Recursive kissing circles: pack the gaps forever, discover a fractal with Hausdorff dimension ≈ 1.3057
Given 3 mutually tangent circles, Descartes' theorem gives the curvature of the 2 tangent circles that fit in each gap. The formula is quadratic — exactly 2 solutions.
Start with 3 mutually tangent circles inside a large enclosing circle. In each curvilinear triangle gap, pack a new circle tangent to all 3. Recurse forever.
The result is an Apollonian gasket — a fractal with Hausdorff dimension ≈ 1.3057...
Remarkably, if the initial 4 curvatures are integers, all curvatures remain integers (Apollonian group theorem, 1999).
Starting with curvatures (−1, 2, 2, 3): generates an integer Apollonian packing. Every integer that appears has special number-theoretic properties.