Lightning
Each bolt here is grown, not drawn. The leader steps down from the cloud one cell at a time, always toward where the electric field is strongest, until it touches the ground or something tall standing on it. Then the main channel flashes. Turn one dial and the same rule grows a fat blob, a branching coral or a bare wire, and the box-counting chart measures the fractal dimension of each one as it forms.
Lightning, river deltas, frost on a window, coral, mineral dendrites in rock and the cracks in a sheet of glass all branch in a family resemblance you can see from across a room. In the early 1980s physicists found a reason. Several of these shapes grow the same way: something arrives at the edge of the growing thing from far away, and wherever it arrives most often, the edge advances. Tips stick out further, so more arrives there. Fjords between branches are shadowed, so they fill in slowly or never.
In 1981 Thomas Witten and Leonard Sander turned that into a rule you can run. Release one particle at a time far away. Let it wander at random until it touches the cluster, and then it sticks for good. Nothing else. The result is a feathery cluster whose mass does not grow with the square of its radius, as a filled disc would, but with about the 1.71st power. It is more than a line and less than an area, and every reasonable way of measuring it gives the same in-between number.
Three years later Lucien Niemeyer, Luciano Pietronero and Hans Wiesmann found the same rule in electrical breakdown. Treat the discharge as a conductor at the cloud's potential and the ground as another. Solve for the field in the air between them, and add one new piece of channel next to the old one, with probability proportional to the local field raised to a power η. At η = 1 the arithmetic is the same as the random walkers', and the branches come out with the same dimension. Raise η and the channel commits to its strongest tips: fewer branches, a sharper and straighter bolt. Lower it to zero and every edge cell is equally likely, so the discharge swells into a compact blob.
The field also explains the lightning rod. A tall, thin conductor squeezes the lines of equal potential together at its tip, so the field there is far stronger than over open ground, and a leader still high in the sky is already being drawn toward it. In this storm the tree and the tower stand on about a tenth of the skyline and take more than half the strikes. Plant a rod, keep the storm running, and the tile underneath counts how often it wins.
A marble floor that covers the plane perfectly and never repeats — tap any tile and watch its neighbourhood light up all over the floor.