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martes, 6 de marzo de 2018

INTRODUCTION:



Monuments of human strength and ingenuity, the pyramids have always compelled admiration and aroused curiosity. Erected on the Giza Plateau, Pharaoh Khufu’s pyramid (Cheops in the Greek designation) or Great Pyramid, as it is called, is a masterpiece of construction. Together with the pyramids of the pharaohs Khafre and Menkaure, the Sphinx, mastabas and satellite pyramids, it is part of the funerary complex of Giza.

This work – regarded as the first wonder of the ancient world – is the result of the evolution of 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 Zahi Hawass and 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 this evolution of construction, two requirements stand out that were of significance to the pharaohs:
a) The requirement of shape: The perfection in the plotting of the shape of the smooth-faced pyramids begins with the pyramids built by Pharaoh Sneferu and reaches levels of excellence in the pyramids of Giza.
b) The requirement of height: The height of the pyramids built gradually increased. In the great pyramids of Giza there is a large increase in height in the order of 50%, which is a feat of ancient engineering.
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 existing proposals do not help to understand or reproduce the astonishing accuracy achieved in the design of the great pyramids of Giza.
The aim of this book is to present my conclusions after four decade’s worth of research on the subject.
There are a wide range of theories as to the building of these pyramids, but they deal exclusively with the requirement of height. These theories basically analyze how the blocks were transported and raised to a great height during the construction of the pyramids.
The surveying of the pyramid is mentioned simply as a side issue and with no solution, due to the lack of historical documents, and with no significant impact on the construction undertaken.
However, the construction requirement of shape is not merely a technical detail. Like all civil works, the construction of the great pyramids of Giza required working out how they were surveyed in the first place.
"To build the Great Pyramid, it must first be plotted."
To think that it is possible to build the pyramid without plotting it first is a misconception of our times that the ancient Egyptians would not understand and the Pharaoh would not authorize.
As we shall analyze later, plotting the pyramidal shape conditions the construction stages of the pyramid. This is the reason why in order to build the pyramid you must first plot it with the required accuracy, and we cannot skip this construction requirement. This point is key to understanding the construction. Trying to build the pyramid without plotting it has led to a maze of theories that basically address the requirement of building the highest pyramid.
The only way to overcome this confusion, the labyrinth of theories and scepticism and be able to understand the construction of the pyramids is by just starting at the beginning and plotting the pyramid.
But, 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. Furthermore, it is a technique that was only applied to the plotting of the pyramids, since it was not used with other buildings.
We also know that the accuracy of the plotting starts with the smooth-faced pyramids, and we can distinguish two types of surveying:
a) Inaccurate surveying, used in the stepped pyramids, consistent with the measurement elements 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, and the clearest clue as to the technique used for this plotting is the deviation with which they plotted the great pyramids in relation to the cardinal points.
This rotation of the bases of the pyramids has traditionally been regarded as an orientation error, but it is not. It is simply the result of applying the technique that they used, which we shall describe in this book.
The application of the technique used will help us to plot the Great Pyramids of Giza with the required accuracy and reproduce the characteristics of the original plotting.
We will then be in a position to answer other old questions:
What was the original height of the Great Pyramid?
What was the original relation between the height and the base of the pyramid?
Finally, only after identifying the construction stages of the building, which are the results of the plotting used, will we be able to analyze the techniques used to raise the blocks in each of them.
Understanding how the pyramids were plotted will enable us to visualize without much difficulty how they were built, with the simplicity and efficiency that characterized the ancient Egyptians.



In the course of each day a gnomon will produce a curved shadow path. In the course of a year, these curved paths will be found between the curves of the winter and summer solstices. The summer solstice marks the highest position of the sun in the sky, while the winter solstice indicates the lowest position of the sun in the course of the year. In addition to the Earth’s rotation and translational motion, there is also the sun’s declination, caused by the shifting position of the Earth’s axis. The sun’s declination movement is continuous throughout the year, except on the solstice days, when there is no declination.  The sun’s declination is noticeable in that the sun rises at a different point in the east and sets at a different point in the west every day. The sun’s path on the solstices is east-west, and there is no declination. For this reason it is always suggested that the pyramids were orientated during the summer solstice. On the equinoxes, the sun rises at the point closest to the cardinal point east, and it sets at the point closest to the cardinal point west. On the equinox, the movement of the sun will have a declination, just as it will have any other day for the rest of the year.


                                                        Figure 27: Shadows cast by the gnomon



The largest sundials are the most accurate, just as the largest pyramids are the most accurate in their plotting.
                                                                                                          Daniel Gerardo


Characteristics of the plotting of the pyramids of Giza:
The Great Pyramid acts like a gnomon, with its summit casting its curved shadow path on a daily basis. The size and plotting of the square base of the pyramid were calculated in such a way that the corners on the north side, points 1 and 2, touch the curved shadow path of October 11th. In the course of that day the shadow enters the pyramid by the N-W corner and exits it by the N-E corner. The same thing happens with the pyramid of Khafre, but on October 8th.On the morning of that day, when the sun reaches point 1 (see Figure), the sun’s rays strike at the same angle of slope as the pyramid’s edge, and the shadow of the pyramid’s summit falls exactly on the N-W corner, point 1. The curved shadow path and the pyramid’s N-W corner touch each other at that moment.


Figure 28: The Sun’s Declination

The curved shadow path and the pyramid’s N-E corner touch each other at that moment. As we explained above, the curved shadow path deviates somewhat towards the north due to the sun’s declination, and the chord that joins two points on this curve will also be tilted towards the north.
By drawing a straight line joining both corners, points 1 and 2, we get the north side of the base of the pyramid, which is also the chord of the curved shadow path.
The side thus obtained will not have an E-W orientation, but will deviate slightly towards the north, as will the chord of the curved shadow path. This deviation is due to the sun’s declination occurring between points 1 and 2, which can be seen in the angle between points 2 and 3. Point 3 in the sky is where the sun would be if there was no sun declination. Point 2 in the sky is the actual position of the sun with sun declination. The sun’s average hourly declination during the month of October is an estimated 0.9 degrees. In the estimated 4 hours that the sun takes to go from position 1 to position 2, the sun’s declination is 3.5 '.
The relation between the height of the pyramid and the side of the base is not accidental, but must clearly have been determined by plotting on a scale model. The curved shadow path is determined by the height of the pyramid and the day on which plotting is undertaken. The length of the side of the base must be such that it makes it possible to intersect the curved shadow path with the vertices of the north side of the square of the base. In addition, the length of the side of the base must be such that it allows the perpendicular to the midpoint of the north side to run through the centre of the base of the pyramid, and distance d must be half the side.
It must also be noted that since the curved shadow path is not symmetric, if the square for the base was plotted in an east-west direction, only one vertex could intersect it, the other vertex being some distance from the curved shadow path (25 cm). For both vertices to intersect the curved shadow path, not only is it necessary for the sides to be the right length but the square must also be rotated by an angle equal to the sun’s declination with the axis of the pyramid as its centre. The rotation is counter-clockwise because the plotting was undertaken in October with the sun declining towards the winter solstice.
The shadow cast by the pyramid’s summit allows us to plot the north side of the base of the pyramid as well as the edges of the face and its apothem. The rotation of the base of the pyramid is observed in the bottom right-hand corner of the figure.
In my view, this is the conceptual origin of the "Indian Circle" method proposed by Martin Isler regarding the orientation of the Great Pyramid. The pyramid itself is the gnomon, and its shadow – cast in accordance with Thales' theorem – was used to plot the pyramid.
When working on the model, they drew a scale circle on levelled ground with a diameter equal to the estimated diagonal of the base of the model. In the centre of the circle is the gnomon with the scale height of the pyramid. By joining with a straight line the points where the circle intersects the curved shadow path, they determined the chord, which is the side of the base. Next, they measured distance d on the perpendicular of the chord to the centre of the circle, which must be half the side obtained.
The relation between the height of the pyramid and the side of the base is unique and has to be determined by means of scale plotting. The casing of the pyramid is then plotted by casting the shadow of the model obtained.
In a 1/100 scale model the deviation is 2.5 mm, while at the base of the pyramid it is 25 cm.
  

The plotting of the pyramidal shape – such as in the pyramid at Meidum – began on the stepped core already built. The ground was then levelled within the perimeter around the core, which supports the casing. The pyramid’s uppermost piece, called pyramidion, is small, and plotting it is like plotting a very small pyramid. The pyramidion is not plotted by measuring. Due to its small size, and for greater precision, the measurements are transferred. In addition, all its dimensions, the diagonals of the base, the edges, the height and – as we will see later on – the apothems which were also plotted are accessible. A rod is used for each of these dimensions. Once the pyramidion has been placed atop the stepped core, the summit point – from which the plotting will be undertaken by means of strings – is defined. The summit point is determined in such a way that if the pyramidion’s edges are projected visually, there is enough room for laying the casing over the core.


Figure 29: Plotting from the Summit

Once the pyramidion has been leveled, it is orientated in such a way that at some point on October 11th, the shadow will enter by the N-W corner and exit by the N-E corner.
The pyramidion is a small model, a small pyramid equivalent to the Great Pyramid because their angles are the same and their dimensions are proportional (see Thales' theorem).
The right scale for the model may have been 1/100, its dimensions being 100 times smaller than the Great Pyramid to be plotted. This model makes it possible to plot the casing of the Great Pyramid in accordance with Thales' theorem. The shadows cast by the pyramidion are equivalent and proportional to those cast by the pyramid, and are used to plot it. The first step of the procedure involves accurately plotting the pyramidion and projecting it towards the base of the pyramid.

The solar shadows cast by the pyramidion must meet the requirement of good sharpness and resolution. To improve the resolution of the shadows, we will use a shadow model rather than the pyramidion.


Figure 30: The Edges form the top.
The model will consist of a square base made of fine wood and a gnomon placed in its centre. On the shadow model, the shadow cast by the gnomon is seen along its daily path. The gnomon will have a sharper tip than the pyramidion to improve the resolution of the shadow cast. The resolution of this model is similar to that of a sundial. The resolution of a timepiece this size is 1 minute of time, which is equivalent to 145 mm. The largest known sundial has a resolution of 15 seconds. The larger the timepiece, the greater the distance between hours and the better its resolution. In the case of both sundials and the pyramids, the larger their size the greater their precision. Although it seemed rather illogical when thinking of plotting the pyramids by measuring, it is now clear that when you measure time with a gnomon, that is just what happens.
The largest sundials are the most accurate, just as the largest pyramids are the most accurate in their plotting.


Figure 31: Example of pyramidion.





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Book: Plotting and Building the Pyramids of Giza


Daniel Gerardo
CopyRight 10/2016