Why is the Apollonian Gasket composed of infinitely many circles? This famously known fractal has infinitely many circles, however I find it hard to find a rigid proof that confirms how or why this fractal is composed of infinitely many circles (and only circles). What if, at one point inside the circle, there comes a point where no proper circle that is tangential to the other circles in the fractal can be constructed?

Every "gap" at any stage of the gasket is a region surrounded by three tangent circles. This is true at the start, as you can tell; it's also true every time you insert another circle. Every such concave region surrounded by three tangent circles has two places you can put a circle such that it's tangent to the first three; one surrounding them, one inside them. (Formulae are known for the placement of these circles.) So you can always place in another circle.