Apollonian Gasket
Infinite circle packing via Descartes' Circle Theorem: (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)
About: An Apollonian gasket starts with three mutually tangent circles and fills every curved triangular gap with the unique circle tangent to all three. Descartes' Circle Theorem (1643) gives the curvature k₄ of the fourth circle: k₄ = k₁+k₂+k₃ ± 2√(k₁k₂+k₂k₃+k₃k₁). If the initial curvatures are integers, all subsequent curvatures are integers — an amazing number-theoretic fact. The gasket has Hausdorff dimension ≈ 1.3057... (Apollonius problem, studied since 200 BCE).