Apollonian gasket — infinite circle packing from four mutually tangent circles
Descartes Circle Theorem: (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)
Descartes' theorem (1643): if four mutually tangent circles have curvatures k₁,k₂,k₃,k₄ (reciprocals of radii), then (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²). Starting from three tangent circles, there are exactly two Soddy circles tangent to all three. The Apollonian gasket fills the curved triangle with infinitely nested tangent circles — its dimension is ≈1.3057, and integer gaskets (Descartes' theorem preserving integer curvatures) exist starting from (−1,2,2,3).