[9 Oct 2011: I originally wrote this article
in 2007 and it was on its own web page. Recently I saw another
television special about the pyramids, and my frustration with the
astounding ignorance of the Ph.D. who wrote it moved me to re-post it
here, update it and fix the formatting.]
How to Build a Pyramid - Giza Style
by David F. Leigh
"Architect! Build me a monument! It must be 280 cubits high and outlast the ages!"
So the Pharoah has appointed you to build his next pyramid. Whe're to
start?
You live in the Bronze Age, in the year 2580 B.C. Is it even possible
to carry out the order with great precision given the tools that you
have available to you? Let's find out. We'll start with an inventory.
Your resources
- Aproximately 20,000 highly motivated manual laborers, stonecutters, foremen, and and support staff.
- Adequate food and shelter for these workers (money is no object, since it doesn't exist anyway)
- The natural resources of the land on which you live, and those of surrounding countries for which you can trade.
- Your intelligence and common sense
Your tools
- Wooden mallets
- Measuring rods and ropes in cubits (about the length of you elbow to your outstretched fingertip)
- Bronze, copper, and stone chisels, adzes, axes, choppers and pounders.
- Wooden Levers
- Rope. Lots of rope
- Wooden sledges
- Oil and water.
- Whatever you can dream up that doesn't involve steam, motors,
iron, plastic, or other modern materials. Copper, bronze, stone, wood,
leather, papyrus reed, ivory, bone, rope, lead, glue, and shell are all
acceptable materials.
- You can use techniques whose discovery is credited to people
living centuries later, so long as they are simply accomplished with
Bronze Age technology. For instance, you can bisect an angle with a
compass and straightedge. After all, improper assignment of credit
happens all the time, even today.
Making a blueprint
There
are plenty of reasons for selecting a pyramid to be our monument.
First, Pharoah's father and grandfather built pyramids.... it's a family
tradition. And pyramids echo the timelessness of mountains. And with
its sloping sides, a pyramid looks much taller to someone standing at
its base than it really is.
But there are practical
reasons, too. Two hundred eighty cubits is really, really tall. So tall
that we're not going to be able to build an obelisk that high. It would
fall over or break with the stress of trying to raise it. We can't
build in clay or brick either, since that will crack and buckle under
the weight that's placed on it. Besides, bricks crumble over time. So
we're going to be building in stone. And we're going to need a wide
base. A pyramid is
extremely stable.
Nonetheless,
we have some limits. Build it too steep and the weight will crack the
base. Too shallow and it won't be impressive. This needs to be extremely
well built and precise, both because we take pride in our work and to
celebrate the perfection of nature and our faith in the infallibility of
our ruler.
So we start with an architect's drawing on
papyrus. This will be a scale drawing we'll use to get approval from
Pharoah, and it will be important for other reasons to be seen later.
We start with our height:
two hundred eighty cubits. That's what Pharoah ordered, that's what we'll deliver.
[The
height here is completely arbitrary. We could have as easily said that
we're going to fill up a certain area, and the calculations will work
out exactly the same. However, there's no actual problem with the fact that it's arbitrary: the Pharoah wants something bigger than has ever been built before, and it should be divisible by two for the simple reason that we want, in this early age, to avoid fractions. As you'll see, even though the value of pi will be inherent in the pyramid, all of the numbers we'll manipulate are integers.]
Now for the base. We know that it'll be big, big, big, so
we're going to need to be able to measure it exactly when we scale it
up. Measuring with rope is no good... it stretches. Measuring rods have
to many opportunities for slippage. Pacing it off is a joke. I choose
to roll a wheel because an odometer is the best low-tech way of
measuring the long distance exactly. So, in laying out my blueprint, I
measure the base with a wheel, a little scaled-down odometer, one unit
in diameter. It looks a little like a pizza cutter. As it happens, 140
turns of the wheel gives us a pretty pleasing shape, just shy of 440
cubits wide at the base. The pyramid is broad but imposing. It's stable,
with a slope that's not too steep or shallow. And the math is pretty
easy, too. All integers. I'll be able to pace off 70 turns in either
direction from the center to find the edges.
I have no
intention of sticking with just a papyrus blueprint. We'll build a
number of scale models of wood or stone. Heck, we'll build a limestone
model for Pharoah to gaze upon while we're building the real thing.
[Now,
with a real pyramid we'd include a bunch of other features... a
gallery, tombs, conduits... I'll describe those at a later date. But for
right now, for the purposes of this web page, I'm building a
featureless mountain.]
Our project plan is to finish the work in about 20 years, before Pharoah
dies. To do this we're going to have to shift over 2 million stone
blocks into place. Working 360 days a year for 20 years (I'm leaving off
a few days for contingency and festivals, or whatever), we're going to
need to move about 300 blocks per day. With an average of maybe 2.5 tons
per block we're going to need teams of 20 to 25 men per block.... let's
say 25. That means 7,500 unskilled men at a time on the gangs
if each gang only delivered one block per day.
More likely you'd be able to shift a number of blocks per day. Add
rampbuilders, quarrymen, and more workers to place the stones, then some
masons to finish the blocks. I'm going to estimate 15,000 people
working on the pyramid at any one time, but I'll call it 20,000 to
include support workers... cooks, potters, doctors, etc. And I think
that's conservative. This is doable. We don't have hundreds of
thousands of mouths to feed, and given our population, we can do it
with part-time conscripted labor.... sort of like a term in the army.
At least we won't need slaves. And we can scale up or back for certain
portions of the construction.
[Just
after I first wrote this I saw a TV special about the Pyramids on the
National Geographic channel, narrated by Avery Brooks (talk about
timing!). They came to the same reckoning regarding total personnel, but
for different reasons. The producers figured that 2,000 laborers would
be sufficient to the task. If so, they'd each be shifting an awful lot
of blocks each day. As I've personally engaged in labor, and I calculate
the length that these blocks would be pulled, I conclude that you'd be
doing good to walk the distance in the time allotted per day, much less
pull a 2.5 ton weight. I think we'd need at least double the number of
laborers they're suggesting. Likewise, there are too many support staff.
People worked harder and longer... we're not talking about a 40-hour
week with weekends off. I think they were extremely light on common
labor and extremely heavy on support staff, but their total figure of
25,000 was reasonably close to mine.]
Selecting a location
The
criteria for the location our monument are going to be critical. We're
going to need a sturdy, level rock base, near a river and limestone
quarry. We need limestone instead of granite because we're using Bronze
Age tools, and granite's too hard for us to work with exclusively. The quarry will be the
source of our raw materials, but the river will bring us wood and
supplies, and (very importantly) and unlimited supply of water.
Proximity to the river means we'll have farmland on which to grow the
food for our workers. Good climate is a plus so we can get in as much
work as possible per year. Look at
this satellite image of the Giza Plateau (29°58'51"N 31°09'00"E).
Zoom out and see how it meets every one of our requirements. It's a
limestone plateau on the edge of a fertile river valley cutting through a
desert with a stable climate. Furthermore, we already own it. Isn't
that a great location? We'll use it.
Leveling the foundation
This
is a fun task! How do we level a rock the size of many professional
football fields? Well, I'd do it with a bucket of water.
Have
you ever poured water on a concrete floor? It fills every dip and
shows off every bump. So we're going to take some of that awesome
manpower we have and splash buckets of water all over the plateau. Then
it's scrape... level... scrape with our adzes, gavels, and granite blocks
until the water doesn't run or pool. It's just a lot of hard work.
However, there's nothing technically challenging to a Bronze Age
engineer. I think the scrubbed and level plateau would have been one
heck of a sight, impressive in its own right.
[The
TV dramatization had the Pharoah pounding a stake into the thick sand
to indicate True North, as he's surrounded by scrub and creosote bushes,
or whatever it is that grows in Egypt. It's a wonder they could they
didn't trip over all the plants. The Architect dangled a pendulum from
his outstretched hand to site the circumpolar stars as the pendulum
swayed to and fro. Why lay a stake that's going to be swept away when
the site is subsequently cleared? Or did they simply plan on dropping
multi-ton rocks on the plants and dunes, squashing them flat? I don't
ask for perfection, just common sense.]
Measuring the base
This is where we use our full-size odometer
one unit (a cubit) in diameter. first we do some gross measurement to
figure roughly the center of the plateau. Then we mark off right-angle
centerlines North/South and East/West. The North/South line was easy to
determine, as we have the clear desert air to help sight the North
Star. There are a number of low-tech ways to determine a perfect right
angle. The easiest is probably the compass method you learned in high
school. Our "compass" is going to be pretty big, though... we can use a
length of very stout rope for it. We would not use it to mark the long sides
of the structure since it would stretch, but the shorter length we'll
use here can be accurate enough.
[My
survey would look entirely different from the TV dramatization. Imagine
the site completely cleared of scrub and sand . Only bare limestone
remains. This might resemble a huge deserted Air Force concrete paved
storage field for B-52 bombers. There's no point in pinning down True
North until the ground has been prepared to this degree. Very small
people in the midst of this vast stone dance floor would survey the
centerline to True North. The plumb bob would be suspended from a wooden
armature and shielded from the wind. These are professionals.
Note also that we're perfectly fine using an odometer even though some might argue that the ancient Egyptians didn't use the wheel. News flash: they did. Even before the introduction of the chariot, they used plenty of wheels... just not for vehicles.]
Getting the angles right
One
of the most important things we're able to derive from the blueprint is
the angles at the base. In fact, we're going to use the drawing to
create some tools to help us in building. Namely, our angle gauge. It's
basically like a crude protractor with a plumb bob. There are a number
of variations of this tool that would work. We'll make some similar
T-shaped or A-shaped tools to make sure our blocks are level.
During building we can put this thing against the side of the pyramid
at any time and sight along it to ensure that our angles are proper. It
doesn't matter what the angle actually is in degrees or radians... all
that matters is that we exactly match the drawing. If it does, then all
four sides will meet at the exact center at exactly the right height,
assuming we measured the base correctly. There's no need for us to
devise a method of measuring the height during building, since it's
necessarily determined.
Ramping up ** (see update)
[Cutting
stone blocks isn't something that's generally disputed in a discussion
of pyramid building, so we're going to take it as a given that our
experienced Bronze age quarry workers can actually do the job we know
they did. At a later date I'll discuss the how we determined the size
of the blocks we're to use.]
We're using a ramp to
get our blocks up to the plateau and to hoist them onto the structure.
Yep, that's it. No hoists, no cranes. See, there are three things we
have plenty of as a result of our quarrying and location: sand,
limestone rubble, and water. This ramp is going to be huge, and we
really don't care, because there are far too many advantages to the ramp
to give it up. First, there's ease of transportation. With a big
enough ramp with a shallow enough slope, we have the the energy-saving
benefits of an inclined plane. Trying to lift the blocks vertically will
take far too much effort and will be much slower by comparison. Second,
we can get more workers on the task of moving each block. You can't
crowd very many people around a crane. Third, there's safety. There are
no cranes to collapse. The block is never off of the surface; if a rope
breaks, it won't fall on someone. And the shallow slope prevents a
block from sliding back very far or fast. Finally, we can depend on
unskilled labor to do the work. The architects may be professionals, but the laborers are just citizens doing community service. When
they've done their time they'll go back to farming or fishing or herding
goats.
A good portion of our crew will be engaged in a
never-ending road paving project. As we build up each level of the
pyramid, the ramp is extended to the next level. It becomes longer, and
longer as it gets higher. At the end of the project, this ramp will be
completely dismantled. We might use the material for other construction
projects, roads or paving, or we might just use it to back-fill the
quarry.
We're going to be moving our large stone blocks
with ropes, wooden sledges, and brute force provided by gangs of workers
who drag them up the ramp. We
could use some other clever techniques,
such as attaching rockers to the sides of a block to turn the block into
a wheel. The problem here is with safety. Pushing a wheel up the ramp
is ironically tougher than moving the block on a sledge. Why? Because
with the sledge friction isn't just your enemy; it's your friend. You
can rest, change crews, etc without expending a lot of energy to keep
the sledge stationary. If a rope breaks it's not a big deal in the grand
scheme of things. To overcome friction you'll be using water or oil on
the sledge runners. With the wheel rockers, you'd have to expend a
great deal of energy just to keep it stationary. It means constant
tension on the ropes at all times. And it means that a rope breaking
could be a devastating event as two tons of rock goes careening down the
ramp, killing workers as it goes, causing them to release their ropes
on
their blocks, which go careening down... you get the picture.
Ironically, for our construction purposes, dragging is the superior
technique.
The work goes something like this: some
workers build the ramp up. At the same time lots of quarrying is going
on and stones are awaiting transport. When the ramp is finished, then
the ramp-builders lay off or are reassigned while stones are dragged up.
When the level is completed, it's the same thing all over again. The
number of workers available for quarrying varies depending on how many
are needed for transporting the stones or working on the ramp at any
particular time.
** UPDATE 8/20/2013: It's unlikely that the actual pyramid had a huge external ramp. A new theory, which has plenty of credibility as it makes testable predictions, is that the pyramid contains an internal spiral ramp. This is well-thought out, and explains a number of features of the great pyramid that are otherwise puzzling (to say the least). In addition to requiring very little additional material (a ramp would only be used for the bottom third of the pyramid), it explains the "grand gallery" as a channel for a counter-weighted sledge used to hoist the granite slabs used for the roof of the King's Chamber. Also, this technique agrees with my assumption that safety is a major concern, and describes how the limestone cladding would have been placed first, back-filled with the sandstone blocks. As you can read below, I have always maintained that the cladding would have been placed as the construction progressed rather than separately, but Houdin's theory not only ups the safety, but makes construction much easier. Click on the cover of Archaeology to read more about it. The major point here is that neither magic, divine intervention, nor super-science is in any way ever required for construction of this impressive structure.
Casing the Joint
Some people might consider cladding the pyramid
with casing stones after the main structure is built. I prefer to plan
the cladding as each level is done. Again, this is a safety issue. The
casing stones are smooth. How would you possibly plan to put the stones
there after the fact? You really don't want to wind up sliding on an
unstoppable path to the plateau floor to be crushed by a falling casing
stone. Cladding as we go keeps it safe, and allows us to better track
our angle as we move up the structure.
It also allows us to build and dismantle our ramp only once.
** UPDATE 9/1/2022: I recently saw a detailed video (here's a link) of the construction including the casing blocks, and was pleased to see further confirmation that this is exactly what they actually did. In fact, the casing stones were laid first, and then the level was filled in with rough-hewn stones and mortar. We know this because the rough-hewn interior limestone lies on top of the casing stones. The Egyptians did not take the time to finish any of the interior stones (except in the corridors and burial chambers) as no one would see it. There are significant gaps which were filled in with limestone and gypsum mortar. The vast bulk of the interior is barely better than rubble. It was moved straight from the quarry without significant shaping.
Capping it off
The
capstone ("pyramidium") is the last thing we'll add. Prior to putting
it in place the architect and each team leader might sign or mark the
base. After all, this is something to be truly proud of. The Pharoah
has a surprise for us... he's provided specialists to apply gold leaf to
the capstone. The reflection of the sun can be seen for miles!
Cleaning up
Goodbye ramp. It's dismantled and hauled away by the millions of
bucketloads.. The white limestone casing is washed and polished on the
way down. The polishing is done by scrubbing the casing with large flat
stones, using the ever-present sand as an abrasive.
** UPDATE 9/1/2022: Again, this is exactly what they did. Polishing stone was done with other stones, simple as that.
What Did We Just Build? Math and Wonder in Hindsight.
Isn't this
an amazing structure? And though the construction took sweat and
determination, the design itself was dead simple, requiring only integer
math, a wheel, and some plumb bobs and string. But it was only simple
because we looked at the problem through the eyes of a Bronze Age
engineer. Imagine we show the finished product to a mathematician who
had no experience with the engineering challenges that faced us. What
does he see?
- The ratio of the base to the height is pi/2.
Comment: Sure it is! We
used half the number of turns of the odometer wheel as the number of
cubits in height we'd planned. Since each full turn is of length pi,
then the ratio of the base to height is one-half pi, or 1.570796326795.
Our design is theoretically exact. Using the values of height and base
from Wikipedia ( b=440, h=280), the calculated ratio for the actual
Great Pyramid is approximately 1.571428571429. The difference isn't
worth stressing over.
- The perimeter of the base equals the circumference of a circle whose radius is equal to the height of the pyramid.
Comment:
Naturally! The circumference of a circle is 2 * pi * the radius. We'll
just call the height "one unit", so it simplifies to 2*pi. Since each
side is to the ratio pi/2 (above), and we have four sides, then the
perimeter is 2*pi. It couldn't possibly be otherwise. Of course, that's
true for ours, but the actual Great Pyramid is an approximation.
- The stones at the base are placed with high accuracy. The variance of the length of each side is on the order of scant inches.
Comment: This is a result of
using an odometer to do the measuring. On the actual Great Pyramid, the
length of the sides vary by as much as 8 inches, which is fantastically
good for a structure of this size, but hardly supernatural.
- The angles are 51°45'27" (see note)
Comment: Well, of course.
This is because we took great care to build our angle surveying
equipment (the angles and plumb bobs we used to sight with) from the
original blueprints, which were created with a scale of our circular
odometer. Using the measurements, 200 cubits of height and 100 turns of
the cubit wheel at the bottom, we can calculate the angle as follows:
base angle = ASIN((1/2 * base)/ height)
base angle = ASIN ((1/2 * 140 cubits * pi) / 280 cubits) = 51.75751851602 degrees = 51°45'27"
This is purely determined by the
ratio of the height to the base. If it weren't this, then the
height/base ratio we noticed above wouldn't be accurate. Again, our
design is theoretically exact. Interestingly, the numbers given for the
real Great Pyramid vary according to the source. Wikipedia gives it as
51°50'40". However, if we took the reported height (280 cubits) and
width (440 cubits) values as accurate, then the calculated value should
be 51°47'12.4" . Various sources give the value as any of these, plus a
dozen more. Some are the result of approximation, poor observation, or
are simply calculated using whatever values for height and width strike
the fancy of the observer.
Remember
that the top of the Great Pyramid is missing and has to be calculated.
so either the base/height ratio is off or the angle is. So much for
supernatural accuracy.
NOTE: Actually it's going to be a little bit off due to the curvature of the Earth. See below.
- The geographical location is the "center of balance" of the landmass of the Earth.
Comment: We had some
requirements for the location, none of which involved surveying the
entire Earth. Besides, we live where we live. It's not like we migrated
to Egypt just to build this one pyramid. Go back to Wikimapia.org and
look at the location again. How many other locations in Egypt match our
requirements as well as this one? None. That's why so much of this
megalithic work is centered at Giza. And there is no bloody
"center of balance" for our globe, unless you count the center of
gravity in the core. What's described by this observation is an artifact
of the map projection being used.
-
The coup de grace: The sides bow inward ever so slightly. In fact, the arc of the bow is equal to the curvature of the Earth! Surely this couldn't possibly be a coincidence!
Comment: If it were noticeable, this would be such an amazing effect that it's tempting to claim you actually were
cleverly encoding your "astounding knowledge" of the exact curvature of
the Earth. The reality is much more mundane. Our technique for
leveling the plateau involved the flow of water under the influence of gravity.
It ensured that all points on the plateau were equidistant from the
center of the Earth. On the other hand, our method of measuring the
sides (measuring out from a centerline), is valid only for plane geometry.
On a curved surface like the Earth, though, the best definition of a
"line" is a Great Circle route. Try this experiment yourself. Cut a
perfectly square piece of paper. Then place it on a globe and pin down
the corners. What happens to the sides? They appear to bow inward, to a degree that mirrors the curvature of the surface of the globe!
So this "encoding" has nothing whatsoever to do with actual knowledge
of the curvature of the Earth... rather, it's evidence that the engineer
didn't even take it into account (and probably is totally ignorant of
it!) This would be noticeable only in a monument of stupendous size.
Update: OK, I've been asked to explain this one further, so I'm including a graphic to help folks visualize this: Start at the center and measure a line North/South, and another East/West. Now lay out a grid, starting from those center lines and working outward in both directions, making sure that all of the N/S lines are parallel to the original N/S line. Likewise, all the E/W lines are parallel to the original one (like lines of latitude on both sides of the Equator. When you look at the result you'll see that only the center lines trace Great Circle routes. The lines on the edges are bowed compared to a Great Circle route when viewed from above, just like lines of latitude. An azimuthal stereographic map projection illustrates this perfectly (think the Pan Am logo).
Compare this to Great Circle routes, which is what you'd get if you measured your straight lines with a line or surveying equipment. Great Circle routes are straight as you get on a globe, but appear to bow outward, like lines of longitude converging at the poles and spreading at the equator. Now, measuring with a wheel from a centerline might not be how the Pyramids were actually laid out, but it explains so much so easily that I double-dog-dare you to find a more elegant approach that matches the actual geometry of the Great Pyramid.
Closing Thoughts
Prior
to writing this page I looked up the dimensions of the Great Pyramid,
and but pretty much discarded everything except the height in designing
my building approach. Happily, the approach then pretty closely matched
what was actually built for the Pharoah Khufu. Though they didn't have
our advantage of 5,000 years of accumulated technology, the Pharoah's
builders were no less intelligent than modern engineers. And if I can
figure this out on my own, they could, too.
I didn't do
any particular archeological research before writing this How-To. It
wasn't necessary. My goal here isn't to tell you how the Egyptians
did build the Great Pyramid, but to show that it
could
have been done with the technology of the time. And it absolutely
could. No advanced technology, no alien visitors, no divine intervention
need apply. In fact, the argument that the Egyptians conceived and
built it themselves is so conclusive that to refute it you'd have to
actually produce a witness to the intervention. Good luck.
That
said, the building of such a structure is an astounding feat. And the
coincidental relationships of its measurements, driven as they were by
necessity, truly interesting. I don't care if somebody reads whatever
meaning they want into them, and uses them to illustrate whatever
message they have to give. Understand, though, that this is an exercise
in retcon -- retroactively assigning intent where there was none by the
builders. There's no evidence that the builders intended to encode
anything beyond what we know of their culture...
at all.
Postscript (2007):
I
think I ought to explain that, since this page has been online for half
a day and I'm getting multiple feedback from multiple directions.
I
think that, coincidental or not, such numerical relationships can be
used in an instructive way, much as
St. Patrick is said to have used the shamrock to explain the Trinity or Richard Middleton used a
deck of playing cards as a prayer aid. Our ability to use patterns in this way does not provide evidence that this is the
purpose
of the object we're using as illustration. For instance, we are a race
of people that see bunny rabbits in the clouds, yet we don't imagine
that clouds were designed to display images of bunny rabbits. That said,
the fact that something is used to illustrate a message in no way
diminishes the message itself. The illustration should help you to
understand the message; it is not the message itself.
Of
course, just because there's no evidence of something doesn't
necessarily mean it's not so, either. But it's always good to be clear
about what we can and can't prove. In the case of the Pyramids, we can
reasonably expect them to designed with the symbology of their culture
in mind (and the same would hold true for structures built in
Mesoamerica, or China, or anywhere else). I don't think it's necessary
to ascribe to the builders of these objects an understanding of concepts
that are foreign to what we know about their culture, anymore than we
can reasonably say they were incapable of work that can clearly be
demonstrated as possible using (again) what we know about their culture
and technology.
- Must the Egyptians have known the exact value of pi? NO.
- Must they have had help from aliens? NO.
- Is it valid to use the Great Pyramid to illustrate the joining of the
finite to the infinite? Definitely. Illustrate whatever you like. Just don't confuse your analogy with the builders' intent.
- Is the technique I describe
sufficient to disprove that space aliens built it? Unless you introduce me to a space alien, I'll take that as my working assumption: YES. Extraordinary claims require extraordinary evidence.
Postscript (2011)
On Hulu now is a miniseries entitled "
The Pyramid Code".
Although there is mixed in here some facts and a few valid conjectures,
In my opinion this is, in aggregate, a huge honking sack of bovine
excrement. In it is postulated all sorts of magical thinking under the
very thinnest veneer of scientific language. This does a grave
disservice to the ancients by taking their very real accomplishments,
dismissing them as impossible, and replacing them with rainbow unicorn
fantasies of resonant crystals and 'subtle energies'. Yes, it was
written by a Ph.D, Carmen Boulter, and that's a very sad thing for her university.
Copyright 2007-2011 by David F. Leigh
