Apollonius Circle & Gasket

Given three mutually tangent circles, Apollonius (262 BC) found two tangent circles. Iterating produces the Apollonian gasket — a fractal with dimension ≈ 1.3057.

Controls

Statistics

Circles: —
Dim: 1.3057

Descartes Circle Thm

(k₁+k₂+k₃+k₄)²
= 2(k₁²+k₂²
+k₃²+k₄²)

k = 1/r (curvature)

k₄ = k₁+k₂+k₃
±2√(k₁k₂+k₂k₃
+k₃k₁)

Fractal Dimension

The Apollonian gasket has Hausdorff dimension δ ≈ 1.3057 (McMullen 1998). The residual set has measure zero.