Push three coins together on a table and they leave a small curved gap. Exactly one circle fits that gap and touches all three. Draw it, and now there are three smaller gaps, each with its own perfect fit. Keep going and you get a foam that never finishes. The strange part is the numbers written inside: tap any circle and add them up yourself.
For the curious kid.
Every circle here wears a number. It is the circle's bend: how sharply its edge curves. A big lazy circle has a small bend, a tiny tight one has a huge bend. If the big outside circle has radius 1, a circle marked 23 has a radius of exactly one twenty-third. Press "Start from four", then "Kiss again" a few times, and watch the rule at work: every gap between three touching circles gets the one circle that fits, and every new circle makes three new gaps. Four circles, then 4 more, then 12, then 36. It never ends, because a gap never closes: it only splits.
Now the trick. Tap any circle. On the right you will see it with the three circles it was squeezed between, and a sum. Add the four bends and square the total. Then square each bend, add those, and double it. The two answers always match. That one rule is strong enough to work out the size of every circle in the foam without measuring anything. And if the first four bends are whole numbers, every bend after them, forever, is a whole number too. Nobody put the numbers in. They fall out.
Deeper.
René Descartes sent the rule to Princess Elisabeth of Bohemia in 1643: four mutually tangent circles with bends a, b, c, d (bend = 1/radius) satisfy (a + b + c + d)² = 2(a² + b² + c² + d²). A circle that encloses the others counts with a negative bend, which is why the outer circle here is marked −1. The chemist Frederick Soddy, who won a Nobel Prize for isotopes, rediscovered it and published it in Nature in 1936 as a poem, "The Kiss Precise", whose punchline is that the sum of the squares "is half the square of their sum". The construction itself is far older: Apollonius of Perga was finding circles tangent to three others around 200 BC, and the foam carries his name, the Apollonian gasket.
Fix three of the circles and Descartes' rule is a quadratic in the fourth bend. Its two roots are the two circles that fit: one on each side. The roots of a quadratic have a known sum, here 2(a + b + c), so if you already hold one root d, the other is simply 2(a + b + c) − d. No square root is ever taken, so whole numbers stay whole. The same line of arithmetic works when you replace each bend by bend × centre (Lagarias, Mallows and Wilks, 2002), and that is how this page places every circle: it adds and subtracts. Nothing is solved, measured or nudged into contact, and yet every circle lands touching its three parents.
The circles fill the disc: the ones with bend up to 6,000 already cover 99.8% of it. What is left over is a dust with zero area that is still more than a curve. Its dimension is 1.30568… (computed by Curtis McMullen in 1998), and the same number runs the census: the count of circles with bend at most T grows like T to that power (Kontorovich and Oh, 2011). That is the chart on the right. The dashed line has the slope theory gives; only its height is pinned to the last point. The slope the page measures from its own circles sits beside it.
Which whole numbers show up as bends? There is an easy filter: in each foam, bends only ever land in six or eight of the 24 possible remainders when divided by 24 (Elena Fuchs, 2011). For twenty years the belief was that, past some point, every number passing that filter appears. In 2023 Summer Haag, Clyde Kertzer, James Rickards and Katherine Stange proved the belief false. Press "Another four" until the first four read (−3, 5, 8, 8). Before writing this I counted that foam's 3,328,609 circles with bends up to 600,000. Every one of the 129 perfect squares that pass the filter (36, 144, 324, …) is missing. The filter allows them; a deeper law of reciprocity forbids them. The foam everyone had stared at for centuries was still keeping a secret three years ago.
Honesty notes, museum policy. Bends are exact integers. Centres are floating-point sums and were checked headless: 137,000 circles across four foams, each touching all three parents to better than one part in a billion, and the 700 largest compared pair by pair: no two overlap. Circles smaller than half a pixel are left undrawn until you zoom, so the gold dust in the cusps is always a little finer than shown. The census on the right always counts the whole foam up to bend 6,000 times the outer circle's, not just what is in view. There is one more room behind this one, for people who like fractions.
Four circles touch, and everything after that is addition.