Topic · Geometry

Apollonian Circle Packing

Circles inside circles inside circles.

Start with three mutually tangent circles and a rule for what counts as tangent. The Explorer recursively fills every curved triangular gap with a new circle, then fills the smaller gaps in turn — pick the starting curvatures and watch a gasket emerge that is fractal forever.

Discover the story → Open the Explorer

First encounter

Three circles. Fill every gap. Forever.

The big idea

Draw three circles that all touch one another. Between them sits a curved triangular gap. Drop the largest circle you can into that gap — it touches all three. Now you have three new, smaller gaps. Repeat. The picture you get, the gasket, is a fractal coral that grows from a single rule: every gap is a new seat for a new circle.

A concrete example

Take the seed (−1, 2, 2, 3): one big outer circle of radius 1, two equal interior circles of radius 1/2 side by side, and one little circle of radius 1/3 perched between them. Each number is the curvature k = 1/radius. The outer one is negative because it encloses the others — that minus sign is the only piece of bookkeeping you need.

k = 1/r · the curvature

k < 0 · circle encloses the others

Why it matters

Apollonius asked which circles touch three given ones around 200 BC; Descartes locked the answer into algebra in 1643. But which integer curvatures actually appear in any given packing is still an open problem in arithmetic geometry. A picture of touching circles turns out to hide a piece of number theory we haven't finished writing.

Below: compute the fourth curvature yourself, then watch the gasket grow.

Section 01 · The starting position

Three circles touching

Draw three circles, each one tangent to the other two — three kisses between them, three points of contact. Around 200 BC Apollonius of Perga asked the natural follow-up: which circles are tangent to all three? The answer is exactly two. A small one inscribed in the curved gap between them, and a large one circumscribed around the triple. Add either to your three, and you have four mutually tangent circles. That quadruple is the seed of everything that follows.

Section 02 · Descartes' theorem

(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²)

Write each circle's curvature as k = 1/r, taking the outer enclosing circle's k negative. Descartes worked this out in his 1643 correspondence with Princess Elisabeth of Bohemia, showing that any four mutually tangent circles satisfy a single quadratic identity. Solving it for the unknown fourth gives k₄ = k₁+k₂+k₃ ± 2√(k₁k₂ + k₂k₃ + k₃k₁). The two signs are Apollonius' two answers: + drops the small inscribed circle into the gap, − gives the other tangent circle on the opposite side of the triple.

Descartes, 1643

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

Two solutions: the + sign drops the small inscribed circle into the gap; the − sign gives the other tangent circle on the opposite side of the triple.

Interactive · the formula by hand

Type three curvatures, watch the fourth appear

Descartes' formula is a quadratic in k₄, so it has two roots. Type any three signed curvatures into the boxes below and the calculator returns both roots, the inscribed and the circumscribed. Pick the presets to feel how the algebra behaves on the canonical integer seeds — and notice how often the answer is itself an integer.

Interactive · Descartes calculator

Three curvatures

k₁k₂k₃

Try a known triple

seed triple

Two solutions for k₄

+ inscribed (small)

k₄ = 3

− circumscribed (large)

k₄ = 3

Pick three curvatures — positive for ordinary circles, negative for an enclosing one. The formula gives the two circles tangent to all three. Try (−1, 2, 2): the + solution is k₄ = 3, the − solution k₄ = 3 as well — the seed (−1, 2, 2, 3) is symmetric.

Section 03 · Recursive filling

Every gap is a new seed

Once the seed quadruple is in place, every curved-triangular gap is itself bounded by three mutually tangent circles — exactly the configuration we started from. Apply Descartes' formula again with the + sign to drop the inscribed circle into the gap. That new circle splits its gap into three new, smaller curved triangles, each of which is again a seed. The process is self-similar: the rule for filling the first gap is the same rule that fills every gap forever.

Interactive · the recursive picture

Pick a seed, turn the depth knob

Choose one of the four classical integer seeds, then slide the recursion depth up and down. At depth 1 the first generations already fill the gaps; at depth 6 a coral of thousands of circles. The renderer is the same one that powers the full Explorer — only the depth ceiling is lower so the story page stays responsive.

Interactive · grow the gasket

Seed packing

Recursion depth

3164 circles

Each step fills every curved-triangle gap with its inscribed circle. The colour sweeps with the log of the curvature: rose for the outer ring, then amber, cyan, violet for finer and finer detail.

Section 04 · The integer surprise

Pick integers — get integers forever

Choose the four seed curvatures to be whole numbers. Then a small algebraic miracle happens: every subsequent circle in the gasket also has integer curvature. The reason is that (k₁+k₂+k₃+k₄)² = 2(k₁²+…+k₄²) is symmetric in the four variables, so swapping the old k₄ for the new k₄' gives k₄' = 2(k₁+k₂+k₃) − k₄ — no square root, just integer arithmetic. The packing (−1, 2, 2, 3) generates curvatures 6, 11, 14, 15, 18, 23, 26, 27, 30, 35, … Every single integer Apollonian packing does the same. The picture is, at heart, a number-theoretic object.

Integer seed packings

seed (k₁, k₂, k₃, k₄)	first new curvatures
(−1, 2, 2, 3)	3, 6, 6, 11, 14, 15, 18, 23, …
(−2, 3, 6, 7)	7, 10, 15, 19, 22, 27, 31, 34, …
(−3, 5, 8, 8)	8, 12, 12, 20, 21, 29, 32, 44, …
(−4, 8, 9, 9)	9, 17, 17, 20, 33, 36, 41, 56, …
(−6, 11, 14, 15)	15, 23, 26, 35, 42, 47, 51, 59, …

Every curvature in every one of these packings is an integer — forever, at every recursion depth.

Section 05 · Hausdorff dimension ≈ 1.3056867

Between a curve and a region

Keep filling. The union of all the circles you ever draw covers more and more of the original disc, but never all of it. The leftover dust — the points that belong to no circle — has Lebesgue measure zero, yet it is everywhere dense. Its Hausdorff dimension was computed numerically by Boyd and rigorously bounded by McMullen at about 1.3056867. The number is universal: it does not depend on which seed you start from. Every Apollonian gasket has the same fractal weight.

Hausdorff dimension

≈ 1.3056867

Strictly between a curve (dimension 1) and a region (dimension 2). The Apollonian residual set has measure zero but is fractally dense at every scale.

Section 06 · Open problems and connections

Bourgain–Fuchs, hyperbolic geometry, and what remains unknown

The integer curvatures in an Apollonian packing form a set of natural numbers, but which numbers appear? The local-to-global conjecture predicted: a curvature appears in a fixed packing iff it appears modulo 24 in that packing — beyond a finite list of exceptions. Disproved in 2023 by Haag, Kertzer, Rickards and Stange (Annals of Mathematics): there are infinite families of curvatures that are locally allowed but never globally appear. Bourgain and Fuchs had already shown in 2011 that the integer curvatures still have positive density — a fixed positive fraction of all integers really do show up. Behind this lives a deeper story — every Apollonian packing is an orbit of a group acting by Lorentzian reflections on a quadratic form of signature (3,1), and the same packings reappear as boundaries of hyperbolic 3-manifolds. Geometry, number theory, and group theory meet in a picture of touching circles.

Take it further

Open the Explorer.

The Explorer lets you push the recursion deeper, toggle curvature labels, switch palettes, and try every integer seed packing. Everything you just read is one click away.

→ Open the Explorer

Where you meet it

How and where this technique lives in the world today.

Granular packing

How sand, gravel and pharmaceutical powders fill containers is modelled with Apollonian-style sphere packings — important in concrete, drug tablets and powder metallurgy.

Number theory

Integer Apollonian packings are studied by analytic number theorists — papers by Sarnak, Bourgain & Kontorovich produced new results on prime curvatures.

Foam & emulsion physics

Foam structures (beer head, food emulsions, lung alveoli) inherit Apollonian-style packing constraints during their formation.

Graphic design

Logo, poster and tattoo design uses Apollonian gaskets for striking radial geometry.