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Ken Stoops
UH 350
Engineering the Pyramids
The Great Pyramids of Egypt have been a subject of fascination to the
entire world for thousands of years. For over four millennia people have marveled
at the size and height of the pyramids and pondered how such massive
monuments could be built without the existence of iron tools or wheels. From
visitors from ancient Greece to modern twentieth-century tourists many have
wondered how such projects were undertaken and accomplished so long before
even the industrious Romans or the clever Greeks. Clearly the Egyptians had a
firm grasp of engineering to enable them to build such massive structures which
did not collapse under the weight of stone. The methods they used were simple
but effective and are a lasting legacy to all generations.
There are over seventy pyramids of varying size and style throughout
Egypt.1 Most are clustered in a small region just below the Delta on the north end
of the Nile. With few exceptions they are located on the west bank of the Nile.
The pyramids range in size from 484 feet tall with over 2,200,000 stone blocks
weighing up to 70 tons2 (averaging 2.5 tons) each to small mud-brick stepped
mastabas long since eroded into odd shapeless mounds of weather-worn dirt.
It is well-known that the pyramids were built as fabulous tombs for
Egyptian pharaohs to assist in their survival in the afterlife. Construction was
most likely done by hand. The interiors of the tombs were lavishly furnished with
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as much care as the rest of the pyramid. Building, furnishing, and maintaining a
pyramid was enormously expensive, as well as time-consuming and labor
intensive, so most of the pyramids were built during the Old Kingdom when labor
was plentiful (labor during later periods was diverted to the military). Later
pharaohs were buried in much simpler tombs cut into rock cliffs.
Two distinct styles of pyramids can be distinguished from their shape and
time of construction. The oldest kind is called a stepped mastaba and is basically
composed of layered pedestals stacked to form a tapering structure.3 These are
roughly pyramid shaped but do not always have a square base (the square base
became universal as the true pyramid form became more popular). The
development of the stepped mastaba marked the shift to building in stone from
baked mud-brick. Working in stone required a much greater labor force and also
required the use of stonecutting tools to shape the blocks. Most of the step
pyramids date from the Early Dynastic period and are generally somewhat
eroded due to their advanced age. The step style died out fairly quickly and
progressed to the adoption of the more familiar pyramidal style, although it
appears that all of the pyramids contain a step pyramid inside as their structural
core.4 The original and most famous example of the step pyramid is Zoser’s
pyramid, built by the pyramid’s inventor Imhotep.
The second style of pyramid is the familiar square-based triangular prism
pyramid. Containing the same internal structural design as the step pyramid,
these pyramids are smoothly sloped on the outside but tend to be much larger
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than the earlier step pyramids (this may be a testament to the spectacular
success and popularity of the design). The overwhelming majority of the most
famous pyramids are of this style, such as the Great Pyramid of Khufu (Cheops)
at Giza. A variation of this style is found in the Bent Pyramid of Seneferu at
Dashur, which features a strange change in angle partway up its height (note that
this pyramid is an oddity and may have been altered during construction to avoid
a structural failure as is discussed later). Most of these pyramids are well
preserved; the best example is the Bent Pyramid.5 Pyramids built during the
Middle Kingdom tended to be poorly constructed and many of them are badly
eroded, such as the mud-brick pyramid of Amenemhat III at Dashur, which has
largely crumbled into a colossal heap.6
The construction of the pyramids was done entirely by manual labor. Draft
animals were not used at the time and so all heavy labor must have been done
by hand7. Contrary to popular assumption, there is little evidence of the use of
slavery to build the pyramids. There was very little slavery in Egypt at the time,
and most scholars agree that the pyramids were built by Egyptians in service to
their god-king the pharaoh. As Mendelssohn noted, it would be difficult to
maintain an enslaved population in good working order for the amount of time
needed to build the dozens of major pyramids.8 The workers were most certainly
Egyptian farmers/peasants who were drafted or possibly volunteered to work an
the pyramids during the time of the Inundation when the fields were underwater.
The workers were housed and fed by the government and there are records of
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the workforce being supplied with grain, beer, and clothing in return for their
labor.9
The pyramids were built almost completely out of limestone, with the
exception of certain parts of galleries and passageways made of granite. Granite
is much harder and more durable than limestone as a building material but may
also have been used for its more exotic appearance. The main structural support
of the pyramids comes from a massive core built from low-grade limestone
quarried somewhere near the pyramid. This stone is roughly cut in many cases
but carefully fitted on the top and bottom to improve structural integrity. The
polished outer covering, most of which has been removed by stone robbers, was
always made of high-quality limestone brought from Tura. Granite, when used,
was quarried near Aswan.10 The huge holes left from quarrying the stone for the
pyramids can still be seen at Tura, Ma'sara, and Beni Hasan along the Nile
River.11 The large-capacity pit quarries, such as one at Giza near the major
pyramids, may still exist but are probably mostly filled in with sand.12
Stoneworking in ancient Egypt was difficult at best. The hardest metal
available was copper, although a very hard rock called dolerite was also used to
chip away other stone. Surface work and finishing was done by chipping with
copper chisels and dolerite pounding-balls.13 Dolerite could also be used to
polish the surface by grinding. Quarrying was done in two main methods, both
using the time-honored technique of wooden wedges soaked in water. When
pounded into crevices dug out with copper chisels, the wood was soaked with
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water, causing it to swell and break the stone, hopefully along an even plane.14
The lower-grade limestone was quarried using the open-cast method. In opencast quarrying the stone was dug in pits on the open ground and from
outcroppings. This method was used for the majority of the stone and had the
advantage of being able to provide abundant amounts of stone continuously
because the mining could take place over a large area on the surface.15 The
better limestone at nearby Tura was painstakingly carved in deep mines out of
the cliffsides, leaving behind huge hollows which can still seen today. 16 The
granite, when used, was brought by boat from cliffsides at far away Aswan, over
500 miles away. The granite was much harder to quarry than limestone due to its
extreme hardness. While speculations vary, the granite was probably chipped out
using dolerite pounding-balls and possibly using copper chisels, although the
copper would have fared quite badly against the hard granite. The good granite
could have been exposed by heating the surface with fire and the throwing water
on it to flake off the surface layers.17
Theories on hauling the huge stone blocks seem to vary more than any
other subject. Animals were not used for labor, and there is no evidence that
either the block and tackle or the wheel were known in Egypt during the Old
Kingdom. Timber was available but not plentiful, and in most cases would have
been insufficient for use in any derricks or cranes capable of lifting such heavy
blocks.18
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Most scholars agree that the blocks were brought to the pyramids’ site
either by dragging overland or by boat via the Nile River. It seems most probable
that the blocks were wrestled onto wooden sleds after quarrying and simply
dragged by teams of men to their destination.19 While the exact amount of men
used for pulling is not known, teams may have numbered up to several hundred
for some of the larger blocks. Drawings have been found from other Egyptian
stoneworking projects showing some kind of lubricant being poured ahead of the
stone.20 To make the travel easier, they may have laid some kind of permanent
or semi-permanent path over which to do the dragging. It has also been
suggested that the stones were dragged on top of rolling logs as opposed to a
simple sled. This, too, sounds reasonable but was still greatly labor-intensive.
A considerable amount of the stone used came from some distance away
or on the opposite site of the Nile. The Egyptians were skilled boatmen, as their
entire country was aligned along the Nile River, so transport by boat was an
obvious choice.21 As the majority of the work on the pyramids was done during
the time of the Inundation of the Nile, this was also fortuitous. With the swollen
Nile flooded well past its banks, it would have been possible to float the stone
almost all of the way up to the base of the pyramid! Ancient drawings of giant
Egyptian stone obelisks mounted aboard ships have been found, so it is known
that they used boats as bulk transport.22
Most scholars also agree that the pyramids could only have been
assembled through the use of enormous ramps such as the Romans used much
later at Masada. Probable designs for these ramps vary from spiral-shaped
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ramps curling around the perimeter to a single large ramp going up the side,
probably in the direction of the source of stone.23 The ramps would have to have
been made of rubble or compacted dirt. Certainly desert sand would not have
been practical. The surface of the ramps may have been laid with logs or
planking to prevent the heavy blocks from sinking in to the soft ground. Many
people were probably injured and work lost from collapses of these artificial
mountains, but very little historical information is known about the Egyptians'
methods. Another more unlikely theory suggests that the stones were levered up
the sides of the pyramids using wooden levers, but this would doubtless have
been even more difficult and unreliable than the effort of building a giant ramp.
Much has been speculated about the Egyptians’ relatively advanced
geometrical and mathematical skill which was necessary to plan and properly
execute the construction of a pyramid. While the Egyptians were clever in a very
practical way, they were apparently largely unaware of most of their genius. For
example, all but two of the pyramids are built at an angle of 52 ½ degrees, an
angle which leads to some astonishing values in PI, a transcendental number not
to be understood until thousands of years later. While numerous scholars and
crackpots have devised all sort of explanations for this, it can be very well
explained by the use of a circular measuring tool. The most reasonable
explanations stem from the use of a circle in the design or measuring of the
pyramids. The rope measuring standards of the day were known to stretch, so for
critical measuring the more reliable use of a rolled log or drum could be used for
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more consistent measurements. This would lead, almost accidentally and
probably unknowingly, to perfect numbers in PI.24
The Egyptians had easy access to a number of very simple but effective
means of ensuring that the pyramids were built according to exact specifications.
There were standardized measurements known as cubits, approximately 52 cm
long (the length of a forearm). Long-distance measurements were done using
flax or palm fiber ropes.25 Despite the obvious inaccuracies they would have
faced, the pyramids are accurate to within 8 inches in 755 ft, a less than . 01%
margin of error.26 First of all, the perimeter and base of the pyramid had to be
squared out and leveled and the sand was cleared away to expose the
underlying rock, which served as a natural foundation. Next, the base had to be
measured to a perfect square. While it is easy to form a perfect rhombus through
the simple measuring of all four sides it is more difficult to ensure that all of the
corners are at 90. The actual method used for this is uncertain, but it is known
that the Egyptians had discovered the use of so-called Pythagorean proportions
to form right triangles. To form a level base they dug trenches and filled them
with water. The surrounding rock was then chipped away to establish a very
consistent level foundation.27
Interestingly, all of the pyramids are carefully oriented with respect to the
solar axis. Each side faces one of the cardinal points, and most of the entrances
face north. The alignment was likely one by sighting the path of the sun or stars.
The Egyptians seemed to be aware that the stars and sun rotated about an axis
in the northern direction and may have plotted their paths to establish the point of
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the compass (actually, the compass was not known in Egypt until much later) at
the center of the arc of stars, conveniently perpendicular to the rising and setting
of the sun28
The problem of making sure that all four sides of the pyramid came to a
perfect point was also difficult. Constructing four concurrent angled planar
surfaces in such a way that they ,meet at a point with no visible error or option for
correction after initiation is tricky even today. Nevertheless, this was solved in a
way that would seem almost evolutionary to a historian. All pyramids begin as a
stepped mastaba design. First, successive layers of square platforms are
assembled to form a solid structural core. A tall pole like a surveyor's rod was
probably placed at the easily found center at the top. The sides were then
sighted to this standard to eyeball the correct angle for the outer layers. 29 The
gaps in the steps are then filled in with packing stones, and the final surface was
done with fitted and polished Tura limestone to form a smooth angled plane on
all four sides.
Working in stone raises a number of critical problems. Most importantly, it
is extremely heavy. While this is a concern for the laborers, it is of foremost
importance that the design be carefully planned in order to handle the millions of
tons of crushing weight bearing down on the foundation and lower parts of the
pyramid. The pyramids are not simply stacked stone in a simple toy block
arrangement. This oversimplified model is too small and lightweight to
demonstrate the enormous internal forces involved yet also very easily shows the
weakness of such a childlike design, as it can be easily knocked over with the
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application of even a small load. Depending on the fit of the blocks, significant
internal lateral forces can accumulate, which can be strong enough to cause
slippage and even a collapse if not dealt with carefully.30 This is the same
principle that causes buckling in steel beams in modern buildings. If the stones
are perfectly squared and smoothed, then it might be theoretically possible to
simply stack the stones like toy blocks, but this is hardly possible without modern
precision measuring. Even then it would be of questionable architectural wisdom,
since this structure more accurately resembles a cluster of independent towers.
Lateral forces will develop when a stone is laid in such a way that it is not
perfectly level or square. A slight slope will cause a small tendency to slide to the
side, resulting in a plastic flow of the entire structure.31 Although the frictional
forces are usually sufficient to hold the stone in place, if enough load is placed on
the stone, the stresses developed can cause a cracking or possibly even a
shifting of the stone. In addition, an unfinished or poorly dressed stone will have
much smaller contact points resting on the tops of bumps and irregularities,
resulting in load concentrations tens or hundreds of times larger than normal
which can be enough to cause crumbling, cracking, or even splintering of the
stones.32 Certainly such small inaccuracies would have been common in
stoneworking in ancient times and in sufficient quantity could have caused a
general (and often catastrophic) failure of the structure. Note that although the
stones in the pyramids are extraordinarily well done and nearly perfectly fitted,
this does not ensure that any or all of them are either level or square. Also note
that while the visible outer stones are perfectly fitted, especially the decorative
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casing stones, it would be much more time-consuming to perfectly fit the millions
of structural stones that compose the bulk of the pyramid.
The Egyptians used a variety of different techniques to counteract these
unseen internal forces. Perhaps the most useful and certainly the simplest was to
stack the blocks in such a way that they overlap and interlock with one another.
Even without the use of any kind of binding or glue, the frictional forces alone
were enough to give the structure a much improved overall stability and integrity.
This method is seen in all examples of masonry found today. This design is even
strong enough to allow open spans to be built out of relatively small blocks, a
major improvement over block-and-lintel construction, which requires a single
gigantic piece of stone to bridge a gap.
The second technique, quite possibly a stroke of genius in such primitive
times, was to angle the stones inward.33 Contrary to common sense, an angled
base is far safer than a perfectly level base. Angling the stones inward quite
effectively counteracts the internal lateral forces by compensating for their
altering of the direction of loading, much as bends on the highways are banked to
allow higher speed travel. By ensuring that loads are transmitted normally (at
right angles) to the stones, the lateral frictional forces are eliminated (another
consideration in modern roads preventing motorists from unexpectedly flying off
of bends on slippery roads). The main structural use of this technique is in heavyduty buttress walls in the central core.34 Although these were hidden, they
functioned just the same as buttresses in much later Gothic cathedrals by
channeling outward pressures downward into the foundation. These are a
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fundamental element of the design of the pyramids, a lasting architectural
contribution from lmhotep in the design of the first pyramid.
It is also important to note that in many cases angling the stones will
cause an increase in compressive loading. As with concrete, stone is extremely
strong in compression and yet quite weak in shear and (in the case of block
construction without mortar) practically useless in tension. Angling the stones
situates them in such a way that as much load as possible is compressive and all
other stresses are minimized. In engineering this is described by maximum
normal stress and Mohr's circle. This is also the fundamental element in the
strength of the arch, which was understood until nearly 2000 years later in the
Roman period.
The chambers and passageways inside the pyramids are also subject to
tremendous pressure from the stone above. To redirect the weight around the
cavities, special layers of big blocks were placed above them. The entrance to
the Great Pyramid of Khufu at Giza clearly shows layers of gigantic rectangular
blocks in an inverted “V” above the passageway. The tomb chamber in at least
one pyramid has a roof composed of horizontally laid layers of converging
blocks. Both of these methods redirect the load sideways into the surrounding
stone. This works on the same principle as inward angling except that these are
outward sloping. Thus they divert the weight outward instead of inward, which for
an internal void is desirable.35
The chambers and passageways were most certainly built into the
pyramids' substructure during construction. This allows for the roof blocks to be
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situated on top of their supports and integrated into the design and also quite
obviously means that the chambers were built without any load on top of them. In
some cases granite was used in place of limestone for the inner passageways.
Cedar logs have also been found braced horizontally across the tapering roofs of
a few tomb chambers in unfinished (unsafe/failed?) pyramids, but speculation
holds that these were not structural but merely spacers used during
construction.36
It is a credit to the engineering genius of lmhotep that the very first
pyramid built appears to have been completed flawlessly without any
adjustments needed during the course of its construction. His basic design was
followed in just about every pyramid built thereafter. There were, however, some
significant failures.
The ruined pyramid at Meidum is the best example of a catastrophic
failure in ancient engineering. The second pyramid ever built, it was such a
disaster that the surrounding area was abandoned and the site shunned until
modern times.37 This pyramid appears as a large square tower built upon a small
hill This is no a hill at all but the collapsed remains of the outer shell of this illfated venture. While speculations on this vary (some early scholars attribute the
entire affair to stone robbers), it seems that the pyramid was built in several
stages. First completed as a stepped mastaba, the first stage very much
resembled lmhotep's original design with one small deviation. Very close in size
and design, this pyramid contained half as many internal buttress walls as the
other and also contained poorly squared stones.38 This is not a critical difference,
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as most of the first stage still stands, but it was a poor foundation for the
subsequent additions. The next stage, which only partially remains, was simply
an enhancement of the first and added to its size. The final stage, which has
entirely fallen away, gave the monument a true pyramidal shape by smoothing
the outline and filling in the steps.39
There were some rather obvious mistakes made during the ongoing
construction. First of all, each stage was fully completed before the next was
begun. That is, the outer surface was smoothed to give a pleasant finish. The
next stage was built against this surface but had no structural connection to the
underlying framework. This created disastrous slip planes along which a collapse
was inevitable.40 In addition, the steps at each layer were outward sloping
(apparently for drainage during rain) and thus directed the load in an entirely
unsuitable direction.41 Second, the last stage extended beyond the bedrock
foundation and was built upon a stone foundation laid in the sand.42 The exact
cause for collapse is not known, but any one of these factors alone could have
destroyed the pyramid. It does seem that the most likely impetus was a buckling
in the outer mantle (third stage), causing all of the stone in the third stage to
come crashing down, partially destroying the second stage as well. This could
have been caused by the lateral forces from the poorly squared internal blocks
pressing outward against the thin shell on the outside.43 Because of pile of rubble
it is difficult to tell if the failure was caused by a shifting in the foundation of the
outer mantle. In any case, the Egyptians were quite aware of the causes and
took drastic steps to avoid this kind of accident in the next few pyramids they
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built,44 before returning to lmhotep's original design and also improving the
quality of their construction to the point where they could safely modify his design
to create the larger, angle-sided true pyramids at Giza.
The next pyramid to be built was the Bent Pyramid at Dashur. One theory
holds that this pyramid was nearly completed at the time of the collapse of the
one at Meidum. Frightened that another disaster could happen, the design of the
Bent Pyramid was suddenly altered from 51' 52" to a safer, lower angle of 43 1/2'
in hopes that this would avoid another calamity. Another, possibly corollary,
theory suggests that dangerous cracks or problems were forming and the slope
was lessened to ease the strain.45 Of course, other theorists simply state that the
pyramid was rushed to completion or the unusual design was even deliberate.46
Nevertheless the pyramid immediately following the Bent Pyramid was the Red
pyramid, built entirely at the lower angle of 43 1/2', which also suggests a sudden
turn to the cautious after the pain of the catastrophe at Meidum.47 All other
pyramids were built at 51' 52", so probably the Egyptians learned from their
experiences and were quite successful from there on, including the excellent
Great Pyramids at Giza built shortly afterward.
Although much has been said about "an ancient civilization as advanced
as our own" there appears to be little acceptable evidence to support this. Such a
project as a pyramid is not really unusual except for its enormous size. It seems
that many people would prefer to believe in some alien or supernatural aid rather
that comprehend that ordinary men would dedicate so much work and effort to
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the exaltation of their god-king. The building of the pyramids is an entirely
attainable project and simply requires a dedication not seen in our world for
centuries. The ancient Egyptians' grasp of mathematics, geometry, engineering,
and common can-do is impressive, even if they themselves were not fully aware
of the methods and principles they employed. Similar feats have been
undertaken by other civilizations worldwide and simply demonstrate the capability
of a unified society with the dedication and commitment to do what they felt was
important in life.
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Bibliography
Mertz, Barbara, Red Land, Black Land, Coward-McCann, Inc, New York, 1966
Silverberg, Robert, Empires in the Dust, Chilton Company,Philadelphia,1963
Mendelssohn, Kurt, The Riddle of the Pyramids, Praeger Publishers, New York, 1974
Weeks, John, The Pyramids, Lerner Publications Company, Minneapolis,1977
Pace, Mildred Mastin, Pyramids Tombs for Eternity, Peter Bedrick Books, New York, 1981
Fix, Wm. R., Pyramid Odyssey, Mayflower Books, New York, 1978
Andreu, Guillemette, Egypt in the Age of the Pyramids, Cornell University Press, Ithaca, 1997
Johnson, Paul, The Civilization of Ancient Egypt, Atheneum Press, New York, 1978
Macaulay, David, Pyramid, Houghton Mifflin Company, Boston, 1975
1
Weeks, p 7
Fix, p 12; Johnson, p 50; Silverberg, p 35
3
Mendelssohn, p 40
4
Mendelssohn, p 117
5
Pace, p 56; Weeks, p 5
6
Andreu, p 42; Mendelssohn, p 18, 138
7
Pace, 44, 63; Weeks, p 30
8
Johnson, p 50-51; Mendelssohn, p 147
9
Johnson, p 51
10
Mertz, p 222-223; Pace, p 61; Weeks, p 10
11
Weeks, p 22, 25, 44
12
Andreu, p 40; Weeks, p 44
13
Weeks, p 23-27
14
Mendelssohn, p 125; Pace, p 61; Silverberg, p 35; Weeks, p 24
15
Weeks, p 25
16
Macaulay, p 18-21; Weeks, p 24-25
17
Weeks, p 26
18
Mendelssohn, p 121; Weeks, p 30
19
Andreu, p 41
20
Andreu, p 31; Martz, p224; Pace, p 63; Weeks, p 30, 32
21
Martz, p 232-234; Weeks, p 28
22
Andreu, p 37; Johnson, p 49
23
Macaulay, p 38-39; Mendelssohn, p 142-143; Mertz, p 224-225; Weeks, p 38-39
24
Mendelssohn, p 64-74; Pace, p 57
25
Weeks, p 20
26
Johnson, p 50; Mertz, p 220
27
Macaulay, p 25; Pace, p 58; Weeks, p 17-18
28
Macaulay, p 22; Pace, p 59-60; Weeks, p 19
29
Mendelssohn, p 116
30
Johnson, p 49; Mendelssohn, p 98-99; Weeks, p 34-35
31
Mendelssohn, p 110
32
Mendelssohn, p 98
33
Mendelssohn, p 99
2
18
34
Johnson, p 49; Pace, p 64
Macaulay, p 35, 40-41; Mendelssohn, p 124
36
Mendelssohn, p 92
37
Mendelssohn, p 79
38
Johnson, p 49-50; Mendelssohn, p 89, 101
39
Mendelssohn, p 82-88
40
Mendelssohn, p 82-83, 101
41
Mendelssohn, p 103
42
Mendelssohn, p 102
43
Mendelssohn, p 103
44
Mendelssohn, p 121
45
Mendelssohn, p 118; Pace, p 54
46
Johnson, p 53; Mendelssohn, p 118
47
Mendelssohn, p 120
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