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Same material. Turn it sideways and it’s a different structure.
Cardboard is secretly a bundle of tiny tubes, all running the same way — that is its grain. Stand those tubes up and the weight runs straight down their walls to the table. Lay the same tubes on their sides and the walls have to bend instead of carry, so the block flattens. Same trick as a drinking straw: rock solid on its end, flat the moment you press its side. (A material that is strong one way and weak the other is called anisotropic — cardboard is one of the most extreme there is.)
For your stoolPoint the grain up and down in every leg and wall.
Strength resists breaking. Stiffness resists bending — and depth is how you buy it.
A flat sheet has almost no depth, so a load folds it easily. Fold that same sheet up into a channel and you move material away from its centre line, which is what actually fights bending. (Engineers measure that with a number called the second moment of area.) Stiffness climbs with the cube of depth: make a beam twice as deep and it becomes roughly eight times harder to bend. You paid nothing — it is the same piece of cardboard.
For your stoolFold flat panels into channels. Depth is stiffness, for free.
A rectangle is a mechanism. A triangle is a structure.
The four corners of a rectangle can hinge, so it folds over sideways into a parallelogram without a single side changing length — nothing has to stretch or break for it to collapse. Add one diagonal and every part of the frame becomes a triangle. A triangle cannot change its shape unless one of its sides actually gets longer or shorter, so it locks.
For your stoolBrace every open rectangle with a diagonal.
One strip alone takes everything. Woven together, they all take a share.
Press one spot on a single sheet and that spot takes all of it — until the board creases and gives way. A woven mat is two layers thick everywhere the strips cross, and it spans in both directions at once, so pushing on one strip drags all of its neighbours in with it. The same push becomes a small share for every strip. In engineering that is redundancy: not wasted material, but many routes to the ground instead of one. It is the opposite of what the word means in everyday speech — here it is exactly what keeps you safe, and it is how a chair seat works.
For your stoolWeave or grid the seat so every strip takes a share.
A column doesn’t fail by being crushed. It fails by going sideways.
A long thin column almost never runs out of material strength — it runs out of stability. Past a certain load it becomes easier for the column to bow sideways than to stay straight, and once it starts it cannot stop. What sets that load is shape, not amount: a flat leg has one direction it is easy to bend, so it picks it. Roll that same sheet into a tube and there is no weak direction left to pick.
For your stoolRoll your legs into tubes. Shape beats thickness.
Loose layers slide. Glued layers can’t — so they act as one thick board.
Stack three strips and bend them: each one bends on its own, and their ends slide past each other like a fanned deck of cards. You have three thin beams, not one thick one. Glue stops that sliding, and the instant it does the three layers have to bend as a single deep board. Glue’s real job is not holding pieces together — it is stopping the slip. And since stiffness goes with the cube of depth, refusing to slip is worth far more than the glue weighs.
For your stoolGlue every layer face to face — and cross the grain, like plywood.
Weight needs a straight road to the ground.
Every force you put on a structure has to travel somewhere — through the parts and down to the floor. That route is the load path. Put a leg directly under the load and the force drops straight down the tube; the seat hardly knows it is there. Move the legs out to the ends and the force has nowhere to go but sideways through the seat first, bending it the whole way, before it can find a leg. Same seat, same weight, same cardboard — nothing got weaker. Only the path got longer.
For your stoolPut legs directly under where you sit — including when you lean back.