Three mutually tangent circles leave two curvilinear triangular gaps. Each gap admits a unique inscribed circle tangent to all three (Apollonius of Perga, 2nd century BC). Fill recursively forever — you get a fractal with Hausdorff dimension ≈ 1.3057.
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Descartes' kissing-circle theorem (1643)
If four circles are mutually tangent and their (signed) curvatures (1/radius) are k₁, k₂, k₃, k₄, then
(k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²)
Given three of them, the equation is a quadratic in the fourth — giving the two solutions for circles tangent to all three. The gasket recursion swaps each circle for its conjugate partner: k₄' = 2(k₁ + k₂ + k₃) − k₄.
Remarkable fact: if all four starting curvatures are integers, every circle generated by the recursion has integer curvature. The result is an integer Apollonian packing — a fractal whose curvature spectrum is purely arithmetic, related to deep questions in number theory and the Riemann hypothesis.