DIRECTORY / SANDBOXES / SANDPILE.ABELIAN

ABELIAN SANDPILE MODEL

Bak-Tang-Wiesenfeld self-organized criticality exhibiting power-law fractal avalanche distributions.

RUNNING

// GRAIN INJECTION & CRITICAL PRESETS

// POINTER BRUSH TOOL

// COLOR PALETTE

⚡ PLUS // ABELIAN ALGEBRA & SOC POWER-LAW ACTIVE

// LATTICE TOPOLOGY (NEIGHBORHOOD)

// CREUTZ IDENTITY & ABELIAN GROUP ADDITION

SOC POWER-LAW HISTOGRAM P(s) ~ s^-τ τ ≈ 1.22

SUB-STEPS / FRAME 25x

TOTAL TOPPLES: 0

DISSIPATED GRAINS: 0

💡 TIP: Click the canvas to drop sand at the cursor, drag to pour a trail. Shift+Drag to draw wall barriers.

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Self-organized criticality, 1/f power-law avalanches, and Apollonian fractal gaskets.

ACADEMIC & ALGORITHMIC REFERENCE

// HISTORICAL ORIGINS

Introduced by physicists Per Bak, Chao Tang, and Kurt Wiesenfeld in 1987 as the foundational model of Self-Organized Criticality (SOC).

// GOVERNING EQUATIONS

z(x, y) \ge 4 \implies z(x,y) \leftarrow z(x,y) - 4, \quad z(\text{neighbors}) \leftarrow z(\text{neighbors}) + 1

// BROWSER IMPLEMENTATION

Iteratively relaxes grid cells with 4 or more grains by distributing to 4 orthogonal neighbors until the lattice reaches a sub-critical equilibrium.

// GUIDED EXPERIMENTS TO TRY IN THIS SANDBOXNORMAL

1Apollonian Circle Emergence

🎯 Action:Select '⏳ DROP AT CENTER' and click '💥 +10K GRAINS' to observe toppling cascades.

✨ Observe:Observe the spontaneous formation of self-similar fractal circular patterns and Sierpinski triangles.

// CURATED PAPERS, RFCS & RESOURCES

The landmark paper introducing self-organized criticality in statistical physics.