Work Like an Egyptian

Photo courtesy of http://liberato.org Ricardo Liberato

I just saw yet another documentary on building the pyramids. Once again, it was all about the age-old question of how they got all those blocks up there. It’s been a favorite subject of archaeologist and crackpot alike since the return of the scholars who went to Egypt with Napoleon’s army in 1798.

These notes are about how implausible most of the last couple of hundred years of theories are, and how strange is the persistence of the same basic idea, given that simple solutions that would actually work are fairly obvious.

A more plausible way is presented here.

The Problem

The Great Pyramid of Cheops, like the others, is mostly made of stone blocks roughly the size of refrigerators and the weight of a Cadillac Escalade. A few of the blocks weigh as much as 80 tons, but most of them are about 2.5 tons, and there are something like 2.3 million of them.

How did the ancient Egyptians get them up there without machinery? The hard-core obsessed also fret about how they got the big ones manhandled into place, and about how they got the massive pyramidal cap-stone up there with nowhere to stand, etc., but the canonical question is ever how they got all the little stones up there.

The Problem

The image you still see the most often is of thousands of guys in diapers dragging the blocks on rollers up a mile-long earthen ramp.  I don’t think anyone takes this seriously anymore.

Cheops is about 500 feet tall, which is 210 courses of blocks. The first problem is that a five hundred foot high earthen ramp would bulk larger than the pyramid itself–a crazy amount of dirt to move around even if you could engineer a pile of dirt strong enough to move that kind of weight on, which is extremely doubtful.

Another problem is that for the higher layers, the ramp would have to be wider than the narrowing pyramid itself in order to bear the pounding of workers dragging millions of tons of stone over it. This means that the ramp would have to more or less bury the pyramid.

So earthen ramps are out.

On the positive side, the majority of the mass in a pyramid is near the bottom, so the average block is not as high up as one might guess.  With 210 layers, half the blocks are below the 50th layer, and  80% are below the 70th layer. According to Euclid, the center of mass of a pyramid is h/4 above the base, so the average block is only going to ascend 1/4 of the height of the pyramid. Nevertheless, any ramp still has to go at least near to the top, and even the top 1% of the pyramid volume is still 25,000 blocks.

The fundamental problem is time, not the difficulty of moving a block. The project took about 23 years and the number of blocks in the pyramid divided by the number of work-days and daylight hours, etc. works out to raising a block to its final place every 90 seconds, all day, every day, for the duration.  That would rival the rate at which a modern container-port with multiple specialized cranes and thousands of trucks can empty a container ship.

Ninety seconds per block is very optimistic because big sections of the work, such as building the foundation, cladding the structure, tearing down and hauling away the hypothetical ramp at the end, etc. had to come either before or after the stacking phase. It also assumes nonstop traffic, that is, that teams don’t rest on the way up, ropes never break, and they never have to stop traffic to raise the ramp or do maintenance, etc. Another inconvenience is that a single ramp also forces you to complete as you go up because you can only get off at the top.

It’s not a plan.

Modern Solutions

More up-to-date proposed solutions dispense with the huge earthen ramp by winding the ramp around the pyramid, or building switchback ramps that lean against the pyramid. There’s another variant that has the winding ramp inside the pyramid. These ideas eliminate the giant earthen ramp that wouldn’t work anyway, replacing it with a more economical narrow ramp that leans on the building, but they don’t touch the bottleneck problem.

You can check out some of the many variants of the ramp scheme here: Building the Great Pyramid

You might be able to use multiple earthen ramps for the lower portion, and save the winding single ramp for the upper layers, but with a winding ramp you’re limited to one path. There are more than a million blocks higher than a hundred and twenty feet, which is already extraordinarily high for anything made of earth.  Most of these schemes also have lots of 90-degree turns that would slow things down even more.

Even if you overlook the bottleneck problem, packed dirt isn’t very strong. You can’t build a narrow pile of packed earth hundreds of feet high because it won’t bear its own weight, let alone many tons of moving stone, men, and draft animals. Mud-brick is stronger, but the tallest mud-brick structures in the world are not much more than 100 feet high, and few are as much as even 50. Moreover, you can be sure that the owners of 100-foot mud brick buildings don’t allow grand pianos on the top floor.

An Easier Way

The model pictured below is based on a different principle.

This proposed method uses each layer of the pyramid as a base for a short ramp that is the height of a single layer of stone. A block is dragged up a greased ramps one level high, then levered sideways the width of a ramp to put it in position to dragged up the next ramp.

The ramps would be made of wooden timbers, and each block would be strapped to a low sled like a shipping pallet. Grease would ease the sled’s slide up the ramp.

We take it as a given that it is not difficult to construct wooden ramps of adequate strength. today you could make it out of 2×6 lumber from the big box store.

The issues open to question are

• Is it practical to drag stones weighing two or three tons up such ramps, and how would it be done?
• Is it feasible to raise 2.5 million blocks to their ultimate destinations within the required time?

Dragging Stone Up The Ramps

The ramps in the model have a 1:10 slope. In practice, builders would use whatever slope experience shows optimizes the efforts of the workers. 1:10 might be a little shallow.

The ramps modeled here would be about thirty-five or forty feet long, plus a few more feet of flat area at each end.

A grown man could easily drag 150 pounds up a forty-foot ramp. 150 pounds is the weight of a two-sheet bundle of 4×8 5/8″ sheetrock, and such bundles are routinely carried away by a single worker when unloading a truck. (I know because in my youth I unloaded I don’t know how many tons of them.)

If three tons, or 6000 pounds, could be lifted by 40 men, then clearly 40 men could drag such a weight up a greased 1:10 incline in a single surge of effort.  Note that this would be an anaerobic effort–just thirty or forty seconds of hard pulling, not a long slog up thousands of feet of ramp.

Rollers under the skids might or might not make the job easier. They would certainly make dragging stone easier on a flat surface but they would be difficult to manage on a narrow incline, especially with work taking place on the layer below.  So let’s assume the worst case, that dragging the stone on greased wood is the best they can do.

Style of Use

A path of ramps up the pyramid would function like a bucket-brigade. Each team would drag a block up their assigned incline by brute force, then lever it a few feet sideways to position it at the bottom of the next ramp.  Levering it sideways would only take a few of the forty men. Most of the team would rest while the next team took the block on up another level, and they waited for the team below to supply a fresh block at the foot of their own ramp.

Is It Fast Enough?

A few facts and assumptions:

• Construction took about 23 years
• Cheops is 481 feet tall
• The base is square, 756 feet on a side.
• I has 210 layers of stone blocks.
• The blocks vary in size but they average about 17.52 square feet on top and bottom and are 2.29 feet thick.
• The average number of layers a block must be lifted is 52 (because half the blocks are above and half are below that point.
• Archaeologists believe the building employed about 10,000 workers.

If we think of the task in terms of units of one block lifted up one layer, we have an average of 52.1 lifts/block * 2.3 million blocks, or about 120 million lift units to get the job done. That implies average one block lift every 5.26 seconds, round the clock, for twenty years.

The key to understanding the power of this scheme is that the block moves are fungible, i.e., any block moving up a level is of equal value, from the second layer to the 210th.

Suppose the entire cycle of hauling a block up a forty-foot ramp takes five minutes: thirty seconds of all-out hauling, plus 4.5 minutes of low-effort preparation and rest. Our teams don’t take care of moving the block across the flat top of the destination level to their final position. Other teams would do that. We’re only concerned with getting them up there.

The first layer requires no lifts. Lifting blocks from the ground to the second layer produces a pretty grim block movement rate.  The path of ramps has a length of only one ramp, so it is only moving a block every five minutes. At that block-lift rate, it would take centuries to do all the lifts.

However, by the time the pyramid is 11 layers high, it’s not a block every five minutes anymore, it’s a block lift every thirty seconds because you have ten lifts going on in parallel.

The pyramid is 756 feet wide, so you could fit up to approximately twenty ramp paths per side, but for the moment, suppose we use four per side.  With ten layers, we’re doing sixteen lift paths each doing a lift every thirty seconds, so now we’re doing a lift every 1.875 seconds on our sixteen paths.

Suppose we allow a maximum of 8400 workers available for raising stone. We choose this seemingly random number because we have 10k workers, and a single path of 210 levels, times 40 workers per level, equals 8400. So that’s the number you’d need on a single path that goes all the way up.

With 8400 workers, we can get sixteen paths 13 levels high, which is a rate of about 1.42 seconds per lift, which is 3.7 times as fast as the nominal 5.26 seconds/lift that we said we need to get it done in twenty years.  So, the stacking could take as little as 5.4 years of night and day effort. Or 10.8 years of 12 hour days.

This is good, because there is more to do than move the blocks.

Do We Have a Problem?

With our 8400 workers we can only operate all 16 paths for 13 levels. After that, we have to start closing down paths as we go higher, for want of workers to serve the ramps. Eventually, we must winnow it down to a single path for the final layers up to 210 levels.

As the number of paths goes down, is progress going to choke because there are not enough blocks entering at the bottom?

Interesting question, but no.  Every lift of one block up one level is of equal value, so we are still chewing away at the required 120 million block/level moves at the same speed because throughout the stacking (at least since level 13) we have maintained the same number of active ramps (not paths.) The blocks spend more time rising per block, but the rate at which individual lifts are occurring remains constant. It’s unintuitive, but the remaining pyramid is shrinking even as the total rate we add blocks is dropping.

Note that there is a lot of stone in those first 13 levels.  We were talking about four lifts per side, but we could have used a lot more paths on the lowest levels to keep the full 8400 available workers busy. Therefore, we actually could start on day one with the full compliment of 8400 workers, but it’s simpler to explain by keeping four paths per side all the way to level 13.

There Are Ways to Up That Efficiency

How this would actually be done is somewhat oversimplified here. The idealized version shown might only be necessary for the outer and upper portions of the pyramid.  It would be easier to wrangle the blocks with ramps that are perpendicular to the sides, rather than parallel.

This would make it faster to move stone up to the second or third layer. And because they would take up much less room, you could have a lot of them.

That’s great for the first two or three courses of blocks, but there are 210 courses.

Remember that they pyramids are huge. The base is 756 feet on a side, and even half way up is still 378 feet wide.  That is about as long as a football field, including the end zones. That is big enough that for much of the mass, it wouldn’t be necessary to use ramps running parallel to the sides. There is enough room for ramps and staging that you could raise much of the bulk of the internal volume up with the same accelerated technique used on the first courses.

The ladder-like ramp arrangement on the outside could be reserved for the upper portions where the flat top doesn’t have space for perpendicular ramps, and to fill in the areas that had to be left unbuilt to allow an outside margin of space for perpendicular ramps.

At some point it would probably make sense to use only two, or even one side, because then you could fill the current course all the way to the outside edge on the other sides.

The process would be much the same, with each team lifting blocks one level where they are picked up by another team, but with room to move, the process would be more efficient.

The Pyramid is Not Just Three-Ton Blocks

One of the advantages of this system is that it breaks the major engineering challenges down to the basic operation of moving one block up one level.  This sidesteps the many issues that come with schemes involving a long ramp.

One other problem this approach simplifies is that of moving giant stones weighing up to 80 tons half way up the pile. No ramp the ancients were likely to be able to build would be adequate for that task. It would not be easy even today.

Rather, just as the problem of moving the 2.5 to 3 ton blocks is reduced to the problem of sliding them up a single block-height, the problem of elevating the monster stones can be reduced to the problem of jacking them up about three feet.

Lifting fantastically heavy stones was clearly a solved problem, as they were able to move them out of the quarry, which requires lifting them high enough to get rollers or carts under them.

The process for raising them to high levels in the pyramid would be to first place them in the appropriate place on the ground level of the base. After that, they need only be jacked up about two and a half feet each time a new course is added. With the large stone jacked up, standard blocks would be rolled under it. It would then be set down on rollers to be moved back and forth as the rest of the stone in the layer are put in place. In this way, the huge stones would be floated up course by course all the way to their ultimate height in the structure.

There is plenty of room to maneuver. Even half way up, you still have roughly the area of two entire football fields, with end zones, side by side.

As a Management Proposition

Civil engineering is as much a management problem as a technical problem, and once pyramid building technology was worked out, it would have become mostly a management problem.

Team Spirit

As described, this would not be a galley-slave level of work. As most of lifting worker’s time would be spent resting between short hauls, men accustomed to farming or construction work would have no trouble sustaining this kind of periodic effort all day.

The construction of the pyramids was a religious and civic exercise, not slave labor. Slaves don’t normally have a stake in the labor, but with free people doing the work, the bucket-brigade nature of the process lends itself to a competitive team spirit because nobody wants their team to be the one that breaks the rhythm. Anyone who has worked on a construction site knows the embarrassment of being the guy who others have to wait for.

Minimal Engineering

There is no fancy engineering required for the donkey work of lifting millions of stones. Building such ramps is rough carpentry, and the hundreds of ramps are identical, so it’s easy to make the process efficient.  At the completion of the building, the last ramp would be zippered from the top down, leaving no trace behind.

As the pyramid is being raised, ramps from retired paths can simply be transferred to higher levels. Once the process is worked out, it’s almost all foreman level leadership–very little real engineering time would be required to supervise the lifting.

Flexibility

One of the most obvious problems with the single ramp approach is that any problem on the ramp brings the entire process to a halt.

With the parallelized approach, a problem with a single ramp need not stop traffic on any other path.  Moreover, it need not stop traffic even on the path is is part of.

This is because a small staging area would be left open at each level so that in the event of ramp having a problem, teams below that point would have space to stash a few stones while they wait for the problem above their level be cleared. Similarly, teams above a blocked ramp could send up stones that would otherwise have remained permanently their level, in order to keep the maximum number of ramp teams moving stone.

The buffering concept described above prevents a local problem from stalling all the workmen ahead and behind. Notice that once again, this works because the system lets you treat any block moving up a level as a fungible unit of work.  As long as a block moves up, it’s good, and blocks never move down.

Seasonality

The method is fast enough and expandable enough that most or all of the available workforce could be applied to loading and unloading stone when river levels are conducive to shipping, accumulating large stores of stone at the base to be raised up the pile when shipping is slack.

Alternatively, most of the workmen could be assigned elsewhere and, periodically, a large number of citizens could turn out for a limited-time barn-raising where huge amounts of stone would be raised in a short burst.

So why stop there? Say the pyramid has a flat top and is thirty levels high when the seasonal water levels get too low to bring in more blocks.  As long as blocks move up you’re making progress, so during the dry spell, you can keep moving up blocks that are in place from one half of the pyramid to make the other half higher.  The teams can keep working, and no work is lost in using the existing pile as a buffer to let your teams keep steadily raising blocks. Work never has to stop.

When blocks again arrive in plenty, the engineers would direct replacement of the stones that were moved up first, in order to get the existing pile back to being flat on top, because that shape maximizes the amount of work that can be done when the supply of new blocks is choked off again.

The schedule demands an average of 10 courses completed per year, but the bottom courses require far more stone than the top courses. Therefore, it probably makes more sense to think in terms of number of paths up.  If we start with 16 and end with one, then we need stone arriving at the base 16 times faster at the beginning than at the end to keep the lift crews busy.

We don’t really have enough information on the relative cost of quarrying and moving the stone to the site to estimate how they would have allocated resources. The critical thing is that the lifting could be made so fast that for much of the time, the bottleneck was probably in getting the stone quarried and delivered to the site, not the problem of lifting the stone.

For this reason, my guess is that especially in the lower layers, they probably used the majority of the available workers quarrying and moving stone to the site, and very likely used the barn-raising strategy to get them to the top of the pile.  This might have taken the form of periodic lifting festivals at times of year when farming wasn’t practical, and when huge numbers of citizens could turn out for brief intervals to lift stones rapidly.  Perhaps teams from administrative areas would compete to lift vast stores of stone that had been banked up during the rest of the year.

The important thing is that if you can reduce the problem of raising the stone to minimally skilled labor that need not take much time compared to the quarrying and delivery, you change the structure of the problem.

Final Words

I’ve moved one-ton stones by myself on flat surfaces. If anything, the workforce allowances I’ve described seem high. Piling up 2.5 million stones is clearly not a hard problem, and wouldn’t have been thousands of years ago. There’s something about the symbolism of lifting that appeals to the imagination profoundly in a way that the phases of the process that are genuinely mystifying do not.

Next—the mystery of getting the cap block on.