Flute

it won't hold itself up. pinch the edge.

pinch the edge of a drooping paper strip into a U and it stiffens into a bridge

Paper cannot stretch, so it can curve one way or the other but not both at once. Curl it across its width and bending it along its length would mean stretching it, so it stiffens: the deeper the U, the more it holds. It is why a pizza slice folded into a U stays up and a tape measure stands out in the air.

Concepts: gaussian-curvature, geometry, bending-stiffness, thin-sheets, second-moment-of-area, theorema-egregium.

An interactive you play with one thumb, on Wundr.