Apollonian Gaskets: Infinite Circles Packed Into a Single Triangle

Take three coins, push them together until each one touches the other two, and look at the little curved triangle of empty space trapped in the middle. Now imagine you had a coin small enough to drop into that gap, sized perfectly so it touches all three original coins at once. Then imagine doing that again, in each of the three even smaller gaps that new coin just created. And again. And again, forever. You end up with an infinite cascade of circles, each one nestled perfectly against its neighbors, getting smaller without ever quite stopping — a lace-like pattern called an Apollonian gasket. It looks like it belongs in a stained-glass window or a satellite photo of soap foam, but it's built entirely from one of the oldest ideas in geometry: circles that just touch.

The Concept

An Apollonian gasket starts with three mutually tangent circles — each one touching the other two at exactly one point, with none overlapping. That arrangement leaves two curved triangular gaps: one in the middle, and one outside if you enclose the whole thing in a fourth, larger circle. Into each gap, you inscribe the one circle that fits perfectly — tangent to all three circles bordering it. That new circle creates three smaller gaps of its own. You fill those too. Repeat infinitely, and the gaps shrink toward zero size while their number explodes toward infinity.

The result is named for Apollonius of Perga (c. 262–190 BC), the Greek geometer best known for his work on conic sections. In a now-lost treatise called Tangencies, Apollonius solved a general problem: given any three objects in a plane — each one a point, a line, or a circle — construct a circle tangent to all three. The specific case of three mutually tangent circles, which has exactly two solutions (one nestled inside, one wrapping around the outside), is the seed of the gasket. Apollonius's original text didn't survive antiquity; we know his problem mainly because the mathematician Pappus of Alexandria described it roughly 500 years later. The recursive, infinitely-nested packing that bears Apollonius's name, though, appears to be much younger — the fractal-style iteration shows up first in a notebook of Gottfried Leibniz from the 17th century, nearly two millennia after Apollonius posed the original tangency problem.

What makes the gasket more than a pretty picture is that the sizes of the circles aren't arbitrary — they're locked together by an exact algebraic relationship, discovered independently more than once.

Why It Matters

In 1643, René Descartes wrote to Princess Elisabeth of the Palatinate about the problem of three mutually tangent circles, and worked out a strikingly clean formula relating their sizes. Define the "curvature" of a circle as the reciprocal of its radius (k = 1/r), with a bounding outer circle counted as having negative curvature since it curves the opposite way around everything inside it. Descartes found that for any four mutually tangent circles, the curvatures satisfy:

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

This is now called the Descartes Circle Theorem. Descartes stated it without much explanation of how he got there, and it was independently rediscovered twice more: by Jakob Steiner in 1826, and by Philip Beecroft in 1842. Its most famous re-appearance, though, came from Frederick Soddy — the British radiochemist who won the 1921 Nobel Prize in Chemistry for his work on isotopes. In 1936, apparently unaware of the earlier history, Soddy rediscovered the formula and, rather than publish a dry paper, turned it into a poem titled "The Kiss Precise," printed in the journal Nature. It opens by describing circles that "kiss" (touch) and ends by handing the reader the exact algebra needed to find a fourth circle from three. There's also a quieter, separate thread: the Japanese mathematician Yamaji Nushizumi stated an equivalent relationship in 1751, working in the wasan tradition of Japanese temple geometry, using the radii directly rather than their reciprocals.

The formula matters practically because it turns a construction problem — "draw a circle tangent to these three" — into pure arithmetic. Given three curvatures, you can solve a quadratic equation to get the fourth exactly, with no compass, no approximation, no iteration. That's the engine that lets you build an entire Apollonian gasket by hand, circle by circle, using only addition, multiplication, and square roots.

Beyond pure geometry, gasket-like Apollonian packings turn up wherever something needs to be crammed efficiently into leftover space. Researchers have used Apollonian-style packings as idealized models of dense granular materials — think how sand grains or aggregate settle into high-strength concrete, filling gaps between larger particles with progressively smaller ones. In antenna engineering, Apollonian packing patterns have been used to design compact, multiband "fractal antennas," where the same space-filling logic that stacks circles into vanishing gaps lets a single small antenna resonate at several separated frequency bands (one such design demonstrated operation around 1.12, 4.65, and 7.75 GHz from a single compact element). And in network science, "Apollonian networks" — graphs built by the same recursive tangency-filling process, but connecting circle centers with edges instead of drawing circles — have been studied as toy models for hierarchical, scale-free structures like transportation networks and social networks, because they naturally produce many small, tightly clustered hubs alongside a few large connective ones.

The Details

Here's a concrete example you can check with arithmetic alone. Start with a large circle of curvature −1 (so radius 1, curving inward around everything else) and two circles of curvature 2 each (radius ½), packed to touch the big circle and each other. Solving Descartes's equation for the fourth mutually tangent circle gives curvature 3 (radius ⅓) — a whole number, with no rounding required. Now here's the surprising part: if you keep filling gaps in this particular gasket, every single circle that ever appears has an integer curvature. This isn't a coincidence of the first step — it falls directly out of Descartes's formula. Because the equation is quadratic in the unknown fourth curvature, if you already know one solution, the other root is found just by subtraction: k₄' = 2(k₁+k₂+k₃) − k₄. If the first four curvatures are all integers, every curvature generated afterward is too, forever. Packings with this property are called integral Apollonian circle packings, and they've become a genuine object of study in modern number theory, not just recreational geometry. Mathematicians including Peter Sarnak have shown that the integer curvatures appearing in these packings include infinitely many primes — and even infinitely many "twin primes" in the sense of two tangent circles whose curvatures are both prime.

There's also a hidden connection to a completely different-looking object: Ford circles, a way of visualizing fractions invented by mathematician Lester Ford. Draw the number line, and above every fraction p/q (in lowest terms) draw a circle of radius 1/(2q²) sitting tangent to the line at that point. Two Ford circles touch exactly when their fractions are "Farey neighbors" — adjacent terms in the Farey sequence, the ordered list of all reduced fractions with denominators up to some bound. It turns out the Ford circles form their own Apollonian-style packing, a slice of the same tangency logic applied along a single line instead of enclosed in an outer circle, linking the gasket to continued fractions and the deep structure of the rational numbers.

Visually, an Apollonian gasket looks like a fractal because it is one — genuinely self-similar-ish, not just picturesque. Zoom into any gap, and you see the same nested cascade of shrinking, kissing circles no matter how far you zoom, because the same rule generated every scale. Like the coastline of Britain or a fern frond, it has a fractal dimension that sits between the whole numbers 1 (a line) and 2 (a filled area). For the Apollonian gasket, that dimension is approximately 1.305688 — a number mathematicians have been chasing more digits of for over a century, since there's no known simple closed-form expression for it. As recently as 2025, mathematicians Caroline Wormell and Polina Vytnova published a technique that computed this dimension to more than 128 decimal places, a task that requires understanding the packing not just as a picture but as a dynamical system with a precise mathematical structure. That a shape built from third-century-BC tangency and seventeenth-century recursion is still generating cutting-edge research in the 2020s says something about how much can hide inside a "simple" picture of touching circles.

Takeaways

An Apollonian gasket is built by repeatedly inscribing the one circle that fits perfectly into each gap left by mutually tangent circles — a rule ancient in origin (Apollonius, c. 3rd century BC) but iterated into an infinite fractal only much later, apparently first in Leibniz's notebook.
The Descartes Circle Theorem gives an exact algebraic shortcut for the process: (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²), where curvature k = 1/radius. It was independently found by Descartes (1643), Yamaji Nushizumi (1751), Steiner (1826), Beecroft (1842), and Nobel chemist Frederick Soddy, who published it as a poem in 1936.
Starting with integer curvatures guarantees every circle in the entire infinite packing has an integer curvature — and number theorists have proven these curvatures include infinitely many primes and twin primes.
The gasket connects to surprisingly distant territory: Ford circles and the Farey sequence link it to fractions and continued fractions, while its fractal (Hausdorff) dimension of about 1.3057 is still being computed to greater precision by researchers today.
The same recursive space-filling logic shows up in engineering and science far from pure math — modeling dense granular packing in high-strength concrete, designing compact multiband fractal antennas, and building scale-free "Apollonian networks" as toy models of real-world hierarchical systems.

Resources: Apollonian gasket (Wikipedia) · Descartes' theorem (Wikipedia) · Apollonian Circle Packings: Number Theory, Graham/Lagarias/Mallows/Wilks/Yan (arXiv)