Apollonius Gasket
A circle-packing fractal built from Descartes’ Circle Theorem. Starting with three mutually tangent circles, each curvilinear gap defines exactly one new tangent circle — and the process repeats infinitely.
Descartes’ Circle Theorem and the Apollonian Gasket
The Apollonian gasket is named after Apollonius of Perga (c. 262–190 BC), who studied the problem of finding a circle tangent to three given circles. The fractal construction begins with three mutually tangent circles inscribed in an outer bounding circle. In each curvilinear triangular gap between three tangent circles, there exists exactly one circle tangent to all three — and this new circle creates new gaps, each of which can be filled in turn.
The key to computing each new circle is Descartes’ Circle Theorem (1643): given four mutually tangent circles with curvatures k₁, k₂, k₃, k₄ (where curvature k = 1/r, and the outer bounding circle has negative curvature), the relationship (k₁ + k₂ + k₃ + k₄)² = 2(k₁² + k₂² + k₃² + k₄²) holds. Given three known curvatures, we solve for the fourth: k₄ = k₁ + k₂ + k₃ + 2√(k₁k₂ + k₂k₃ + k₃k₁). The two solutions correspond to the two circles tangent to all three — one in each of the two gaps.
A remarkable discovery by number theorists is the integral Apollonian gasket: if the initial four curvatures are all integers, then every circle in the infinite packing also has integer curvature. The configuration (-1, 2, 2, 3) shown in the “Integral” preset is the most famous example. This integrality property connects circle packing to deep questions in number theory, including the distribution of prime curvatures and connections to automorphic forms.
The Apollonian gasket has a fractal (Hausdorff) dimension of approximately 1.3057, meaning it is more than a curve but less than a surface. It appears throughout mathematics and physics — in the study of Kleinian groups, hyperbolic geometry, granular materials, and even the structure of certain quantum systems. The gasket is self-similar: zooming into any region reveals the same recursive structure at every scale.