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

I recently 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 take a quick look at how clearly implausible most of the last couple of hundred years of theories are, and how strange is the persistence of variations on the same clearly unworkable strategy, given that simple solutions that would work are fairly obvious, and examples of a better geometry for solving this type of problem are in front of us almost every day.

One such solution is presented.

The Problem

The Great Pyramid of Cheops, like the other big pyramids, is mostly made of stone blocks that are roughly the size of refrigerators and the weight of a Cadillac Escalade. A few of the special purpose blocks weigh as much as 80 metric tons, but the bulk of the pyramid consists of blocks weighing from 2.5 to 3.0 tons. Each course of blocks is about the height of a dining table and there are something like 2.3 million blocks.

How did the ancient Egyptians get them up there without powered machinery, indeed without the use of wheels? 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. Any satisfying solution to the problem of lifting all the little blocks must also allow for, or at least not obstruct, processes for getting the giant blocks and the capstone up there.

The hypothetical technique presented here addresses all three problems with the same approach.

The Problem

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

Cheops is about 150 meters tall, and is laid in 210 courses of blocks. The first problem is that a 150 meter high earthen ramp is impractical bordering on impossible. The engineering problems are well understood, and while this isn’t the place to go into great detail, there are some relevant points.

Skipping all the math, the the most obvious problem is that a ramp that tall would bulk immensely larger than the pyramid itself.  Packed earth is only slightly lighter than the limestone of the pyramids, so the big job would have been moving and then removing all that earth. Actually building the pyramids would have been minor in comparison.

However, even an earthen ramp were practical, the real problem is time.

We know that the project took about 23 years.  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 resting place on average every 90 seconds, all day, every day, for the duration.

The 90 seconds per block minimum assumes moving blocks for every day of 23 years. That’s unrealistic; there is work that must precede the stacking of the blocks, such as clearing and leveling the site, preparing docks and yards for the materials, and building the foundation. There is also work that must follow the stacking the blocks, such as cladding the structure, tearing down and hauling away the hypothetical ramp, etc.

Perhaps worst of all, traffic on a ramp would be single file. The 90 seconds per block also doesn’t include gaps in availability. It makes no allowance for teams needing to rest on the way up, ropes breaking, oxen dropping dead from exhaustion, holy days, etc.  Nor does it allow for the enormous work to add another one-course high layer to the ramp for each completed course.  It also makes no allowance for fluctuating availability of stone.

We could go on for pages–earthen ramps are 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, replacing it with a more economical narrow, winding ramp that is partly supported by the building. Consideration of feasibility aside, these ideas don’t address the bottleneck that is the real issue. It’s still single file up the ramp, and any mishap delays everything behind it.

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

Variants of the idea aren’t very promising either. For instance, you could use multiple earthen ramps for the lower portion, and save the winding single ramp for the upper layers. This could be more powerful than it sounds, because so much of the mass of a pyramid is low down, but there are more than a million blocks higher than 37 meters, which is already extraordinarily high for anything made of earth.

Packed dirt simply isn’t very strong. It’s certainly not strong enough to build a the kind of sloping ramp pictured above. Mud-brick is considerably stronger, but the tallest mud-brick structures in the world are not much more than 30 meters high, and few are as much as even 15 meters. Moreover, you can be sure that the owners of 30-meter mud brick buildings don’t allow grand pianos on the top floor, let alone a continuous file of three ton blocks dragged by teams of oxen.

Any such proposal that relies on one or even several long paths faces an uphill battle (so to speak) against the unavoidable bottleneck problem.

A Final Problem

A final problem with all of these approaches is that they don’t address the difficulty of moving the big components. Monoliths of as much as eighty tons are located far up the completed structure. A single long ramp would have to be strong enough to support eighty tons on one spot. Winding ramps would have the same problem, but would also have to be designed to allow these enormous objects to be turned.

An Easier Way

The model pictured below is based on a different principle.

This proposed method uses each course of blocks in the pyramid as a base for a short ramp that is the height of a single layer of stone, abut 75cm. A block is dragged up the greased ramp then levered sideways the width of a ramp to put it in position to be 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, and instead of landing directly on stone at the top of the ramp, it would land on rails to make it easy to slide it sideway to the next ramp up.

 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 fast enough to raise 2.5 million blocks to their ultimate destinations within the required time?

Dragging Stone Up The Ramps

The ramps pictured in the model have a 1:10 slope. In practice, builders would use whatever slope experience shows optimizes the efforts of the workers. My intuition is that 1:10 might be unnecessarily shallow. The block only has to rise  0.75 meters, which is about thigh-high on an adult. Picture a ramp seven or eight meters long–about the width of a townhouse, rising a block to the height of a dining table.

The ramps modeled here are slightly steeper. In any event, a ramp also needs a few more feet of flat area at each end.  We will elaborate on this later.

The argument for the practicality of this technique is that a grown man could easily drag 68 kilos up a eight meter ramp. 68 kilos is the weight of a standard two-sheet bundle of 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 many a ton of them.) If a man can easily carry that much weight, clearly he could drag  at least that much.

Therefore, 2,728 kilos could be lifted by 40 men, and by the same argument, it would require at most 40 men to drag such a weight up a greased incline in a single surge of effort.  Note that this would be an anaerobic effort–just thirty seconds of hard pulling, not a long slog up kilometers of ramp. This is the level of effort expended when pushing a car, applied for half a minute.

Rollers under the skids might or might not make the job easier. They certainly make dragging stone on a flat surface easier, but they have other drawbacks (see notes) 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 on wooden rails or skids 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 then wait for the team below to supply a fresh block at the foot of their own ramp.  During their work shift, workers never move up or down, except possibly one level to hook up the next block.

Is It Fast Enough?

A few facts and assumptions:

• Construction took about 23 years
• Cheops is about 150 meters tall
• The base is square, 230 meters on a side.
• I has 210 layers of stone blocks.
• The blocks vary in size but they average about 1.6 square meters on top and bottom and are 70cm thick.
• The average number of layers a block must be lifted is 52.1 (because half the blocks are above and half are below that point.
• Archaeologists believe the building employed about 10,000 workers.

Fungible Lifts

If we think of the task in terms of units of one block lifted up one course, we have an average of 52.1 lifts/block * 2.3 million blocks, which is about 120 million lift units to get the job done. That implies average one block lift every 5.26 seconds, around the clock, for twenty years. Note this is one block, one course, not one block delivered to its final destination.  On average, a block-lift is 1/52 as much work as a delivered block.

The key to understanding the power of this scheme is that the block moves are fungible, i.e., any block moving up one course is of equal value, from the second layer to the top.

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. We’re assuming other teams would do that using rollers. We’re only concerned here with the means and the math of getting the blocks lifted.

The first layer requires no lifts–it’s on ground level. Lifting blocks from the ground level to the second layer produces a grim block movement rate.  The path of a block up the ramps has a length of only one ramp, so a path up is only moving one block one level every five minutes. At that rate, it would take centuries to do 120 million lifts.

However, by the time the pyramid is 11 courses high, it’s no longer one block lift every five minutes, it’s a block lift every thirty seconds because a path has ten lifts going on in parallel.

The pyramid is 230 meters wide, so you could fit up to at least twenty ramp paths per side, but for the moment, suppose we use four paths per side.  With ten layers, we have sixteen lift paths, each doing a single ramp-lift every thirty seconds. At that rate, 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 as an example because a single path of 210 levels, times 40 workers per level, equals 8400, which is well below our worker budget of 10,000. That’s the number you’d need on a single path that goes all the way up. In fact, for reasons we will see, the number of workers moving stone up the structure probably varied both seasonally and over the duration of the project.

With 8400 workers, we can get sixteen paths 13 levels high, for a total of 208 ramps working in parallel, giving a combined rate of about 1.42 seconds per lift or 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 a lot 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, near the top, we will only have enough workers to maintain a single path with 208 ramps.

An obvious question is, 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  approximately the same number of active ramps even as the number of paths shrank. Each block spends more time rising as you go up, but the rate at which individual lifts are occurring remains roughly constant. It’s unintuitive, but the remaining pyramid is shrinking even as the total rate at which we add new blocks is dropping, while the lifts per second rate constant.

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, under the unlikely assumption that there is an unlimited supply of stone waiting to be lifted, we actually could start on day one with the full compliment of 8400 workers. However, it’s simpler to explain by keeping four paths per side all the way to level 13.

There Are Ways to Increase That Efficiency

How this technique would actually be applied is somewhat oversimplified above.  There are obvious optimizations that can be made.

While the method described above would be fast enough to get the job done well within the required time limit, it’s not necessary for every block lift. A simpler and faster technique can be used for a large proportion of the 120 million block lifts required.

Consider that the ramps as shown in the model are parallel to the faces of the structure because it is impractical to make perpendicular ramps of any great height. That in fact was the original problem.

With the ground-level course in place, it would be easy to get the second course lifted with the ramps turned perpendicular to the first course. It is no harder to haul stone up a perpendicular ramp, and there is no direction change at the top.

Such ramps be great for the second course of blocks, but what about the remaining 208 more courses?

Stacking Step-Wise

Remember that the pyramids are enormous. The base is 230 meters on a side, and even half way up they pyramid it is still 115 meters wide, more than the length of  an American football field.

The first course doesn’t have to be lifted, because it is on ground level. When a portion of the second level is in place, having been lifted from ground level to the top of the first course using perpendicular ramps, stacking a third course would begin, using blocks lifted from the first course to the second course via perpendicular ramps aligned with each other to minimize distances hauled and direction changes.

Note that the ramps are just wooden furniture. They are not anchored to anything, and can be moved as needed. There would be no difference between ramps for parallel use and ramps for perpendicular use.

Likewise, as soon as enough of the third course is in place, a fourth course can be started, again with the stones dragged on rollers in a more or less straight line from ramp to ramp.

With a reasonable minimum working space allowance, for much of the pyramid’s volume, there is working room for eight or nine courses of blocks to be underway simultaneously, allowing a tremendous number of ramp teams to be operating in parallel. Each time the lowest incomplete course is completed, another course can be started at the top. Parallel ramps are necessary only to get past the completed courses.

It’s an interesting exercise in operations research to determine exactly how many ramps should serve each course.  The object is to maximize the number of ramps that can be in use.  Fewer ramps would serve each of the incomplete courses at higher levels so that lowest level course can be closed out while there is still plenty of working space above.  Blocks would be added to the upper courses slowly, but as they cease to to be the uppermost, their block adding rate would accelerate.

It’s hard to say precisely what proportion of the total volume could be lifted with the optimized perpendicular ramps, because the exact amount depends on the ideal ramp angle, how much ramp room and elbow room people need to move efficiently, etc., but certainly the majority of the 120 million lifts could be executed in that more efficient way.  To see why intuitively, note that half of the volume of a pyramid is below 1/4 of the height, where there is a tremendous amount of room to work. Above that 1/4 line, there is a smaller pyramid, half the volume of which is below 1/4 of the remaining height, and so on. Repeating this reasoning, we find that 119 courses up, a little more than half way, comprises 7/8 of the volume, and at that height there is still more than a hundred meters square of horizontal working space available.

While most blocks travel a portion of their way up on parallel ramps, the final ten or so courses up for each block takes place on the faster, perpendicular ramps. So the first several courses are almost entirely lifted on perpendicular ramps, and thereafter the proportion of ramp lifts to total lifts declines slowly.

Therefore, the pyramid would almost certainly have risen unevenly, with stepped layers stacked using the faster, perpendicular ramps. Eventually, near the top, perpendicular ramps would no longer be practical, but throughout the courses where most of the volume is, a large proportion of the lifts can be done using the perpendicular version of the process.

More complex arrangement of the perpendicular ramps allow for even more to be in service in parallel.

The Pyramid is Not Just Three-Ton Blocks

One of the advantages of this system is that it breaks the major engineering challenge of block stacking down to the basic operation of moving one block up one level, and provides two similar ways to do the lifts for different situations. This sidesteps the many issues that come with schemes involving a long ramp.

However, there is also the problem 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.

Instead, just as the problem of moving the 2.5 to 3 ton blocks as much as 150 meters up the pile can be reduced to the problem of sliding one block at a time up a single 0.75 meter-high ramp, the problem of elevating the monster stones up fifty or a hundred levels can be reduced to the problem of jacking the component up about a meter high. Consider that lifting the monster stones to a modest height was clearly a solved problem, as they were able to move such stones onto rollers to transport them from quarry to barge and from barge to the building site.  We can even make an educated guess at how they did it.

Lifting the Monster Blocks

The process for raising the monster stones to high levels in the pyramid would be to first place them in appropriate places on the ground level at the beginning, before the stacking process starts. After that, they need only be jacked up about one meter each time a new course is added.

When the layers of the next course approach the location of a large stone, the stone would be jacked up one meter, and 0.75 meter-high standard blocks would be rolled or skidded under it. The large stone would then be set down on rollers so that it can be rolled out of the way to make room for completing the portion of the lower course that is to cover it’s former location. When the stones are in place, the large piece would be rolled back to it’s former location, but one course higher.

In this way, the huge stones would be floated up to their ultimate height in the structure one layer at a time, using no special techniques that were not already used in quarrying and delivering them.

Something like two or three hundred workers could lift an 80 ton block a meter high within an hour or two using levers and wooden blocks. Fewer workers would be required for subsequent lifts for reasons that will be made clear.

The lifting would not be as arduous as it might sound because it is possible to use the stone itself as a counterweight, minimizing the amount of lift required.

Assuming a long block, levers under the block are used to raise one end ten or twenty centimeters. Low blocks would then be slid under it near to the center line, say, 2/3 of the way to the center.  The process is then done in mirror image, placing blocks a similar distance from the center line on the far side, leaving space between them. When doing the second tilt, the stone overhanging the block on the far side acts as a counterweight, make it easier to lift the near side.

The process is repeated several times, rocking and lifting the block several centimeters each time. After the first lift on the first level, fewer workers are needed because the stone will always remain on blocks it can rock on.  When it is high enough to slide stone blocks under it, plus a few more centimeters for rollers, it is set down.

With the stone blocks in place, and rollers positioned, it can be rolled out of the way, allowing work to be finished in the area it formerly occupied. When that work is done, it is rolled back to its original position, but one course higher.

Topping Out With The Capstone

The problem of lifting the enormous capstone onto the top of the pyramid would probably be unsolvable using any technology available in that era. Even today it would be a significant task, as a crane would have to reach halfway across the pyramid, 115 meters, which is beyond the radius of all but a handful of the biggest cranes in the world.

Therefore, we can safely assume that they didn’t lift the capstone onto the pyramid. Instead, they would have raised the capstone in the same way that the other giant stones were raised. It would have been rolled onto the very first course and floated all the way up to within three layers of the top. Three layers would add up to about 2.25 meters.

From the third to last course,  the capstone would be jacked up one meter, and blocks would be placed under it. On that base, it would be lifted another meter, and then another. For a brief period, the capstone would be perched on a narrow pier 2.25 meters high, while the last three courses of the pyramid were filled in to meet it from below.

Note that the final few layers of the pyramid would not provide much space for teams pulling large stones up ramps. Wooden cranes would have been used to lift the stone for the last few layers from the highest layer that could be easily served by ramps.  A crane capable of lifting three tons two or three meters high was well within the technology of the day.

Temporary conventional scaffolding would have been used to provide working space around the final blocks under the capstone.

As a Management Proposition

Civil engineering is as much about management problems as it is about solving technical problems, and once pyramid building technology was worked out, it would have become more about management than engineering. Throughout the project, they would have needed highly skilled surveyors to ensure that the corner block placements were correct, but relatively little skilled craftsmanship would have been required to place the the vast majority of the blocks.

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, probably paid, and certainly 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. 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.  It’s a basic male group dynamic.

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.  Other than the real engineers ensuring that the corners of each course are in exactly the right spots, it is basically brick-laying at a giant scale.

Flexibility

One of the most obvious problems with all of the single ramp approaches 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 and workers are available for lifting.

Alternatively, most of the workmen could be assigned to quarrying and delivering most of the time, 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 to clear the deck for another season of stone delivery.

So why stop there? Planners know when shipping season begins and ends. The block-laying strategy can be such that the pyramid has a more or less flat top when the 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. The lifting 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. A simple variations of this system would work even with the alternate system of perpendicular ramps being used.

The schedule demands an average of 10 courses completed per year, but the bottom courses require far more stone than the top courses. Keeping the number of lifts constant would mean that a much higher arrival rate for stones would be needed at the beginning of construction than at the end, assuming a constant number of lift workers. With anything like the lift-rates hypothesized here, quarrying and moving the blocks to the site would limit building speed, not the task of lifting them to their final place. Therefore, very likely the proportion of workers doing quarrying and shipping v the proportion doing lifting would invert over the course of the project.

We don’t really have enough information on the relative cost of quarrying and moving the stone to the site to estimate exactly 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 almost certainly 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 great majority of the available workers quarrying and moving stone to the site, and very likely used a seasonal barn-raising strategy to get them up on the pile when a large number had been accumulated.  This might have taken the form of periodic lifting festivals at times of year when farming wasn’t practical. On such occasions, huge numbers of citizens could turn out for brief intervals to lift many stones rapidly.  Perhaps teams from administrative areas would compete to lift stones that had been banked up while shipping conditions were favorable.

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 alone on flat surfaces in the studio using nothing but levers, blocks, and rollers. We actually call it, moving stone “Egyptian style.” It’s easier with two people, because one person pries the stone up and another puts blocking or a roller under it.  If anything, the workforce allowances I’ve described seem high. Piling up 2.5 million small stones and a few huge ones is a big problem, but it clearly would not have been a hard problem thousands of years ago.

There is something deep here about how humans think, that modern people in a world full of zig-zagging stairwells, still try to improve on the same single ramp idea that clearly cannot possibly work. I’ve seen a couple of two story flights of steps in my lifetime but I don’t know if I’ve ever seen a three-story flight.  Yet someone in Napoleon’s generation must have published their first thought, “ramp”,  and for more than 200 years, people have contorted that original idea of a single long ramp into countless variations, rather than applying the zig-zag pattern which we almost exclusively use for manually lifting ourselves and our packages up multiple levels.

That tantalizing idea that a person can somehow make that ramp work fires the imagination profoundly in a way that the other phases of the building process that are genuinely mystifying do not.

Notes

Hauling: The exact means by which the blocks are moved up the one-course ramps is immaterial. A sled on greased wood pulled by a large team is one possible way, but there are other candidates. Rollers would work, but they might be hard to manage and more dangerous on an incline.  Block and tackle arrangements would greatly reduce the number of workers required to haul a block up one level. The higher end would be anchored by a bronze pin slotted between two blocks. Drilled holes or similar measures would not be necessary.

Who Does the Hauling: The quarry work and the transportation of the stone would require most of the labor, both skilled and unskilled. Pulling on a rope does not take much skill or training. As the process is clearly capable of lifting stone at greater than the required rate, the Amish barn raising model seems likely.  The 10k available workers would mostly quarry and transport stone and stage it at the base of the rising pyramid. Periodically, festivals would be held where thousands of volunteers would temporarily augment a core team to move up a great deal of stone in a short time.

 Single File:  We have referred to the bottleneck problem affecting single ramps because ramps limit access to single file. People and animals would have to descend as well , of course. But even if the path up were not strictly single file, e.g. you had two paths up side by side, it does not fundamentally change the problem. A single ramp remains a bottleneck, and no plausible number of parallel paths on a single ramp would greatly changed the basic problems.

Vertical Travel Time: Time spent traveling becomes significant as the structure grows. With a single ramp system, the average height to which a worker would have to haul a block up a single ramp over the course of construction would be the height of a 12 story building, and farther into the project, each trip up would be a climb to the height of a forty to fifty story building. This would add up to an enormous expenditure of worker-hours, and it’s tiring for the workers. A bucket-brigade pattern is far more efficient in terms of travel time per individual. Even with the many-ramp system, it might be advantageous for a camp to be maintained at the upper levels where workers would stay for days at a time, with food, water, and sewage being handled by bucket-brigade up and down the face. The alternative not only wastes commute time, but requires many more flights of steps be maintained to allow thousands of people to travel up and down.

Moving Blocks Horizontally:   We have mostly ignored the need to move blocks horizontally once they arrive on a given level because moving a block on rollers on a flat surface is relatively easy and takes fewer workers. A dozen men or fewer could roll stones of that size from the edge where they land to their ultimate resting place or to another ramp.  The average block travels vertically 52 courses in a process that requires 40 * 52 = 2080 workers; a dozen more to roll it across the top is essentially noise in the equation.

Perpendicular Ramps:   The faster strategies alluded to above would have the layers above the lowest completed course be stacked in a pattern winding around three sides. This would allow a smaller proportion of total block-lifts to be made using parallel ramps.

Next—the mystery of getting the cap block on.