Plotting and Building the Egyptian Pyramids

New Book 2022

Introduction

This book deals with the plotting of the Egyptian pyramids. The purpose of my research on this subject is to recreate the technique used in ancient Egypt.

The surveyors of ancient Egypt worked out the plotting of the Great Pyramid with such precision and accuracy that it was only possible to reproduce them with the use of modern surveying instruments. The answers to the burning questions about the pyramids should not only be the conclusion of logical reasoning but also lie within the historical context and be supported by scientific evidence.

For instance, it is common to hear this reasoning:

Those who plotted the Great Pyramid did so with astonishing accuracy. The accuracy achieved is similar to that which would be obtained by using modern surveying instruments. Since the ancient Egyptians did not have any precision instruments, the conclusion is that they did not plot or build the pyramids.

This is a wrong conclusion, for it lies outside the Egyptological context, which indicates that it was the ancient Egyptians who did.

This work – regarded as the first wonder of the ancient world – is the result of the evolution in construction occurring in the pharaonic tombs of the Ancient Egyptian Empire.

Those who constructed the pyramids learnt to build them in Egypt based on their own experience. As the Egyptian civilization built the pyramids, the pyramids built the Egyptian civilization. The discovery of the builders’ village in Giza by Dr Zahi Hawass and Dr Mark Lehner as well as the Diary of Merer – foreman of the Great Pyramid works, which describes how the casing stones were transported, helps us to place the work in the historical context.

In my view, the logical conclusion is that if the ancient Egyptians did not have precision instruments to plot the pyramids so accurately, and they did not plot the pyramid by “measuring” but by using another technique that was within their capabilities.

Considering the limited archaeological information available on the topic, we shall move forward by considering the possibilities at the time and the particularities of the plotting obtained. With that goal in mind, we must first shed any modern surveying skills, procedures and tools, and place ourselves in the context of ancient Egypt.

The surface of the ground under the pyramids of Cheops and Khafre were not levelled, which indicates that they did not survey the square for the base either, as it was impossible to draw the diagonals or see the opposite corner from any of the corners.

When plotting a smooth-sided pyramid, the edges had to be straight and converge on the summit point. The edges cannot have been plotted as construction work progressed because, again, it would have required the use of precision instruments and the necessary surveying know-how.

"Careful surveying during construction was essential. Otherwise, a twist might occur and the edges would not meet at a point at the top."

Surveying the pyramidal shape as the pyramid was built would have resulted in snowballing errors which would have prevented the four triangular sides from meeting at the summit point. Let us add that once these inevitable errors had been committed, it would have been impossible to correct them, since they would have become noticeable with construction at an advanced stage, close to the summit.

How do you plot the Great Pyramid with that amazing accuracy without measuring with accurate instruments? How do you plot the base of the pyramid with 230-metre sides with an average error of 15 mm in length and 32 seconds in the angles? How do you achieve such high precision in measurement without using optical instruments?

In addition, logic and the very foundations of metrology (the scientific study of measurement) say that the greater the distances to be measured, the greater the errors. However, the opposite is true of the Egyptian pyramids. The largest pyramids are the most accurate.

Another amazing fact is that there is no large building in Egypt that has the accuracy of the great pyramids. It is quite obvious that if they had developed instruments to help them measure with such precision and accuracy, they would have used them on other buildings as well.

These amazing and seemingly illogical facts are a sign that the pyramids were not plotted by measuring, and they give us certain insights into how they did it. For instance, we know that with the method they used, the larger the pyramid, the greater the accuracy achieved. This phenomenon is noticeable in the case of sundials. The largest sundials are the most accurate, just as the largest pyramids are the most accurate in their plotting.

I will therefore work on the hypothesis that the pyramids were not plotted by measuring but that the technique that was employed was based on the use of the sun.

I will resort to an architectural software program to reproduce the shadow cast by the top of the pyramids under study. I will also turn to astronomy software to obtain and analyze the position of the sun for every pyramid plotting.

We also know that the accuracy in plotting starts with the smooth-faced pyramids, and we can distinguish two types of surveying:

a. Inaccurate surveying, used on the stepped pyramids, consistent with the measuring tools existing at the time.

b. Accurate surveying, developed through the plotting of smooth-faced pyramids.

Accurate plotting began to develop with the smooth-faced pyramids.

In the case of Meidum, the casing was laid over a very inaccurate core. Everything points to remeasuring after construction of the core, before laying the casing. Something similar can be observed in the Pyramid of Cheops, in which the core looks somewhat misaligned with the final shape of the casing. The courses are not perfectly horizontal, as they should be if the pyramidal shape had been plotted from them.

Unlike the stepped pyramids, in which errors can be absorbed in each step, the edges of the smooth-sided pyramids are supposed to be straight and meet at the top. Any deviation or warp in an edge would be clearly visible. For a pyramid the size of the Pyramid of Cheops, a deviation of 2 degrees at the base grows to 15 metres at the top, so the edges would not meet at the summit point.

Plotting the pyramidal shape without determining the summit point would require the use of precision instruments that the ancient Egyptians were certainly not familiar with. First, it is necessary to build the stepped core to establish the location of the summit point, which is essential for plotting the pyramidal shape from it. (4)

Accuracy of the smooth-sided pyramids lies in their casing, which was laid over an inaccurate stepped core that was constructed first. This is logical and usual in all constructions; completion is achieved by laying the casing over a structure that has been built first.

The scenario of the Egyptian surveyors, and Pharaoh Sneferu in particular, was a very different one. He simply considered how to turn the stepped pyramid of Meidum into a smooth-sided pyramid. It was then a matter of plotting the pyramidal shape over an existing stepped pyramid. The pyramidal shape was plotted from the summit point down. This problem is altogether different from plotting the pyramid from the ground up, and it is the one that the ancient Egyptians solved.

The plotting of the pyramids determines the construction stages, so it is essential to understand the plotting in order to understand the construction of the pyramids.

One of the main research findings is that the slope angles used for the pyramids match the maximum solar elevation (azimuth 90°), according to the latitude where they are located.

The east and west sides of each pyramid were plotted with a slope equal or approaching to the maximum solar elevation at that latitude for azimuth 90° and 270°.

The slopes of the pyramids increase towards the south, as does the solar elevation. They go from 52° at Abu Rawash to 53° at Giza, 54° at Dahshur, and finally 73° at Meroë. The 73° slope of the Nubian pyramids at Meroë is less than the maximum solar elevation, which is 80°, due to limitations for structural reasons.

The technique that I am going to describe is only useful for plotting pyramids, and more specifically their casings, which accounts for why there are no other buildings with that stunning accuracy.

I shall first describe the procedure for plotting the basic lines of the pyramidal shape, that is, the edges, apothems and auxiliary lines. Then I shall use the shadow cast by the plotted edges and apothems, as well as the auxiliary lines, for the final check and adjustment of the pyramidal shape.

Daniel Gerardo Di Matteo

Book Plotting and Building the Egyptian Pyramids