Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar?
What about if we apply the Apollonian gasket construction inside of a fractal like the Koch snowflake? (I think that will still be countable.)
What if you did the Apollonian gasket construction between f(x) = sin(1/x) and g(x) = 2 - sin(1/x) between -1 and 1?(I think that will still be countable too, but it's not matching my intuition... which says "no way is that countable!")
Is there any closed curve that would result in the number of circles being uncountable?
What if we consider the Apollonian gasket made of spheres in $\mathbb{R}^3$?
(Please keep in mind I have only had two courses in Analysis. My apologies if any of this is too naive.)