Drop grains of sand on one square. When a square holds four, it topples — one grain to each neighbour — maybe toppling them in turn, an avalanche. Two miracles: the mess always settles, and the final pattern is a fractal that doesn’t care what order you toppled in. Slide the grains and watch it self-organise.
A cell with height ≥4 topples: subtract 4, add 1 to each of its 4 neighbours (grains off the edge are lost). Keep toppling until every cell is <4 — it always terminates. The deep fact (the ‘Abelian’ in the name): the FINAL stable configuration does not depend on the order you choose to topple. A fail-loud self-check drops the same pile and stabilises it two different ways — forward and reversed toppling order — and throws unless the two final grids are byte-for-byte identical.
A finite grid loses grains off its edge (an open boundary); the Abelian property and termination are exact theorems (Dhar 1990), reproduced here on that grid. The emergent fractal is the self-organised-criticality signature, shown not derived.