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Theory 3
Construction Time
It is interesting to realize that over 70% of the time required
to complete the entire pyramid went to bring the pyramid to just one
third of the final height.
For the pyramid to reach 1/3 of its final
height
70.37% of the total blocks of the pyramid had to be in place.
Here is a simple formula for calculating proportion of
the volume of the unfinished pyramid Vh
to the volume of the finished pyramid V:
Where x = h/H is the ratio of the "unfinished" pyramid
height to the final height (note: 0 < x <1)
H= Height of the finished pyramid
h= height of the unfinished pyramid
Vh is the volume of the
unfinished pyramid at the height h
V is the volume of the completed pyramid at the height H
This graph credit goes to "Draw
Function Graphs":
http://rechneronline.de/function-graphs/
Here are a few interesting results of the
equation above showing relationship between the height
of the unfinished pyramid (h) and its volume (Vh
)
|
x: |
h: |
Vh |
| 1/4 |
1/4 H |
57.8% |
| 1/3 |
1/3 H |
70.4% |
| 1/2 |
1/2 H |
87.5% |
Using this simple equation we can see that 57.8% of the total
pyramid volume (equivalent of its mass) is concentrated in
the pyramid that reached only 1/4 of its final height.
When the pyramid had only 1/3 of its final height, 70.4% of the
total blocks of the pyramid were already in place!
Assuming
even rate for adding blocks to the structure, it took twice as
much time to reach 1/3 of the final height of the pyramid than to
finish it from that point in time.
According to many estimates it took 30 years to build the Great
Pyramid.
The above reasoning indicates that after 20 years of construction
the pyramid height reached only 1/3 of the final height.
* * *
Proof of the equation:
The laws of solid geometry result in a common equation for
the volume of the pyramid:
V=A*H/3
(V = Volume of the entire pyramid, A = area of base,
L = length of the square base, H = height from base to apex)
A=L*L=L2
therefore
V= L2
* H/3
Volume of the pyramid (Vh ) at the height
h = x*H:
Vh = V - Vt
where Vt is the volume of the
"missing" part of the pyramid:
where C is the length of the square top of the pyramid
at the height h
(it is the base of the "missing" part of the pyramid
h =DE on the drawing below.)
From the drawing above we can easily see
that:
Note: above the C is the length of the base of the "missing" pyramid
= DE
Therefore
The volume concentrated in the lower (finished so
far) part of the
pyramid at
the height h (h=x*H, where H is the final height) compared to the
volume of the finished pyramid can be expressed as:
Therefore
Pyramid Construction Theories - Introduction
The task of building a pyramid consists of four distinct stages:
- leveling the base (foundation)
- alignment
- cutting the stones
- transport of the stones to the construction site
- moving and placing the stones in place (construction method)
Cutting the Pyramid Blocks
An average stone block used to build the GP weighs 2.5 ton.
There are few theories how these stone blocks were made:
- Cut in the quarries
- Casted like a modern concrete
Looking towards Khufu's Pyramid an area in the forground shows
where blocks of limestone have been quarried.
Source:
http://www.egyptarchive.co.uk/html/khafre_09.html
Moving Big Stone Blocks
Moving big stone blocks requires energy to conquer:
- friction (horizontal movement)
- gravity (lifting)
- both (e.g. when using ramps)
To make this process efficient this energy should be as small as
possible.
Transport of huge stone blocks in the ancient times was
very likely accomplished by using basic laws of physics.
Long distance transport was made possible by use of boats and/or
floating rafts.
Papyrus boats with the stones tided up underneath, waiting for
high level of the Nile's.
Copyright 2005-2006 by Andrzej Bochnacki.
This image is based on information from David Jeffreys,
Institute of Archeology, University College, London; Oriental
Institute Computer Laboratory, University of Chicago and
Archeological Graphic Services. NOTE: Vertical scale exaggerated to
show ancient Nile river channel (on the left).
Moving stone blocks on land was possible by use of sleds
and ramps (with lubricating agents).
Cedar sled from Lisht.
© Copyright Dieter Arnold, Building in Egypt, p. 276
Moving a statue in 12th Dynasty Egypt. Please note the person
pouring a "lubricant" in front of the sled.
Effectiveness of Lubrication
The Racetrack is a unique attraction of Death Valley National
Park
where rocks slide across the ground all by themselves. Although no
one has recently seen them slide across the ground, it's the wind
blowing through the canyon that makes them move after rain wets
dried up silt.
With a friction level reduced to nearly to zero, even a finger could
push a very heavy rock.
Pyramid Construction Techniques
How the Great Pyramid at Giza was constructed, with resources
available to the ancient Egyptians? Was this a new method different
from the methods used to build other pyramids? Here are few
intriguing images which can suggest the answer.
It seems that over the ages, architects and builders used a
different form of construction in the pyramids.
Although the use of ramps is commonly accepted method of getting
the heavy stone blocks to their final location within the pyramid,
the arrangement of the ramps is in much dispute.
The earliest form of pyramid, the step pyramid, dates back to the
3rd Dynasty, and consists of several steps. This pyramid type was
common in the Mesoamerica.
The Step pyramid of King Netjerykhet Djoser at Saqqara
The Temple of Kukulcan (El Castillo), Chichen
Itza, Mexico
In building this pyramid Senusret II's architects took advantage
of a natural stump of yellow limestone that they cut down into four
steps to serve as the pyramid's base core. Mudbrick was used to
build the upper part of the core, and as several pyramids before,
wings were built out from this core and cross walls within the wings
were built to form a framework. The resulting sections were then
filled with mudbrick.
Ruins of the pyramid Senusret (Senusret Shines)
Also like some prior 12th Dynasty pyramids, the casing was set
into a foundation trench at the base of the pyramid. Most of the
casing was carried off to build a structure for Ramesses II, though
parts of the black granite pyramidion that set atop the pyramid have
been found. There was also a cobble filled drainage ditch around the
pyramid that was filled with sand to channel rain water.
(Source:
http://www.touregypt.net/featurestories/senusret2p.htm )
The early locals of this century called the Meidum Pyramid el-haram
el-kaddab, meaning "false pyramid" and because of its form, it
attracted attention as early as the Middle Ages from travelers. At
the beginning of the fifteenth century, the famed Arab historian
Taqi ad-Din al-Maqrizi thought it looked like a huge, five stepped
mountain. However, it eroded so badly that when Frederik Ludwig
Norden visited it in the eighteenth century, the pyramid seemed to
have only three levels. But it was not weather that eroded it so,
but human beings.
Snefru's Meidum Pyramid in Egypt near the
Fayoum
The pyramid was originally a seven step structure built on a rock
foundation, but perhaps even before it was finished, an eighth step
was added. Each of these first two stages, designated E1 and E2, was
intended to be the final structure. Yet, the pyramid was eventually
rebuilt in order to transform it into a true, smooth sided pyramid.
However, in contrast to E1 and E2, the extension designated E3 did
not rest on a solid bedrock foundation, but on three layers of
limestone blocks laid on sand.
Even more strangely, while the E1 and E2 stage blocks were angled
toward the middle of the pyramid, as in the case of Djoser's Step
Pyramid complex at Saqqara, thus significantly increasing the
structure's strength, the E3 blocks were laid horizontally. (Source:
http://www.touregypt.net/featurestories/meidump.htm )
According to Charles Rigano, the architects and builders of the late
Old Kingdom used interior step pyramid method first (before applying
final casing) in order to mark the true apex which allowed greater
control over the true shape of the pyramid: http://egyptstudy.org/Ostracon/RiganoStepPyrWinter2003-04.pdf
The pyramid and mortuary temple of Sahure, at Abu Sir ©
This new method of pyramid construction can be illustrated by
the Great Pyramid and by later built Pyramid of Khafre.
Khafre was a son of Khufu and his is
the second largest known pyramid in Egypt, only approximately 10
feet shorter than the Great Pyramid. Remnants of its original casing
are still apparent at the top of the structure. After the
accomplishment of the building of the Great Pyramid, King Khafre had
a hard act to follow. Khafre rose to the occasion by building his
pyramid on higher ground giving the illusion that his pyramid was
taller. He also encased the lowest two courses in granite. The
pyramid itself lacks the degree of precision that was present in the
Great Pyramid. Its angle is slightly sharper and the four corners
are not as well aligned to accurately meet the apex. Therefore it
exhibits a slight twist at the top. (
Source )
Pyramid of Khafre (Chephren).
Original Height: 143.5 m (470.79 ft)
Current Height: 136.4 m (447.50 ft)
Length of Side: 215.25 (706.19 ft)
Angle: 53° 10’
In 1837 the English army officer Howard Vyse searched for a
Southern entrance to the Great Pyramid by using dynamite to blast a hole.
He found nothing and soon gave up. The resulting scar has come to be
know
as 'Vyse's hole'.
Source and Copyright
http://www.egyptarchive.co.uk/html/great_pyramid_04.html
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