🔮 The Surprise
Start with just three mutually tangent circles.
Using only Descartes' quadratic formula from 1643, you can find a fourth circle
that kisses all three. Apply this recursively and infinite
circles emerge, filling every gap with perfect tangencies. From finite
ingredients, an infinite fractal structure materializes—bounded in space yet
containing infinitely many circles.
📐 The Mathematics
The key insight is curvature (k = 1/radius).
Descartes discovered that for four mutually tangent circles, curvatures satisfy
an elegant quadratic relation. Even more remarkable: certain starting configurations
produce integral gaskets where every single circle
(infinitely many!) has integer curvature. Try the "Integral" preset!
🌀 Circle Centers
Finding circle radii isn't enough—we need positions too. The
Complex Descartes Theorem extends the formula
to complex numbers representing centers: (k₁z₁ + k₂z₂ + k₃z₃ + k₄z₄)² =
2(k₁²z₁² + k₂²z₂² + k₃²z₃² + k₄²z₄²). This gives the exact position of each
new circle in the complex plane.
∞ Fractal Properties
The Apollonian Gasket has Hausdorff dimension ≈ 1.3057—
more than a line but less than a plane. The gasket is not
strictly self-similar (unlike the Sierpiński triangle), yet it has
well-defined fractal dimension. Related to Kleinian groups, hyperbolic geometry,
and number theory—a crossroads of modern mathematics.