Was the construction of pyramids related to the enclosed volume? Bigger volume = more prestige? In a regular building, a tetrahedral structure would enclose a smaller volume but with a larger surface area. Greater heat/cooling losses from an equivalent height?
The shape might simply be somebody’s good idea. The pyramid shape is extremely stable because it narrows as it rises and, because it has its origins as a square base, super simple to build. It doesn’t take an exceptional intellect to envision the pyramid shape, and it is entirely possible that different people in different parts of the world envisioned the same shape totally independent of one another.
Totally agree. Of the Seven Wonders of the Ancient World, the Pyramids of Egypt is the only one that hasn’t fallen down, because they really can’t. Although they’ve been stripped of their beautiful limestone facings, they’re about as fallen down as they can get.
Said another way, they were built in their fully fallen-down configuration.
Although given a large enough earthquake or meteor strike nearby they would be … reconfigured. They are not indestructible; far from it.
He may be referring to the advantages that triangular rigid framing provides in terms of strength as I mentioned earlier. There is no added strength to piling loose fitting stones in a three, four, or more sided pyramid. Someone mentioned that a large cube would be less stable than pyramid shapes, and that is true if the base is unstable, and issue the Egyptians had to deal with because of weak foundations. Assuming the foundation is stable enough a large cube is no less stable than a pyramid. Gravity holds it altogether. The earth is generally not stable enough to build structures like that though.
Nonsense. Or so I think.
That’s true for small cubes where the total weight on the lowest course is well less than the compressive strength of the stone material. But for a big enough cube, the individual stones in the bottommost course will quickly be crushed into irregular piles of sandgrains by the total weight of the column of stones directly above them. Then the whole thing collapses unevenly.
Sure, at some point, but the bottom stones at the center of conventional pyramid have to hold same amount of weight. I could be wrong, but I’ve never heard of any structures with stones that were crushed under the weight. It would be the slight movements from the earth or changing temperature, rain, snow etc. that make large stone pyramids unstable.
That’s absurd in this context. For what it’s worth, I mentioned tetrahedral rigid structures much earlier – Alexander Graham Bell championed them as a cheap and sturdy method of construction. But it’s got nothing to do with rectangular or triagomnal stones piled on top of each other.
Yes, they would, but there’s no inherent advantage to using a tetrahedral shape in that use case, the way tetrahedral pylon structures are more stable than cuboidal ones. Nothing “absurd” about (re)highlighting that.
Sure, that’s why I said “tend to be”. There are of course exceptions to any case.
A lot of those glacial cirque-formed peaks in the Middle East, are there?
I assume Persia had glaciers historically? And probably Abyssinia as well, although I don’t know that that’d be considered Middle East.
Sure, I guess it might still have some, it certainly has alpine mountains. But also - nowhere near where the pyramids were built.
You provided the context by mentioning two different types of tetrahedral structures which have a notable difference in their structural properties. I also pointed out how the rarity of the shape itself was a notable matter aside from that. The tetrahedral shape does have characteristics common to both rigid and stacked structures, they require less material for a stable structure of the same height. The Great Pyramid could have been made in a tetrahedral shape with a similarly impressive profile requiring a lot less stone and labor if that had mattered to them.
What the hell does that mean? Just because I mentioned that there exist tetrahedral structures that are tall and thin doesn’t excuse saying that squat tetrahedral structures are less appropriate than four-sided pyramids. There’s no contextual connection at all.
I didn’t say that at all. All I did was point out the structural differences between rigid and non-rigid structures which I think is germane to the discussion and not a disagreement with what you are saying.
I am curious why tetrahedral pyramids are so rare, as you stated in the OP and the structural differences don’t explain why.
But the real world does not have solid bases and no movement. Several structures from ancient Egypt (Abu Simbel temple, for example) have been damaged by earthquakes. The Bent Pyramid had its anglechanged part-way up when the edges started shifting. The Black Pyramid and several smaller ones are now just piles of rubble because they tried building a shell with rubble inside, and it collapsed eventually.
The closer to vertical and the taller a construction is, the more likely that over time, things like earthquakes, or weathering, will cause a small section to fail and then it progressively fails. Humans are also complicit, a lot of stone from ancient buildings was taken for new construction in any location, since taking a pre-cut stone is a lot easier than carving/splitting one off the solid face of a quarry.
(For example - If you visit Istanbul (not Constantinople) you can see in a large underground cistern there that some of the support column chunks are pieces from old temples, with the carvings still on the blocks.)
I’ve been there - fascinating place, and there’s a cafe too!
The problem with large structures reaching some form of limit comes from a range of issues.
In the limit there is a maximum height of mountains that comes from the strength of the rock versus density in our gravitational field.
Structures built from individual components fail earlier than this limit because of the limited cohesive forces between the components. A pyramid or cube made of blocks of stone is limited by the shear forces trying to push the blocks apart. That is balanced by the friction between blocks, which may vary depending upon a lot of second order problems. Nothing is incompressible, and friction may decrease if the surfaces start to fail.
This is where the maximum angle of repose comes in. The mass of the upper parts of a pyramid don’t just impart a vertical force on the blocks below. There is a a lateral force trying to spread the structure. Eventually this force will overcome the friction between blocks and the structure will start to move, spread, and collapse. A cube made of stacked stone blocks will fail at a much lower height than a pyramid, and steeper pyramids will fail at lower heights than shallower ones.
A structure made of stone blocks isn’t vastly different to a sandcastle. The regular shape of blocks and stacking improve things, but this is a matter of degree, not an intrinsic difference.
My physics is at junior high-school level, so…
Why would friction decrease? Are you suggesting the rock/rock surface might become rock/sand/rock?
My physics is at the grad school level, and all I can tell you is that friction is weird, and I wouldn’t expect anyone to give a first-principles theoretical explanation for it, and you’d be best off just doing the experiments and seeing.
Which is why I marvel that the Washington monument is 554 feet high, yet made of stone held together with poured lead. Could the Egyptians have built that or was some modern technological advance necessary?