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
| Discovered | Apollonius of Perga (3rd century BCE) / Descartes (1643) |
|---|---|
| Hausdorff dimension | ≈ 1.30568 (McMullen, 1998) |
| Topological dimension | 1 |
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 dimH ≈ 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.