Weight and joint integrity aside, what shape is most likely to fail?
I’d guess that a square would be. I’m no ME though.
How about a circle? Imagine a bunch of tiny sticks, attached end to end to end, all the way around. I don’t know, but it would seem like the logical opposite.
It’s all about how the structure is loaded. A triangle makes a good truss since it can’t deform without compressing or stretching any of the sides where a rectangle needs rigid corners to prevent deforming its shape. There’s also the wobbly table syndrome. Three leg tables won’t wobble even if all the legs aren’t exactly the same which isn’t true of having more legs.
Circles can be very stable. Look at any compressed gas cylinder. A sphere is stable because it’s the smallest surface area for a given volume. If any of the surfaces were flat they would bulge under pressure.
I think you have to define your terms better to get a definitive answer. Are you talking about just a self supporting structure or a load bearing one. Is it in terms of weight supported/weight of structure? Obviously there are many structures which will fall over because their centre of balance is wrong - do they count?
A very long thin beam
I believe some littlle-known ancient greek manuscripts identiify it as the legendary “f*cked-up-a-hedron”. 
I can’t imagine a (non-square) parallelogram being able to support much.
Hilarious.
A long, thin column would be worse.
im not entirely sure what was fully intended by the question, so ill answer it by my initial impression. the revised question (read: the way im answering it) reads
If we built something in many different shapes, which would take the shortest amount of time to destroy by hitting it with sticks repeatedly
corners are weak points. this is obvious once brought up, so i dont think a cite is necessary. ergo the less corners a structure has, the harder it will be to raze. triangular structures are stronger than their quadrilateral counterparts, and building things to be circular can be tricky (especially if you dont have a calculator with that handy pi button) Some castles did have circular towers, but they were ‘a royal pain’ to errect. come to think of it, im not sure why there arent any notable triangle-shaped towers. none, at least, that i can think of
to answer the question, ill say whichever shape you can think of that has the most sides. me, im not sure what its called past a icosagon (20 sides)
How about an upside-down L? You’ve got to beef up the corner of the L, and also stabilize the base.
I know a mathematician who specializes in something called structural stability, which would seem to be the intent of the question. Let me start with an experiment. Place your forefingers tip to tip so that they make, say a 60 degree angle and ask someone to press them down staying in the plane of that angle. The second person should not be able to move them. Oh, he could push your hands down, but he won’t be able to change the angle. Now make the angle 0 so that the fingers make a straight line. Now the second party will have no trouble pushing them down. You will not be able to keep them straight. What you have demonstrated is the first configuration is stable and the second not. If your fingers make a small angle, say 5 degrees, it will still be difficult to maintan that position under a force perpendicular to the direction of the fingers (more precisely to the line between the knuckles). The technical point is that the derivative of the distance between the knuckles with respect to distance of the fingertips from the line between the knuckles is 0 at the horizontal position. This notion of the derivative of the length with respect to the perpendicular position is, according to my informant, the key to structural stability. Sorry for getting so technical, but that is the way it works. He further told me that had created a linkage in which not only the first derivative, but the first five derivatives vanished. Although the structure was theoretically rigid, no material he could find was itself sufficiently rigid that the linkage remained rigid.
I should perhaps emphasize that this all concerns structures that are rigid in principle. Obviously a square with flexible corners cannot be rigid. But the structure I described above was made up entirely of triangles and therefore had no flexible corners.
I don’t know how far the derivatives go, but the textbook example of a “pseudo-rigid” structure is a cube with opposite vertices connected (the vertices are fully articulate on their own). Such a structure can undergo infinitesimal distortions without corresponding stresses. Apparently, in some real-world applications, such structures are the most durable, since they flex slightly in response to earthquakes, high winds, etc., rather than just collapsing as a whole (Aesop’s example of the reed and the oak tree)
The reason a triangle is stable is because, no matter how loosely the corners are joined, the triangle can’t collapse because there is no freedom of movement for any of the joints. For any shape with more than three sides, though, it is possible to collapse the shape. So the answer to the OP is that it’s a tie among all the shapes with more than three sides. So the most stable shape is the triangle, and the least stable shape is all the rest.
I don’t know the rght way to the say this, but I suspect that triangles are the shape that retain their “shape”, i.e are still triangles, with the highest probability when they are transformed via a continuous linear transformation (I guess a point and then a line woudl trump a triangle). Using this “logic” it would seema circle is the least structurally secure shape.
Then why do they use circles to make arches to support cathedrals?
If an object just has to support its own weight and is made of imcompressable material then an arch afaik is the most stable way of distributing the forces. Any straight beam (even in a triangle) is susceptable to buckling or cracking .
however, this is of course true for real 3-D materials, not necessarily a hypothetical 1-D beam structure in the original question
Blancmange.
I was going to give you 10 out of 10 for being able to spell blancmange but you cheated, Mangetout.
Do the pyramids of Giza fall into the realm of triangular contruction?
They’ve held up fairly well.