Why No Square/Cube Law?

JoanDrake

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I just watched a special again on How They Built the Pyramids. In it three guys, An archeologist, an architect and a stonemason, built a little pyramid using small stone blocks. Well, a lot smaller than most Pyramid blocks but still obviously some heavy little things.

It's neat, they go into all the problems they encounter and show how they could be overcome, using only the tech available to the Ancient Egyptians. There's just one problem. What they're doing is not really illustrative at all of what was actually done. It can't be.

The reason is the square/cube law. You've probably heard of it and you can look it up in Wiki if you haven't. All it says is that if you have something and you increase it's LINEAR dimensions by 10, you will increase it's volume, and hence it's weight, by the CUBE of 10, or 1000.

If a 1x1x1 stone weighs 100 lbs, a 10 by 10 by 10 stone will weigh 100,000, yes?

And that much weight difference is a qualitative thing. You won't be able to use wooden rollers of the same size, you'll crush them. You might not even be able to use wood at all.

This is a well-known thing to SF writers. It's why we can't really have 10 foot ants (they couldn't walk or breathe) or 50 foot and 5 inch people. Not without major structural changes that pretty much make them not people anymore.

Now I can see an archeologist not knowing this, and maybe even a stonemason, but an architect? I'm sure not hiring him for any skyscrapers.

Seriously, this was a NatGeo Special and I'm sure some of you saw it. Did they cover this and I just missed it or what? Could be I just don't understand Square/Cube and I should if I want my Giant Crocodile in a Typhoon eating San Francisco to fit in with established science (I'm having him be made of living carbon fiber, that should work). Can anyone help?
 
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If you have a small cube 10cm x 10cm x 10cm and want to stack them to make a 100cm x 100cm x 100cm cube (x10 wider, deeper and higher) then each edge you can count 10 little cubes. You can try this with sugar lumps (but maybe only 3 little cubes wide x high x deep) or similar if you don't believe. It will take 10 x 10 x 10 little cubes = 1000 cubes to make the big cube x10 wider! It's a cube law
2 = 8
3 = 27
4 = 64
5 = 125
6 = 216
7 = 343
8 = 512
9 = 729
10 = 1000

You may be able to use wood rollers. If 21 rollers in use each takes about 1/20th weight (one is being moved from back to front ...)
You may use metal or stone rollers.

I cancelled my magazine sub (little real content apart from photos) and long ago pay Tv as Discovery, History, Animal Planet, Nat Geo etc (the reason for it) had too much junk and too many repeats. We buy occasional decent DVD box sets cheap instead.
 
I can't prove it (as in I can't check it for sure!), but it seems from a variety of sources that the average weight of a single block used in the Great Pyramid was 2.5 tonnes. Or to use ye olde fashioned units about 4,500 lbs. :))) That seems a much more manageable weight with rollers or sledges or whatever they used.

The largest transported stones seem to be the granite stones that were used in the King's Chamber - I'm finding stones in the range of 25-80 tonnes quoted, which brings it into your range. Of course for most of the journey they were floated down the Nile. These were special stones, so no doubt they put extra effort into shifting them (I've seen some interesting suggestions on how they got the big King's Chamber stones up onto the pyramid using pulley systems. My guess is that for the very short haul from the port to the pyramid they just used enormous numbers of people/oxen to haul it using overwhelming muscle power over a prepared track.)

There are numerous huge stones around the world that were constructed at all sorts of different time periods and some were mind blowing weights. Just to take a quick example, Cleopatra's needle that stands in London at the moment apparently weighs over 200 tonnes and was constructed around 1450BC. We're a problem solving race of beings and as is proven in a lot of our monuments around the world we could solve shifting very heavy things with simple technology.
 
The mass of a cube increases with the cube of the increase in the sides, but it's expressed through the faces, which increase with the square of the sides. So a 1 metre cube is 1000th the mass of a 10 metre cube, but that mass presses down over 100 square metres rather than 1 square metre. In effect, the pressure increases at the same rate as the increase in length of side. (Note that the mass is also being applied to more of the log; in fact ten times the length of log, and there will be ten times the number of logs, so again, the mass on a typical section of log has increased by "only" a factor of ten.)

Quick question: are the largest pyramid blocks (some weighing 80 tons) cubes or cuboids? If a cube is doubled in length, but not height or width, the mass per square metre doesn't increase at all on the faces where one of dimensions has been doubled.


If the Egyptians didn't use wooden rollers, what else could they have used? It isn't as if pyramid blocks are the largest stone items they transported. Look at obelisks: the one now in Rome weighs 455 tons. And yet they were usually quarried at Aswan and transported, by river, to their destination. Given that they weren't erected on the riverbank (although I expect they were moved during the floods), that weight did not only have to be moved laterally, but dragged up a slope to permanently dry land. And then they had to be raised to the vertical. (I've stood on an even larger obelisk which was abandoned before being cut from the quarry because of a fault in the rock. Of course, never having been moved anywhere, it's possible that its 1200 tons might have defeated the Egyptians' hopes of transporting it elsewhere.)
 
If the Egyptians didn't use wooden rollers, what else could they have used? It isn't as if pyramid blocks are the largest stone items they transported. Look at obelisks: the one now in Rome weighs 455 tons. And yet they were usually quarried at Aswan and transported, by river, to their destination. Given that they weren't erected on the riverbank (although I expect they were moved during the floods), that weight did not only have to be moved laterally, but dragged up a slope to permanently dry land. And then they had to be raised to the vertical. (I've stood on an even larger obelisk which was abandoned before being cut from the quarry because of a fault in the rock. Of course, never having been moved anywhere, it's possible that its 1200 tons might have defeated the Egyptians' hopes of transporting it elsewhere.)

Precisely this - plus we could add all other cultures that moved large bits of very heavy masonry over difficult terrain. For example, unless they really were just lying about next to site, the big sarsens in Stonehenge would have been a huge challenge to move (perhaps up to 40km according to some theories) for the Brits at the time.

Regarding the big 1200 tonne one - surely the mere fact that they went to all the effort of almost chipping it all out meant they were sort of confident that they could move it if it did not have a flaw. After all they probably were people that had moved huge bits of stone all their lives. There is the monolith at Ramesseum that is estimated to have weighed 1000 tonnes and was allegedly transported 170km.
 
They may have been confident, but were not infallible. One only has to look at the Bent Pyramid to see that they sometimes made mistakes. (With a lot of science not known to them, they often had to rely on trial and error.)
 
An obelisk is no harder really than a cube of same cross-section, as has been pointed out above. Far easier to handle than a cube of same mass.

Wooden rollers are believable.
 
They may have been confident, but were not infallible. One only has to look at the Bent Pyramid and to see that they sometimes made mistakes. (With a lot of the science not available to them, they had to rely on trial and error.)

Absolutely true, but they gave it a go, so it says something about their mindset about shifting big things*. Water transport was out in moving it**, so whatever they were going to do it was going to have to be overland.

Other known mega-structure operations - look at the colossi of Memnon, monoliths of about 700 tonnes each and again they couldn't use the Nile, so they were moved overland 675km to be put on site! (According to Wikipedia.)

Hell, when the river Nile silted up and made Pi-Ramessess a non-viable city, they carted all the monuments and bits of stone they wanted to a site 30km away to make a new city, Tanis. That's a real dedication to their stonework. Different mindset.

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* If you are going to hack out a huge rock with crappy copper chisels over god knows how long with hundreds of men who need feeding etc... you don't do it on a whim, they must have thought it was doable. And I would argue that pyramid building and architecture is a much more complex and harder task by a big margin to do right than transporting a big mass of rock from a quarry to a site.

Interestingly I see that it might not been a fault in the rock but the quarrying process itself that caused the crack, so perhaps it is all academic - as perhaps they had reached the limit of how big a single rock they could hack out of a quarry - and therefore quarrying an immovable stone was impossible :)

** EDIT - Taking Ray's comments on obelisks on board, perhaps you can transport something long and thin (and heavy) by water, rather than a cube of the same mass. I've not done the sums.
 
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OK, that does sort of explain it. I was really more distressed by the idea that the methods they were talking about wouldn't just scale up like they seemed to think, but I guess they might at that.

Did you see the one that showed how the Romans moved these 1000 ton stones at Baalbek (sp?) They made two really huge wheels with square shaped holes in their centers, then fit the stone as an axle and just rolled the whole thing. Amazing, how clever they were back then, I certainly would never have thought of that.
 

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