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no solution, due to the lack of historical documents, and with no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?
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 GerardoCharacteristics 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.
