The Apollonian Gasket: Circles Packed to Infinity

The construction: circles all the way down

Start with three circles arranged so that each one touches the other two — three mutually tangent circles, like coins pushed together on a table. The ancient question, Apollonius' problem (after Apollonius of Perga, c. 200 BC), asks which circles are tangent to three given ones. For three circles that already kiss each other pairwise, the answer is clean: there are exactly two circles tangent to all three at once — a small one nestled in the central gap, and a large one wrapping around the outside. These are the two Soddy circles.

Choose the outer one as a boundary. Inside it you now have four mutually tangent circles, and between them sit curved triangular gaps — regions bounded by three circular arcs, called interstices or curvilinear triangles. Each such gap is itself bounded by three mutually tangent arcs, so Apollonius' problem applies again: there is a unique new circle tangent to all three. Drop it in. That single circle splits one gap into three smaller gaps. Repeat in every gap, forever.

The number of gaps triples at each stage, so the count of circles grows geometrically — roughly like 3ⁿ new circles at stage n — while their radii shrink fast enough that the total area they cover converges to the area of the bounding disk. What is left uncovered, the dust of tangency points and limiting arcs, is the Apollonian gasket itself.

Descartes' Circle Theorem: the engine of the cascade

The whole construction is powered by one identity relating the sizes of four mutually tangent circles. Measure size by curvature k = 1/r, the reciprocal of the radius, with a sign convention: a circle is given positive curvature when the others touch it from outside, and negative curvature when it encloses the others (so the outer bounding circle of radius R has curvature −1/R). With that bookkeeping, the four curvatures obey Descartes' Circle Theorem:

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

René Descartes stated it in a 1643 letter to Princess Elisabeth of Bohemia; the chemist Frederick Soddy rediscovered it in 1936 and published it as verse ("The Kiss Precise") in Nature. Treat the equation as a quadratic in the unknown fourth curvature k₄. Writing s = k₁ + k₂ + k₃, it becomes k₄² − 2s·k₄ + (s² − 4(k₁k₂ + k₂k₃ + k₃k₁)) = 0, whose two roots are

k₄ = k₁ + k₂ + k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁).

The two roots are precisely the curvatures of the two Soddy circles — the plus sign gives the tight inner circle (large curvature), the minus sign the enclosing or outer circle. Because they are the two roots of the same quadratic, Vieta's formula gives their sum, and this relation is the secret of everything that follows.

Locating the circles: the complex Descartes theorem

Descartes' theorem fixes the sizes of the new circles but not where they sit. In 2002, Lagarias, Mallows, and Wilker sharpened it into a version that also pins down the centers. Encode each center as a complex number z in the plane and track the products k·z. Then the same algebraic shape holds:

(k₁z₁ + k₂z₂ + k₃z₃ + k₄z₄)² = 2(k₁²z₁² + k₂²z₂² + k₃²z₃² + k₄²z₄²).

This complex Descartes theorem is a genuine equation about the centered, curvature-weighted positions, and solving its quadratic for k₄z₄ returns the exact center of the newly inscribed circle. In practice this pair of identities is all a program needs: given any three mutually tangent circles it computes both the radius and the position of the fourth, so the entire gasket can be generated by pure arithmetic, never by trial-and-error geometry.

A fractal of dimension 1.3057

The gasket is the residual set: the closed disk minus the interiors of all the circles you ever draw. Two facts pull in opposite directions. First, the circles fill the disk perfectly in the sense of area — the residual set has Lebesgue measure zero (proved by Boyd), so if you threw a dart it would land inside some circle with probability one. Second, the residual set is uncountable and connected, far too intricate to be a mere curve. It threads between dimension 1 and dimension 2.

Its Hausdorff dimension is approximately 1.305686729…. Crucially, this number is not the tidy log-ratio you get for the Cantor set or Koch curve, because the gasket is not built from finitely many scaled copies of itself. It is self-similar only under Möbius transformations (circle inversions), a conformal iterated function system whose pieces are distorted by varying amounts. The dimension is therefore the solution of Bowen's equation — the value δ at which a pressure (thermodynamic) function vanishes — rather than a simple fraction. David Boyd (1973) bounded it between about 1.300 and 1.315; Curtis McMullen (1998) computed δ ≈ 1.30569 with a transfer-operator eigenvalue method. It is the same exponent that governs growth: the number of circles with curvature below T is asymptotic to c·T^δ (Kontorovich–Oh).

The integer miracle: whole-number curvatures forever

Here is the gem that draws in number theorists. Return to the quadratic k₄² − 2s·k₄ + (…) = 0 from Descartes' theorem. Its two roots k₄ and k₄′ satisfy Vieta's sum relation:

k₄ + k₄′ = 2(k₁ + k₂ + k₃).

So if you already know one tangent circle's curvature k₄ and the three it rests on, the other Soddy circle simply has curvature k₄′ = 2(k₁ + k₂ + k₃) − k₄ — no square roots, just addition, subtraction, and doubling. This is a Vieta jump. The consequence is startling: if the first four curvatures are integers, every curvature in the entire infinite gasket is an integer. Each new circle is spawned from three integers by that linear formula, so integrality propagates without end. These are the integral Apollonian packings.

The smallest bounded example starts from curvatures (−1, 2, 2, 3) — a unit bounding circle, two half-radius circles, and a radius-⅓ circle — and you can check (−1+2+2+3)² = 36 = 2(1+4+4+9). Iterating produces curvatures 6, 11, 14, 15, 18, 23, and on forever. Allowing curvature 0 (a straight line, a circle of infinite radius) gives the packing (0, 0, 1, 1): two parallel lines with unit circles between them, whose descendants are the classical Ford circles, one tangent to the number line above each fraction p/q with curvature 2q². The four Vieta swaps that replace one circle by its Soddy partner generate the Apollonian group, a group of integer matrices preserving the Descartes quadratic form.

Which integers appear? Number theory and open frontiers

Given an integral packing, a natural question is: which whole numbers show up as curvatures? Every packing has a unique smallest "root quadruple" that generates it, and it is called primitive when the four seed curvatures share no common factor. Elena Fuchs proved that the only congruence obstruction lives modulo 24: within a fixed primitive packing, every curvature is congruent to one of a fixed handful of residues mod 24, and no others can occur.

The famous local–global conjecture proposed that this congruence condition is essentially the whole story: every sufficiently large integer in the allowed residue classes should actually appear. Jean Bourgain and Alex Kontorovich (2014) proved a powerful version — a density-one result: almost all admissible integers up to N really do occur, the exceptions being vanishingly rare. For years this looked like strong evidence for the full conjecture.

Then in 2023, Summer Haag, Clyde Kertzer, James Rickards, and Katherine Stange disproved the local–global conjecture. They found reciprocity obstructions — subtler constraints coming from quadratic and quartic reciprocity — that permanently exclude an infinite family of integers which nonetheless pass every congruence test mod 24. So the true set of curvatures is thinner than the mod-24 prediction, even though (by Bourgain–Kontorovich) it still contains almost all admissible numbers. The gasket thus remains a live research object: a fractal built by grade-school arithmetic whose integer content encodes deep and still-unfolding number theory.

The Apollonian gasket beside other famous fractals: dimension and how self-similarity works

Fractal	Hausdorff dimension	Type of self-similarity	Rule that builds it
Cantor set	≈ 0.6309 (ln 2 / ln 3)	exact — 2 maps × 1/3	delete the middle third
Koch snowflake	≈ 1.2619 (ln 4 / ln 3)	exact — 4 maps × 1/3	replace each edge with a bump
Apollonian gasket	≈ 1.3057	conformal — Möbius maps, not similarities	inscribe the tangent circle in each gap
Sierpiński triangle	≈ 1.5850 (ln 3 / ln 2)	exact — 3 maps × 1/2	remove the central sub-triangle
Mandelbrot boundary	= 2 (Shishikura)	quasi-self-similar	iterate z ↦ z² + c