There are pyramid-shaped buildings all over the world – most famously the Pyramids of Egypt (and most famously, the three at Giza, although there are plenty of others elsewhere). There are pyramids in Mexico (in Teotihuacan and Cholula and Chichen-Itza, among others), in China (Xi’an). There’s the modern pyramid at Memphis, Tennessee, echoing the one in Memphis, Egypt, and the Pyramid of Luxor in Las Vegas. And the Pyramid at the entrance to the Louvre. And lots and lots of smaller ones elsewhere (here’s a list of pyramidal mausoleums in the US:
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Every one of these, as far as I can tell, has a square base, making them sort of like a regular octahedron cut in half. The shape, it has been claimed, was inspired by natural eroded rock structures throughout the Middle East. But there’s no reason for such an eroded structure to have a square cross-section. Why do all human-built pyramids have square horizontal cross-sections, no matter where they’re built?
In particular, why aren’t there any tetrahedral pyramids, with triangular horizontal cross-sections? You’d expect there to be some natural eroded tetrahedral-shaped structures. The Tetrahedron is one of the five Platonic Solids, and was certainly familiar to the ancient world (Caltrops were common in the ancient world, and have tetrahedral symmetry). Why don’t we see tetrahedral pyramids?
One discussion board on the internet suggested it was hard to build such a structure – “What would the bricks look like?” Well, you can build a pretty good approximation of a tetrahedral pyramid using square or rectangular blocks. But if you want the bricks to fit the shape, you can easily build a perfect tetrahedral pyramid out of courses of equilateral triangular bricks, or a combination of triangular and hexagonal bricks (each hexagon made up of six of the equilateral triangles), or out of trapezoidal bricks and triangular bricks (each trapezoid being made up of three equilateral triangles, or half a hexagon). Or a combination of all three.)
Heck, if you insist on rectangular bricks, you can relax the requirement about sixty degree angles at the apex and create a three-sided pyramid with right angles at the apex made up of rectangular bricks – they just won’t be lying horizontally. A three-sided pyramid is what you get if you have one point of a cube pointing upwards.
I’ve looked through the internet, and can find no tetrahedral structures or buildings. Tetrahedrons HAVE been used as the basis for construction. As the simplest three-dimensional form, it’s also the most rigid, as a triangle is the most rigid 2D form. Alexander Graham Bell made structures made out of linked tetrahedra and claimed they were sturdy and simple to produce – but none of them looked like a simple tetrahedron itself. The Tetrahedron Skyscraper in San Diego similarly exploits the structural rigidity of the tetrahedral form, but it doesn’t look like simple tetrahedron, either.
Via 57 West in New York City claims to be a “tetrahedron”, but doesn’t look it.
This paper suggests building tetrahedral structures on other planets because it’s so structurally stable
..but if it’s so simple, so elegant, and so stable, why doesn’t anybody already do it here on earth? Wouldn’t a stable building be a shoo-in in earthquake zones?