Circle Packing
Watch Apollonian gaskets build themselves circle by circle. Descartes’ Circle Theorem recursively fills every gap with the largest tangent circle that fits — a fractal that connects geometry, number theory, and hyperbolic space.
About this lab
Descartes’ Circle Theorem
Given four mutually tangent circles with curvatures k1, k2, k3, k4 (where
curvature k = 1/r), Descartes’ Circle Theorem states:
(k1 + k2 + k3 + k4)² = 2(k1² + k2² + k3² + k4²)
Given three mutually tangent circles, this equation is quadratic in k4 and
has exactly two solutions — one for each side of the triple. The larger solution
(smaller circle) fills the inner gap; the smaller solution fills the outer gap or
gives the bounding circle (with negative curvature, since it contains the others).
Apollonius of Perga
The problem of constructing a circle tangent to three given circles dates to Apollonius of Perga (~200 BCE). The Apollonian gasket is the fractal produced by starting with three mutually tangent circles and repeatedly filling every curvilinear triangle with the unique circle tangent to all three sides. The recursive process never terminates — there is always room for one more circle — producing a set of measure zero with infinite total boundary length.
Integer Apollonian packings
A remarkable property: if the initial four curvatures are all integers, then
every subsequent curvature in the packing is an integer. The classic example
starts with curvatures (-1, 2, 2, 3) — a bounding circle of radius 1
containing two circles of radius 1/2 and one of radius 1/3. Every circle that appears
thereafter has integer curvature. This connects Apollonian packings to deep questions in
number theory: which integers appear as curvatures? The answer involves quadratic forms and
is not yet completely understood.
Fractal dimension
The residual set of an Apollonian gasket — the dust of points not interior to any
circle — has Hausdorff dimension approximately 1.30568. This is strictly
between 1 (a curve) and 2 (an area), confirming its fractal nature. The dimension is
related to the critical exponent of the Poincaré series of the underlying
Möbius group acting on hyperbolic 3-space.
Connection to hyperbolic geometry
Apollonian packings are intimately connected to hyperbolic geometry. The group of Möbius transformations preserving the packing is a Kleinian group — a discrete subgroup of the isometries of hyperbolic 3-space. Each circle in the packing corresponds to a hemisphere in the upper half-space model, and the packing itself is the shadow (stereographic projection) of these hemispheres onto the boundary plane. This is why Apollonian gaskets appear naturally in the study of hyperbolic manifolds and their deformation spaces.