A grain of sand has no imagination. But let enough grains fall, and geometry appears where no one drew it. Touch the world. Watch it rearrange itself.
An improbable claim: add sand at A, then B; or add it at B, then A. Every intermediate avalanche looks different—but after everything settles, the two worlds will be identical, square for square.
A toppling at site x changes the height configuration by the fixed vector −4δx + ∑y∼xδy, with grains outside the world discarded. These changes add commutatively. The less obvious part is that every legal sequence of topplings terminates in the same stable configuration: the abelian sandpile theorem (often proved using a least-action argument). Because adding grains at A and B also commutes, the two final configurations must match, regardless of which avalanches happened first.
This is a genuine theorem, not a trick of the animation. The experiment compares every square, not just their total mass. The finite open boundary acts as a sink, ensuring stabilization.