www.cheops-pyramide.ch Copyright 2006 Franz Löhner and Teresa Zuberbühler
Alignment of the pyramids and controlling the shape of the pyramid
The pyramid builders carefully chose the building ground for the pyramids. The Khufu-pyramid lies on the best ground existing on the Giza plateau and on top of a rock core which reaches to at least 7.9m height under the pyramid. How important the choice of building ground was, shows the fact, that one of the satellite pyramids (G1c) probably had to be moved closer to the others, because the ground was considered not stable enough.
problems had developed earlier at other building sites, because the strength
of the ground was overestimated when erecting a pyramid. For example the
ground beneath the pyramid of Sneferu (father of Khufu) in Dahshur gave
way under the weight and caused damage in the casing. Another outer layer
of casing had to be added, changing the angle of inclination from 60°
to 54.46°. But again structural problems with the subsidence developed,
the casing slipped and more damage occurred. In the end the angle of inclination
had to be drastically lowered to 43.99° .
Now this pyramid is called the Bent Pyramid, because of its shape.
A survey of the Giza plateau shows that the pyramids are aligned to the north-south axis as well as in relation to each other. The ancient engineers worked very precisely, not only are the three pyramids each perfectly centered, but the angle of inclination is also the same from bottom to top .
A pyramid is a simple shape but when building a pyramid structure as large as the pyramid of Khufu there were a lot of obstacles to overcome.
The ancient Egyptians used cubits, palms and fingers as units for measuring. The cubit was divided into 7 palms and a palm into 4 fingers which resulted in 28 fingers for one cubit.
By stretching a cord between two pegs stuck in the ground, a long straight line was marked out. Then to each peg an equal length of string was tied, more than half as long as the line drawn. By keeping the strings stretched tight and moving the ends around, the Egyptians could draw parts of two perfect circles. These arcs cross each other at two points. By drawing a straight line between these two points the original line is bisected at a right angle and the line is cut into two equal parts.
The Egyptians didn't know the theory explaining the Pythagorean theorem but they knew from experience, that they could form right angles by using a so called primitive Pythagorean triple. One of the most well well-known example of a Pythagorean triple is 3-4-5 where 3²+4²=5² (9+16=25).
We know, that the Pharaoh employed land surveyor, so called harpedonaptai or harpedonapts (rope stretchers or rope-knotters). After the annual flooding of the Nile river, when rich silt was deposited on the valley floor, the property boundaries of the fields were destroyed, so they had to be reestablished. The harpedonaptai used measuring cords with 11 equally spaced knots (or marked with paint) dividing it into 12 parts. With these ropes they could form a right triangle with the sides of 3:4:5.
The same principle could be used to determine the right angle at the
base area of the pyramid of Khufu. These are very exact - the maximum
deviation from 90° is only one minute! 
Since the ropes might have been too bulky to measure such long distances
it is also possible, that wooden cubit rods were used instead.
The base area or plateau of the pyramid of Khufu (including the surrounding
pavement) was leveled with great precision, so only a deviation of 2.1
cm can be measured . In the middle of the base
area a rock core was left.
The only instrument for leveling known to the ancient Egyptians was the square level. This is a right-angled isosceles triangle made from wood. This tool is made in the shape of the letter "A" and looks like a triangle ruler or a builder's square. A plumb-line is suspended from the top of the connected corners. If the plumb bob coincides with a mark in the middle of the crossbeam, the surface area on which the two legs stand is level. By placing this triangle on stone, these could be easily leveled.
Isosceles triangle made from wood with plumb-line and plumb bob, which points to the mark on the cross beam.
For establishing equal levels over large distances, this triangle was laid on top of a beam. First the triangle was checked and adjusted, so it would lie horizontally and level on the beam. Then, by fixing upon a leveling staff in the distance, differences in height could be noticed and corrected. Field tests showed, that up to a distance of 45m a difference of 1 centimeter could be detected . Because of the limited eye sight of humans larger distances could not be covered. But because it is possible to sight on both sides from this kind of measuring station, it probably was possible to correctly level distances of up to 90 meters. Level squares like we described here were found for example in the tomb of Senedjem at Deir el-Medina.
Another suggestion made by some archaeologists is to dig flat ditches surrounding the pyramids and then filling them with water. At the top of the water's surface, the level would be marked along the sides of the ditches as a reference. Other archaeologists object, that the Nile and the Nile channel run about 44m lower and that water evaporates quickly in the desert. They think, this kind of leveling method could not have been used.
The Great Pyramid's north-south axis is nearly aligned to true north, the deviation is only 2' 28''. To achieve this precision the Egyptians had to make an exact survey and do control measurements. Otto Neugebauer  is offering a simple solution to this problem.
A small but precisely worked pyramidion (made from wood or stone) is placed as close as possible to a north-south position on the flattened ground where the pyramid will be built. Then two lines are drawn, taking the edges of the small pyramidion and lengthening them on the side which lies in the shadow. During a day a surveyor notes down, where exactly the tip of the pyramid shadow cuts through those two lines. The pyramidion is precisely aligned north to south, when both lines are cut at the same distance to the corner of the pyramidion. To achieve this, the pyramidion is moved slightly over a period of days, until both segments are exactly the same length. Since the pyramidion can be moved, the process can be repeated at several places on the pyramid plateau, thus checking and rechecking the north-south alignment. Even during the construction of the pyramid, when working on the pyramid frustum, this system can be used again for checking, if the surface is smooth enough, for example on top of the pyramid plateau.
How to determinate the north-south alignment of a pyramid
It is easier to make those measurements during the winter months, when the sun is lower in the horizon, because during the summer months the pyramids don't cast a shadow during large parts of the day.
For further precision you can now build a small but accurately shaped pyramid. South of the pyramid of Khufu lies a small pyramid (G1d) which measures 23 times 23m and has an angle of 51-52° . This pyramid or one of the queen's pyramids could possibly have been used to further adjust the north-south-alignment using again the same process described above but with slight alterations.
Another, a bit more complicated suggestion was made by I.E.S. Edwards. According to his theory the Egyptians built a circular wall whose top was carefully leveled, thus creating an artificial horizon. From a straight pole with a forked top in the center sightings were made to determine, where exactly a certain circumpolar star was rising and setting. Using a plumb line the positions were marked on bottom of the wall. By bisecting those positions, true north was provided . Illustration
 O. Neugebauer
On the orientation of pyramids
Cairo (Egypt): Sun's
location / Overview
www.cheops-pyramide.ch Copyright 2006 Franz Löhner and Teresa Zuberbühler