Apollonian Gasket
An infinite fractal packing of circles, each tangent to its neighbours, filling every gap with smaller circles.
Apollonius’ Theorem
Given three mutually tangent circles, there are exactly two circles tangent to all three — the inner and outer Soddy circles. Descartes’ Circle Theorem relates their curvatures:
(k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²)
where k = 1/r is the curvature (reciprocal of radius). If three curvatures are known, the fourth can be solved exactly.
The seed: a large outer circle and three mutually tangent inner circles. The curvilinear triangle gaps between them wait to be filled.
Infinite Descent
In each curvilinear triangle, we inscribe the unique circle tangent to all three bounding circles. This generates three new, smaller triangles. Repeat forever.
The gasket after several rounds of filling. The residual set — the fractal dust of points never covered by any circle — has Hausdorff dimension approximately 1.3058. If the initial three circles have integer curvatures, every circle in the packing has an integer curvature.