⭕ Apollonius Gasket — Fractal Circle Packing

When the first four curvatures are integers, every curvature generated by the recursion is also an integer, proved directly from Descartes' theorem. The classic example starts with −1, 2, 2, 3 and produces circles with curvatures 6, 11, 14, 15, 18, 23, … Which positive integers appear is an active research question connected to quadratic forms and thin groups in number theory.

The complex extension of Descartes' theorem gives the centre via k₄z₄ = k₁z₁ + k₂z₂ + k₃z₃ ± 2√(k₁k₂z₁z₂ + k₂k₃z₂z₃ + k₃k₁z₃z₁), where z = x + iy encodes each circle's centre. This elegant formula computes both the position and the two tangent-circle choices in one step, discovered by Philip Beecroft (1842) and later by Frederick Soddy (1936).

Apollonius (c. 262–190 BC) was a Greek mathematician from Perga in modern Turkey, best known for his eight-book treatise Conics. His lost work On Tangencies addressed finding circles tangent to three given circles — the Problem of Apollonius — which has up to eight solutions. The gasket named after him was formalised much later using Descartes' curvature formula.

Yes — a deep one. The tangency points of an Apollonian gasket form the limit set of a Kleinian group — an infinite group of Möbius transformations of the Riemann sphere. The gasket lives on the boundary of hyperbolic 3-space, and the packing symmetries are the isometries of a hyperbolic tetrahedron. This has driven major advances in spectral theory and ergodic theory on hyperbolic manifolds.

The recursion is theoretically infinite, but circles below about 1 pixel in screen radius are imperceptible. The simulation halts when a circle's screen radius falls below the minimum radius threshold. Decreasing the threshold reveals more fractal detail at the cost of longer computation; the depth slider adds an independent hard cap to prevent runaway recursion.

Soddy's hexlet (1936) is a 3D analogue: given two spheres internally tangent to a third, exactly six spheres can be simultaneously tangent to all three and to each other, forming a ring. Frederick Soddy discovered this and announced it in a poem in Nature alongside the 3D Descartes theorem for spheres — both the hexlet and the Apollonian gasket are manifestations of the same inversive geometry.

Apollonian packings have deep connections to the Local-Global Conjecture: it is conjectured (and partially proved by Bourgain, Kontorovich and others) that every sufficiently large integer satisfying a finite set of congruence conditions actually appears as a curvature. The packing group is a "thin group" in the sense of number theory, making this one of the most active research frontiers in arithmetic geometry.

Each circle is assigned a generation depth (0 for the seed circles, 1 for the first new circles, and so on). The palette function maps depth linearly to a hue or brightness value, so the oldest, largest circles have one colour and the newest, smallest circles have another. This makes the recursive structure immediately perceptible — each visible colour boundary marks one step deeper in the infinite hierarchy.