One rule, crazy pattern 🤔

This is the abelian sandpile, introduced by physicists Per Bak, Chao Tang and Kurt Wiesenfeld in 1987. The rules are simple: Drop grains of sand, one at a time, on the same square. Any square with 4 grains topples, giving one grain to each of its 4 neighbours. Each topple can trigger another, creating an avalanche. The first grains are easy to follow. The 4th creates a plus, and the 16th triggers the first chain reaction. Then we speed up to 200,000 grains per second until all 262,144 (2^18) grains have landed. It takes over 1.2 billion topples to settle them into a pile 375 squares across. You might expect a smooth, round heap. Instead, the sand draws a fractal: repeating patterns, stripes and checkerboards, forming triangles that shrink toward the centre. ⬛ Black: 0 grains 🟦 Dark to light teal: 1, 2 or 3 grains Every grain plays a xylophone note. Small changes follow a G major chord, while avalanches play higher notes depending on their size. The biggest avalanche you hear? A single grain triggering 145,949 topples. But here's the strange part: The order doesn't matter. Topple squares in any order, or drop all grains at once, and you get exactly the same picture. That's the "abelian" property, established by Deepak Dhar in 1990. In 2013, Wesley Pegden and Charles Smart proved that as you add more sand, the scaled-down pattern approaches a single limit. Later, researchers discovered an Apollonian circle packing hidden within its structure. Yet nobody can describe that limit exactly. When it was first proved to exist, nobody had even proved its outline wasn't a perfect circle. . #AbelianSandpile #Fractal #Math #Sandpile #STEM #MathAnimation #Mathmotion