An Apollonian gasket is built by repeatedly filling gaps with the unique circle tangent to three others. Descartes' Circle Theorem: (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²) where k = 1/r (curvature). If k₁,k₂,k₃ are integers, all subsequent curvatures are integers!