Here's an 'Apollonian gasket'. To get one of these, start with 3 circles snugly packed in a bigger circle. Then keep putting in more circles, each tangent to 3 that are already there.

Each number here is the curvature of a circle - that is, the reciprocal of its radius. Frederick Soddy, a Nobel prize winner in chemistry, showed something wonderful in 1936. If the 3 circles and the big circle you start with have integer curvatures, all the rest will too!

What integers can show up in this game? This turns out to be an amazingly interesting question!

For example, say you pick integer curvatures for the 3 circles and the big circle you start with. Then the other integers that show up can only take certain specific values mod 24.

Yikes! The appearance of the number 24 shows we're getting into deep waters. This number shows up all over math, linking different subjects in amazing ways. I've been trying to write a paper about this, but there aren't enough hours in the day.

Are there any other restrictions on the curvatures? For a long time people thought basically NO. That is: after you pick integer curvatures for the 3 circles and the big circle you start with, all
sufficiently large integers that have allowed values mod 24 actually do show up as curvatures of circles in your Apollonian gasket!

(Well, at least if the curvatures for the 3 circles and the big circle you start with are relatively prime. If they're not, you can divide them all by the same number N, and then all the other circles will have curvatures divisible by N.)

But recently two students doing a summer research project noticed this is WRONG! Check out the new Quanta article:

https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/

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