Apollonian gasket playground

Descartes’ Circle Theorem

Signed curvature k = ±1 / r  (negative for the outer circle).

For four mutually tangent circles with curvatures k₁, k₂, k₃, k₄:

  (k₁ + k₂ + k₃ + k₄)² = 2 (k₁² + k₂² + k₃² + k₄²)

Solving for k₄ given the other three:

  k₄ = k₁ + k₂ + k₃ ± 2 · √( k₁ k₂ + k₂ k₃ + k₃ k₁ )

The complex Descartes theorem gives the new centre z₄:

  (z₁ k₁ + z₂ k₂ + z₃ k₃ + z₄ k₄)²
        = 2 (z₁² k₁² + z₂² k₂² + z₃² k₃² + z₄² k₄²)

Initial config: outer disk + three equal inner circles
  r_in = R · (2√3 − 3)
  recursively fill every triangular gap.

Hausdorff dim ≈ 1.30568   (McMullen, 1998 — no closed form)

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