Beautiful Abelian Sandpiles
4 days ago
- #visual-patterns
- #mathematics
- #group-theory
- Abelian sandpiles are grids where each cell can hold grains of sand; if a cell has 4+ grains, it topples into neighboring cells.
- Toppling continues until all cells have ≤3 grains, ensuring stability. Grains fall off the grid's edge during toppling.
- The 'Abelian' term refers to the property that the order of toppling doesn't affect the final stable configuration.
- Sandpiles form an abelian group, with operations like addition of sandpiles and the existence of an identity sandpile.
- Recurrent sandpiles repeat and form the abelian group, while transient sandpiles (like the empty grid) do not.
- The identity sandpile is a recurrent configuration that leaves another sandpile unchanged when added.
- Identity sandpiles exhibit fractal-like, symmetric patterns, especially visible in larger grids.
- Sandpiles connect to abstract algebra and group theory, offering both mathematical depth and visual beauty.