Circle packing

Apollonian gasket

Start with four mutually tangent circles. In every gap, by Descartes’ Circle Theorem (1643), there are exactly two new circles tangent to the three surrounding ones; place them. Recurse. The closure of all the circles produced is the Apollonian gasket, named after Apollonius of Perga’s problem of finding tangent circles to three given.

At a glance

Descartes’ Circle Theorem

For four mutually tangent circles with signed curvatures k₁, k₂, k₃, k₄ (curvature = 1 / radius, negative for the outer circle enclosing the others):

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

Treating the equation as a quadratic in k₄ gives two solutions, the two new tangent circles in any gap.

Integer Apollonian gaskets

If four mutually tangent starting circles have integer curvatures, Descartes’ theorem implies that every subsequent curvature is also an integer. The arithmetic of such gaskets has been the focus of major number-theoretic work since the 2000s (Graham-Lagarias-Mallows-Wilks-Yan, Bourgain-Kontorovich, and others).

Hausdorff dimension

Curtis McMullen computed dim_H ≈ 1.30568 in 1998 using thermodynamic formalism. No closed form is known.

References

• Descartes, R., letter to Princess Elizabeth, 1643.
• McMullen, C., “Hausdorff dimension and conformal dynamics III: Computation of dimension,” Amer. J. Math., 1998.
• Graham, R. L., Lagarias, J. C., Mallows, C. L., Wilks, A. R., Yan, C. H., “Apollonian circle packings: Number theory,” J. Number Theory, 2003.

Try it

Run an interactive playground at /tools/apollonian.