accurately placed pyramidal block--he suggests the capstone of the pyramid--to be placed in the location of the future pyramid, on accurately leveled ground. The block can be placed with a rough estimation for cardinal orientation, which can be easily achieved by looking at the approximate position for the rising and setting of the sun. The path of the sun is then observed by measuring and marking the position of the shadow of the apex of the capstone block for a period of time before and after noon.

This path will be a curve, as can be seen in figure 6c. It can be seen from the diagram that there will be two points at which the curve intersects with the lines, which can be drawn to extend the eastern and western base of the pyramid. At the point at which the pyramid is orientated cardinally, the lines from the intersection points to the corners of the capstone, AC and BD should be exactly equal. If, as is most likely to occur, the lines are not equal, the capstone can be moved slightly and the process repeated again at noon the following day until the orientation is correct.

Table 1. Orientation of Major Pyramids[2]

Errors in Orientation

Deviations from True North


The pyramids provide a good source of study of orientation error in ancient Egypt. Their significance and importance meant they were carefully orientated and built, and their survival and existence today allows them to be still measured and studied. Once it is accepted that the pyramids were in fact intentionally aligned in the cardinal directions, the measurements taken can be investigated, and the deviations from true north can be compared to establish the errors obtained. Once the order of the errors is established, the various proposed methods of orientation can either be discounted or supported depending on the feasibility that errors of the particular magnitude would be expected. The deviations from true north for the pyramids are shown in Table 1 below.

From this, it can be seen that the pyramids are indeed orientated with an astonishing accuracy. The pyramid with greatest deviation is that of King Djoser. Older than all the other pyramids, it deviates by 3º from true north, double the next largest deviation. For the purposes of analysis and investigation into orientation methods, this result is to be discounted, citing that the orientation procedure was probably not fully realized at the time it was built. [2] Disregarding the pyramids of Djoser, we can calculate that the average deviation of the remaining pyramids is only approximately ±30 minutes from true north.

Table 2. Orientation of the Sides of the Great Pyramid.8

More specifically, the most accurate pyramid was the Great Pyramid of Giza, or the Pyramid of King Cheops, the most famous and studied pyramid of them all. Cole determined values from the variation from the azimuth of the sides of the Great Pyramid of Giza as follows (Table 2).

These values show an error in orientation to the northwest by an average of just over three minutes, which equates to an error of about 12 m over the 230 m side length. It can be said, therefore, that the method of orientation must be able to produce a result as accurate as the great pyramid, yet it is still acceptable for the orientation to produce larger errors.

Observing that the error in the Great Pyramid of Giza is relatively small, even at its maximum, Edwards argues that such an error was too small to have been an error in sun observations.[9] While Edwards believes that star observations are of greater accuracy than the sun observations, the previously discussed flaws in his method still support the fact that sun observations were a viable alternative being far more accessible to the Egyptians. However, at the appropriate time of the year--that is, at the equinoxes--the sun technique could produce the precision achieved.

Table 3. Corner Angles of the Great Pyramid.[3]

Accuracy of Geometry

It is a strong possibility that the errors in the proposed method of orientation essentially occur due to the natural phenomenon on which they are based rather than in any inaccuracy that occurred in geometric operations.[3] It seems likely that the skill of the Egyptians really did produce a high accuracy of measurement, and therefore, the error is more likely to exist within the circumstances they could not control.

Table 1 shows the deviations of each side from true north, which resulted in an average of 0º 4′ 0″ west of true north. Comparing this with each of the corners, as shown in Table 3, we find that the deviation from right angles is only at 0º 1′ 47″, which supports the evidence of greater accuracy in the Egyptians’ geometrical skills rather than the results obtained from their sun observations.

Error from Precession of the Equinoxes

Satisfied that error must lie outside the capabilities of the Egyptians’ methods of orientation, we can postulate that at least part of the error obtained in the orientation of the pyramids was due to the apparent change in the position of the rising and setting sun. This change is explained by precession, a phenomenon that describes the movement of the Earth’s rotation axis in space. The full cycle of precession takes about 26,000 years, with each sign of the zodiac taking 2,100 years for our solar system to pass through. [1]

The precession of the equinoxes, which accounts for a 50.26" west per year movement of the pole, would account for the small changes in the orientation of the pyramids.[10]

Figure 4. This relief is from the Temple of Amun-Re at Karnak and depicts the pharaoh (left) and the goddess Sheshat planting stakes linked by a loop of rope for the “stretching of the cord” ceremony.[11]

Error from Earth’s Tilt

It has been shown how error can occur in the orientation of the pyramids, but particularly in the case of the Great Pyramid, it has to be proven that it is still possible to produce the results found. The apparent change in the rising and the setting of the sun during a year could be such an explanation.

The errors in the orientation are dependant on the time of year that the measurements were taken. As the sun moves through its yearly cycle (from the perspective of the Earth), its position of rising and setting moves, so that only at the point of the equinoxes does the sun truly ‘rise’ in the east and ‘set’ in the west. Therefore, measurements taken at any other time of the year would incur errors, and the further from the equinox, the greater the error. Therefore, if the information for the orientation of the Great Pyramid was obtained on the day of either of the equinoxes, either because of the specific intentions of the Egyptians or by chance, then the orientation would be of a higher accuracy. It should be noted that, as explained previously, it is not believed that the Egyptians were aware of the time of the equinoxes.

Extending Direction

Laying the foundations of a temple or pyramid was important for the Egyptians. A number of rituals took place, beginning with the “stretching of the cord” ceremony (Figure 4) which, over the years, has been analyzed by many Egyptologists and has been associated with building and construction works, surveying jobs, religious festivals and regal responsibilities. It is likely, and quite feasible, that the “stretching of the cord,” known as pedj-shes, was involved in each of these things.

Figure 5. Diagram showing the shadows created by the gnomon and bay to define direction (adapted from Isler[3]).

The purpose of the “stretching of the cord” ceremony was to extend direction. Once the direction was established over a small distance from the observations from the sun, they would extend the lines so that the building would be orientated accurately.

Using shadows as a means of finding direction still meant that the object that created the shadow must clearly identify the exact point in which the direction lies. Isler suggests the use of a bay to create a point on the ground that is clearly identifiable.[3] The bay has been largely thought of as an astronomical sighting instrument. However, as Isler points out using an example of a specimen preserved in Berlin, it would not be accurate for sighting stars, although it would be able to create an accurate line of direction.

The gnomon alone could produce a line shadow. However the thickness of the gnomon itself would alter the line if the center were not obtained. The idea of using the bay is to create two shadow points, from which to extend direction. The bay was held with the notched end to the ground such that the shadow of the tip of the gnomon was not obscured and so that the notch in the bay defines the direction.

Figure 6. Tomb of Senmut. Showing what Isler believes could explain knowledge of the information that could be obtained from using a tall gnomon.[12]

Solstices were more difficult to identify with a shorter length gnomon because the longer the pole, the easier it is to identify the location of the shortest shadow. The ancient Chinese used tall gnomons for this purpose.[3] It could be that the Egyptians did, also. A part of the ceiling of the tomb of the official Senmut (Figure 6) suggests this. The tall triangle shape, Isler believes could be a gnomon, which compared with the height of the figures would be around 6 - 7 m in height.[3] The short object held by the figure second from the left-hand side, Isler says could be a bay, linking the use of the bay to the sun, which is the small disc at the apex of the triangle.

Methods of Leveling

Many ideas have been put forward over the years as to how the Egyptians obtained level surfaces to such precision. Many of the proposals are considerably complex and unfeasible in contrast with the simple existence the Egyptians led. Evidence of a simple leveling tool is perhaps the only clue to unlocking the leveling secrets of the Egyptians.

Despite the mystery that surrounds their techniques, it is evident that the Egyptians obtained great accuracy in their leveling. Largely through the work of J. H. Cole in 1925, we are able to determine the accuracy that could be achieved with even the most basic tools.

Flooding

Had the site of the pyramids been a flood plain, leveling may have been a relatively straightforward task. Despite the fact the pyramids were built on a desert plane, water has still been suggested as the tool for various methods of leveling. Englebach suggested the flooding of the entire area around the pyramids enclosed by an embankment so that the rock could be cut to a certain distance below the water surface, thereby creating a level plane.[2] This would in fact be a difficult task, for the volume of water required would be enormous, and even if the water had been able to be transported, the level water surface would have been disturbed significantly while the rock was being leveled.

Channels

Lehner proposed an alternative method of water leveling comprising a network of channels that could be built and then flooded.[4] The level of the water could be marked on the sides, and once the channels were drained, the channels could be cut down to a standard height.

Lehner’s argument is that the number of channels was also limited and that only the strip around the site of the pyramid was leveled.[13] This was definitely the case in pyramids such as Cheops (the Great Pyramid) and Chephren (both 4th Dynasty), which were built on what was originally a natural sloping surface, some 7-10 meters above the final surface. In each instance, a mass of rock was left in the center, and therefore the surface could not have been completely leveled.[4] Such large rock masses, on ground which naturally slopes, would require extremely deep channels in order to level to an appropriate depth all the way around. This is impractical however, both in terms of its complexity as well as the logistics of the land and moving the water, and hence, the theory is probably very unlikely. In fact, despite the theory of Englebach, Lehner and others, there is no indication the Egyptians used any type of water level or leveling rod in their practices.

Figure 7. This photo shows a level--the A frame with the plumb bob in the center. The other instruments are a square, used for right angles, and a plumb device, which was used to make the gnomon perpendicular. They are from the tomb of Sennutem at Thebes, dating from the 20th dynasty. The mallet was not found with these instruments.[6]

Leveling tool

The Egyptians made use of a leveling device that was based on the principal of gravity.[2] It was a square level, in the shape of an A with a plumb line hanging from the top (Figure 7). A level surface was achieved when the plumb line was in line with a mark that was on the center of the cross bar. In the event that the legs of the level were not exactly equal, calibration of the instrument could take place. The level was placed on a surface, and the position of the plumb line marked on the crossbar. Then the instrument was reversed and placed in the same position, and the plumb line position marked again. If the surface is level, the plumb line will fall in exactly the same position on the cross bar. If there was any variation in the points on the cross bar, then the surface was not level, and the difference between the two points was marked to be the center point. Then the process could be repeated using the center point to define the level.

Accuracy Obtained

Using the leveling device described above, it is clear that with practice and increased knowledge, the accuracy of their work would also improve. This indeed has been observed, with the later pyramids showing increased accuracy.[2] The pyramid at Medium (early 4th dynasty) deviates from horizontal by less than 12 cm; the Bent Pyramid (early 4th dynasty) deviates less than 4.6 cm; and the pyramid of Cheops (late 4th dynasty), less than 2 cm.

Despite these bold statements of deviation, it can be seen that despite average deviations, there was noticeable unevenness over the foundations. This will be illustrated in the closer study of the Pyramid of Cheops in the following section.

Figure 8. Diagram showing the differences in height of the corners of the Great Pyramid. All values are in meters, and the arrows point down hill in all instances. Bearing in mind the tools with which the Egyptians were working, and that the distances between these measured levels was approximately 230 meters (325 meters between diagonals), the results show an incredibly impressive ability to obtain a level surface.

The Great Pyramid of Giza Study

Investigating the Great Pyramid of Giza (Pyramid of Cheops) can give us a more detailed insight into the possible accuracy that the Egyptians could obtain. In 1925, J. H. Cole published the results of a survey of the Great Pyramid, covering its size and orientation. A precise leveling run connected points around the base of the pyramid to two previously established bench marks. The results were given to one thousandth of a meter. Given the year in which the survey took place, and the lack of electronic distance measurement as well as the fairly indefinable line of the edge of the pyramid, it is quite optimistic to quote such accurate figures. For the following calculation, results to the nearest millimeter will be used, though it is recognized that even this is probably optimistic.

In Figure 8, a plan view of the base of the pyramid is drawn. Levels are given for each corner based on Cole’s observations. The levels are that of the foundation stone, which is the rock that would have been leveled prior to the construction of the pyramid. The measurements on the sides and diagonals are the differences between the levels. Calculating between the levels of the foundation rock at each corner, we do not obtain a 15 mm difference, as Cole claims. Instead we can see a maximum difference in level of 60 mm between the northeast and southeast corners, and a 22 mm difference on the western corners. The diagonals are also fairly similar in level, with 27 mm and 11 mm differences northwest to southeast and northeast to southwest respectively.

Conclusion

The evidence of the work of surveyors can be found in the items recovered in archaeological excavations and in the paintings on the tomb walls. Despite this knowledge, nothing exists which has documented the methods of surveying in ancient Egypt, leaving open questions as to how the surveyors obtained their position in order to align buildings with such precision and on a near perfect level surface.

Many have speculated on the surveying methods employed by the Egyptians, in all aspects of building and construction. However, it seems that quite often the specific tasks of the surveyor or how they were carried out fail to be thoroughly thought through.

Using the simplicity that is so evident in all aspects of Egyptian life, we can base the work of the surveyors on something equally simple--the sun and a simple leveling tool. And with these, they managed to create one of the most spectacular monuments mankind has ever produced.


Editor’s Note: For more of Chapman’s insights into the practice of surveying in ancient Egypt, check out the first two parts of this three-part series: Cubits and Cords: The math and tools of ancient Egyptian surveyors [www.pobonline.com/Articles/Article_Rotation/BNP_GUID_9-5-2006_A_10000000000000478038], and Positioned For Prominence: Land administration and the surveyor’s status in ancient Egyptian Society. [www.pobonline.com/Articles/Features/BNP_GUID_9-5-2006_A_10000000000000535138 }


References

1. Crystal, E. 20th Dynasty. 2003. [online] Available: www.crystalinks.com/dynasty20.html [29 September, 2003]; Precession of the Equinoxes. 2003. [online] Available: www.crystalinks.com/precession.html [14 September, 2003].

2. Isler, M. 2001. Sticks, Stones, and Shadows: building the Egyptian Pyramids. University of Oklahoma Press. Norman.

3. Isler, M. 1989. “An Ancient Method of Finding and Extending Direction” JARCE XXVI. The American Research Centre in Egypt. Cairo. pp. 191-206.

4. Lehner, M. 1997. The Complete Pyramids. Thames and Hudson Ltd. London

5. Gingerlich, O. 2000. “Plotting the Pyramids” Nature (Vol. 408). Macmillan Magazines Ltd. pp. 297-298.

6. Glanville, S. (ed). 1942. The Legacy of Egypt. Oxford University Press. London. reprinted 1963.

7. Neugebauer, O. 1983. On the Orientation of the Pyramids. Brown University. Providence.

8. Cole, J.H. 1925. “Determination of the Exact Size and Orientation of the Great Pyramid of Giza” Survey Of Egypt Paper no. 39. Government Press. Cairo.

9. Cottrell, L. 1950. The Lost Pharaohs. Evans Brothers Limited. London.

10. Stecchini, L.C. The Pyramids of Egypt. [online] Available: www.metrum.org/key/pyramids [13 August, 2003]

11. Understanding the Myth of Creation [online] available at www.mythofcreation.com [25 September, 2003]

12. Bauval, R. and Gilbert, A. 1994. The Orion Mystery. William Heinemann Ltd. London.

13. Shaw, I. 2002. Building the Great Pyramid. BBCi History. 2002. [online] Available: www.bbc.co.uk/history/ancient/Egyptians/great_pyramid_print.htm [31 March, 2003.]

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