Photo courtesy of http://liberato.org Ricardo Liberato
I just saw yet another documentary on building the pyramids. Once again, it was the age-old question of how they got all those blocks up there. It’s been a favorite of archaeologist and crackpot alike since the return of the scholars who went to Egypt with Napoleon’s army in 1798. Why people still care, more than 200 years later, is a mystery in its own right because the building of the pyramids presents much more interesting problems, but it is fun to think about if you’ve ever moved anything heavier than a couch.
The Problem
The Great Pyramid of Cheops, like the others, is mostly made of stone blocks roughly the size of refrigerators. Some of the blocks weigh as much as 80 tons, but most of them are about 2.5 tons–something like 2.3 million of them.
How did they 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 Classic Solution
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 layers of blocks. The first problem is that a five hundred foot high 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 as it nears the top, it would have to be wider than the narrowing pyramid itself in order to bear the pounding of workers dragging thousands of tons of stone. The ramp would have to more or less bury the pyramid.
On the positive side, the majority of the mass in a pyramid is near the bottom. With 210 layers, half the blocks are below the 40th 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 about 53 layers. Nevertheless, the ramp still has to go at least near to the top, and even the top 1% is still 25,000 blocks.
There is a more fundamental problem that for some reason you don’t hear much about. I’ll spare you most of the math. The short version is that the project took 23 years and the number of blocks divided by the number of work-days and daylight hours, etc. works out to raising a block every 90 seconds, all day, every day, for the duration. To put that in perspective, 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 actually a gross idealization because a big chunk of the work, such as building the foundation, cladding the structure, tearing down and hauling away the hypothetical ramp at the end, and building the rest of the complex, 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., all of which delays also comes out of the time budget. 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. There’s another variant that has the winding ramp inside the pyramid. These ideas eliminate the giant earthen ramp that wouldn’t work anyway, 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
Conceptually, these schemes are even worse than the long earthen ramp idea, because, with an earthen ramp, you could at least have multiple ramps in parallel for the lower levels where most of the blocks are, whereas with a winding ramp you’re limited to one. These schemes also have lots of 90-degree turns that would slow things down.
Another daunting problem with even with the pyramid providing some support, simple 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, and few are as much as even 50. You can be sure that the owners of 100-foot mud brick buildings don’t allow grand pianos on the top floor, either. With the ramp resting on the pyramid stones and clinging to the side of the pyramid as it went up, it might have been possible to build a ramp that never had to be very tall and could withstand the forces of moving 2.5 ton blocks, but it’s not easy to see how you could pull the same trick with the 80-ton monsters. There’s got to be a better way.
An Easier Way
The persistence of the basic idea of one long ramp, whether straight or winding, is especially striking because steps, the oldest and most common way to get to the top of a building, almost never work that way. Long stairways usually have flights, one for each floor, that alternate in direction, so that the stairway can fit in a space that is taller than it is wide.
Two hundred years of variants on the same plan, despite a fundamental impracticality that can’t be improved away, is a clue to how we think. Making improvements to the previous way of doing something, rather than starting from scratch, is a basic mechanism of human cooperation.
The model pictured below is based on a completely different principle, and self-evidently works; there’s nothing that even needs testing or load calculations. Moreover, you can easily demonstrate that you could raise the stones fast enough with it, and it’s not hard to come up with variations on the theme that would work even better. The triviality of these easy solutions that would work makes our two hundred years cultural fascination with the problem itself seem somehow mysterious and wonderful.
Instead of a single mile-long ramp winding around to a height of 500 feet, you use the steps of the unclad pyramid itself to support a column of little ramps that alternate in direction one layer at a time. The proportions of the wood blocks pictured above are fairly realistic and I’m showing 1:10 ramps, but that is an arbitrary choice—in practice, you’d use whatever slope that experience shows optimizes the efforts of the workers. The ramps modeled here would be about thirty or forty feet long, plus a few more feet of flat area at each end.
The ramps are modeled as if made of the same stone as the blocks, but wood timbers would be better because of the greatly reduced friction of a wooden sled on greased wood. Wood was probably more expensive, but it would have cut down on the labor to install and remove the ramps, and it could have been sold or reused at the end of the project. The Egyptian bean counters would have made that call.
The ramps would be used bucket-brigade style. Each team would drag a block up their incline by brute force, then lever it a few feet sideways to the bottom of the next ramp. Then they 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. A dozen men can easily lever a 5,000-pound block around so most of the team could rest once the block was at the top of the ramp.
Unlike a long 1600-foot slog, forty feet is a quick anaerobic haul so each worker could pull a lot more weight. Any grown man can pull, or even carry, a hundred pounds up a thirty-foot ramp. Clearly, fewer than fifty men could drag a 5,000-pound block up a 40 foot greased incline in a single haul lasting less than a minute. Archaeologists believe the builders probably had about 10,000 laborers at their disposal, which would allow about 200 such ramps working in parallel.
Fifty levels comprise almost 3/4 of the total blocks, and at that height, 200 teams would give you four columns of ramps with teams working at every level side by side all the way to the ground. If each team took one minute to haul a block up a level, then rested for three minutes, you’d have a block arriving a the top every 60 seconds. At the lower levels, the block-arrival rate could be much higher because you could have more columns working side by side with the same 200 teams.
As described, it would not be a galley-slave level of work. With 75% of the time spent resting between short hauls, men accustomed to farming or construction work would have no trouble sustaining this kind of labor all day, and the majority of the blocks could be delivered much faster than the overall rate required, allowing for a slower rate for fewer columns of ramps at the higher levels.
This system would also give a great deal of flexibility. It is fast enough and expandable enough that, for instance, most or all of the workforce could be applied to unloading stone when river levels are more conducive to shipping, accumulating large stores of stone at the base, and moving them 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. At the completion of the building, the last ramp would be zippered from the top down, leaving no trace behind.
Is this how they did it? Who knows?–but clearly, it’s faster and more efficient than any scheme involving a single ramp, partly because there is no practical limit to the number of teams you can have working simultaneously, and partly because a temporarily stalled team wouldn’t have to have much effect on other teams. If you had a small staging area at each level, teams below a temporarily blocked ramp would have room to stash a few stones while they wait for next team to recover, and teams above a blocked ramp could send up stones that would otherwise be used on their level in order to keep the maximum number of men moving stone. The uniformity of the blocks and the similarity of the levels effectively buffers against snags in supply.
The buffering concept described above prevents a local problem from stalling all the workmen ahead and behind. But notice that this works because the system lets you treat a block moving up a level as a fungible unit of work. As long as a block moves up, it’s good. There’s no extra cost to sending up a block that might otherwise have stayed where it is forever. Any block moving up a level is equally valuable.
So why stop there? Say the pyramid has a flat top and is fifty levels high when the seasonal water levels get too low to bring in more blocks. As long as blocks move up and not down, you’re making progress, so you 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.
When blocks again arrive in plenty, the engineers would use the new blocks first 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, yet it doesn’t cost any more in terms of one-level-lifts.
The technique also gives plenty of resting time and doesn’t require the team members to move more than one level up and down during the workday. It also requires no real engineering—just rough carpentry for the ramps. It’s important to remember that the workmen weren’t slaves. Psychologically, the bucket-brigade nature of the process inherently lends itself to team spirit because nobody wants their team to be the one that breaks the rhythm.
Obvious minor modifications of this basic idea also allow the exceptional huge blocks to come up by the same method.
I’ve moved one-ton stones myself Egyptian-style. If anything, the workforce allowances I’ve described seem high. It’s just 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.
Time and Material 