Deflection grows with the cube of length
The single most useful proportionality in mechanical design.
Deflection grows with the cube of length because a longer beam has both more length to bend along and a longer lever arm working against its stiffness at every point along that length, and those two effects compound, so doubling a beam's length can multiply how far its tip droops under the same load by roughly eight times rather than two.
Leverage and length, multiplied together
A beam supported at one end and loaded at the other resists bending because the material near its fixed end has to generate an internal resisting force strong enough to balance the load acting at the far end, and the further away that load acts, the larger the leverage it has, exactly the way a longer spanner lets the same hand force turn a stiffer bolt. Making the beam longer increases that leverage in a straightforward way, but it also gives the beam more length over which the same small amount of bending, spread out along every slice of the beam, can accumulate into a larger total droop at the tip. These two effects, greater leverage and more length for the bending to accumulate over, both grow with the beam's length, and because they multiply together, the total deflection at the tip grows with the cube of the length. Counted strictly there are three factors of length at work: the leverage at the root grows with length, the bending spread along the beam gives the tip a slope that grows with length again, and that slope carried out over the whole distance to the tip adds the third. Said in one sentence, the tip deflection of a cantilever is proportional to the load multiplied by the length cubed, and divided by both the material's stiffness and the cross-section's own resistance to bending.
A diving board twice as long
A short diving board barely dips at all when someone stands on the end of it, springing back almost as soon as they step off. Extend that same board to twice its original length, using the identical material and the identical thickness throughout, and under the same person's weight the tip droops about eight times as far, because both the increased leverage acting on the fixed end and the extra length available for the bend to build up along have doubled together, and their combined effect on deflection is what the cube of the length actually captures. A springboard used for diving is deliberately built long and flexible for exactly this reason, since the same modest push from a diver's legs, multiplied by the long board's disproportionate deflection, returns far more spring at the end of the board than a short, stiff platform of the same material ever could.
Shorten the span before thickening the beam
Because deflection grows so steeply with length, shortening the unsupported span of a beam is an unusually powerful way to stiffen it compared with almost any other single change available to a designer, since a modest reduction in length produces a disproportionately large reduction in droop. For a given material and cross-section, the unsupported span is what most strongly predicts how stiff a beam will feel in practice, since the same beam that feels reassuringly stiff over a short span can feel alarmingly soft the moment its supports are moved further apart. Thickening a section to cure flex adds weight along the whole length of the beam, while moving a support closer attacks the cubed length term for the cost of a single bracket. Halving the length of a cantilever cuts its tip deflection to an eighth. Doubling the thickness of a rectangular section achieves the same eightfold gain, since stiffness grows with the cube of thickness too, but it doubles the weight of the beam to do it.
A bookshelf spanning between two brackets and loaded evenly along its length does worse still. A longer shelf carries more books as well as bending over a longer span, so its sag grows with the fourth power of the span, and a shelf half again as long sags about five times as far under books packed just as tightly.
Small errors in span, large errors in stiffness
The cube also explains why small manufacturing errors that leave a support slightly further from where it was meant to sit can cause a part to feel disproportionately floppier than the drawing predicted, since the error is cubed. A support sitting a tenth further from the load than drawn gives about a third more deflection, and one a fifth further out gives about 70% more. A mounting bracket built a few millimetres taller than the drawing specified, moving its supported point slightly further from the load, can end up springier than the same bracket built correctly by an amount that looks out of proportion to such a small error until the cube is remembered.