I worked with a guy who catalogued all of the geometric ratios used by early stonemasons (like, 10th and 11th centuries), and it would be pretty far-fetched for me to believe most of them weren’t understood back in Greek and Egyptian cultures.
He (John James) published a booklet called “The Ratio Hunter” and if you’re into that sort of thing any given page will simply blow your mind.
For example
I got to spend a summer with him at Chartres and by golly he demonstrated how many of these were used to lay out tiny details in the stonework.
The glass atrium of the Rock and Roll Hall of Fame (designed by the same architect as the Louvre pyramid) is (mostly) triangular-based, if you want a modern example.
I think that the regular tetrahedron is far more interesting as a physical object if it is something in one piece that you can pick up and turn around. We know from the ancient device of caltrops that this configuration was long understood. But a large pile of stone blocks is another matter; the 3D-ness of the symmetry is more incidental, one face always being invisible and fixed to the ground.
Tesselation, using square and other regular shapes of tiles was used as far back as 4000 BC.
The Sumerians used clay tiles to compose decoration features in their homes and temples. From there, tessellation found its place in the art of many civilizations, from the Egyptians, Persians, Romans and Greeks to the Byzantines, Arabs, the Japanese, Chinese and the Moors, so it’s reasonable to assume that the Egyptions were familiar with triangles and hexagons at least.
I think that’s a lot of it. Aside from that, if we’re talking about the very large pyramids most of us are familiar with, as viewed by people at the time they were built, a 3-sided and 4-sided pyramid look the same unless you’re standing on top of it or from some higher vantage point.
I bet the stonemasons would have shit a brick if they’d seen that drawing. “Oh no!, we gotta make a jillion of those, from 4 separate materials. I quit; I’d rather shovel out stables.”
I assume this too. Cutting squared pieces is probably simplest. Otherwise, they’re cutting prism pieces with flat tops, or else stacking becomes an interesting nightmare. Everyone everywhere seems to work with rectangular blocks.
You can see the original pyramid, the Step Pyramid at Saquarra. Originally, the graves of the early Egyptian nobility were brick rectagular buildings built of mud brick. Imhotep (architect) made the original mastaba for Djoser (pharoah) the standard rectangular shape, then extended it to square and added more, smaller square constructs on top to form the steps of the first pyramid. There’s interesting studies on how the construction was extended, piece by piece, to end up with the final result. So, rectangular to square. Not much of a leap.
Later pharoahs copied and refined the idea. The Giza pyramids used much larger blocks, rather than the small-ish limestone blocks of the step pyramid; when you’re working in volume, why do extra stonework? They also conceived of covering the outside with smoothed limestone (vsible remnants near the top of the middle pyramid in Giza.)
Later, when the Old Kingdom was in decline and resources were less plentiful, they tried things like filling the interior with rubble instead of solid blocks. Yo can see the result in the Black Puramid, near the Bent Pyramid. All that’s left is a central core and a pile of rubble, like several lesser pyramids from the time.
It occurred to me that there was at least one triangular structure with (I believe) an equilateral triangle cross-section – The Trylon, one half of the iconic symbol of the 1939 World’s Fair (the other half being the Perisphere).
Of course, it’s not at all what I had in mind – it was an incredibly tall “pylon” (“Trylon” is supposed to come from “TRiangular pYLON”), with its height far exceeding any of the sides. But at least it’s the right sort of shape.
Edited to add: Evidently the name “Trylon” has come to mean a three-sided pylon, even it it wasn’t at the 1939 World’s Fair. There’s a company that manufactures them today, but I have no idea where any might be.
They are pretty common as masts for radio antennas. A lot of cellular towers are trylons. IME they’re more common in flat rural areas where you want more height to get more range, and there are no existing buildings to hang antennas on.
When I was back in Boy Scouts and trying to get my Pioneering merit badge (along with a bunch of other guys in my troop, at summer camp) we found ourselves running out of time when we were building a signal tower. It was all long poles lashed together. We made a command decision and decided that it would be a three-sided signal tower, instead of a four-sided one.
Necessity and a truncated schedule is the mother of invention.
I wonder if that’s how they came up with the idea for a “trylon”.
Tetrahedrons make sense for compression structures like masts or signal towers. They have no advantage for mass gravity structures like pyramids, and involve extra work in design and build. What would be the point?
Why? Erosive forces tend to be either uni- or bi-directional, they would not tend to produce tetrahedrons unless there was a particularly strong ground fabric favouring that. And while joint systems with 60° dihedral angles are common enough, the nature of jointing would favour lozenges not triangles. You can get triangular conjugate joints but they tend to be small-scale and not affect regional fabric enough to influence large-scale landform shape.
I have no idea what you mean here. Squat tetrahedra would be “mass gravity” structures just as a four-sided pyramid. Or a five-sided pyramid (which even Martians can build). There’s absolutely no reason you couldn’t build a stable three-sided pyramid, as long as its height was consistent with engineering stability.
Glacial Horns (AKA Pyramidal peaks) form when three or more glaciers diverge from a single point. If there are three, you get a three-sided peak from natural erosion forces
Mont Ventoux on the Tour de France and Triple Divide Peak in Montana (which sends water falling on it into three different watersheds, and ultimately, three different oceans) are examples of three-sided peaks
