The Apollonian Gasket: Packing Infinity Into a Circle

Three circles, one guaranteed fourth

Take any three circles that are mutually tangent — each one touching the other two at exactly one point, none overlapping. There is always a circle that fits snugly into the curved triangular gap between them, tangent to all three. This is not a coincidence of the specific circles you chose; it follows from a beautifully compact piece of algebra known since antiquity and rediscovered by René Descartes in a 1643 letter: Descartes' Circle Theorem.

Descartes' Circle Theorem

The theorem is stated in terms of curvature, k = 1/radius, with a sign convention: a circle that encloses the other three (the outer boundary of the whole gasket) is given a negative curvature, since it bends the opposite way relative to the small circles nested inside it.

(k1 + k2 + k3 + k4)^2 = 2·(k1^2 + k2^2 + k3^2 + k4^2)

solved for the new circle's curvature:
k4 = k1 + k2 + k3 ± 2·sqrt(k1·k2 + k2·k3 + k3·k1)

k = 1 / radius   (negative for a circle that encloses the others)

Given any three mutually tangent circles, this formula hands you the curvature — and from curvature the radius — of the fourth tangent circle directly, with no trigonometry, no iteration, no root-finding. The ± reflects the fact that there are always two circles tangent to a given trio, one on each side; a gasket construction keeps the smaller root, the one that nestles into the existing gap rather than swallowing the whole configuration.

Recursion: filling the gaps of the gaps

Start with an outer circle and two circles packed inside it, mutually tangent to each other and to the boundary — three circles, three pairwise tangencies. Descartes' theorem gives you a fourth circle filling the curved triangle in the middle. That fourth circle now forms three new curved triangles with its neighbours — one against each pair it touches — and each of those gets its own new inscribed circle by applying the same formula again. Every new circle spawns three more gaps, forever. The construction never terminates because a curved triangular gap, however small, is topologically the same shape no matter how far down you recurse — there is always room for one more circle.

A fractal with a strange, precise dimension

After infinitely many generations, the circles cover the entire area of the outer boundary except for a residual set of points that belong to no circle at all — the gasket itself, the boundary curve you actually see. That residual set has zero area yet is far more intricate than any ordinary curve, and its Hausdorff dimension — the fractal dimension measuring how it fills space between one and two dimensions — has been computed numerically to be approximately 1.3057. Unlike the dimension of, say, the Koch snowflake, this number has no known closed-form expression in terms of familiar constants; it emerges from the multifractal spectrum of how circle sizes are distributed across generations, and mathematicians have spent decades refining ever more precise numerical bounds on it.

Integer curvatures: Apollonian circle packings and number theory

If the starting four circles have integer curvatures, Descartes' formula guarantees that every single circle in the entire infinite gasket also has an integer curvature — a striking fact, since the formula involves a square root. It works because starting from an integral Descartes quadruple, the recursion only ever needs the linear relation k4 = 2(k1+k2+k3) - k4' (the sum of the two roots of the quadratic), which preserves integrality automatically. This turns the gasket into a genuine object of study in number theory: which integers appear as curvatures, how densely, and with what statistical distribution, are still active research questions connected to the theory of quadratic forms and the arithmetic of Apollonian groups.

Frequently asked questions

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