I posted this question last night right before I went to bed and woke up with this exact idea in mind. :-) Great observation!

This is a really interesting perspective that makes a lot of sense. Thanks so much for sharing it!

This is a wonderful intuition - thank you for pointing out the flaw in my initial reasoning about the L-shaped tiling of the square! In that sense, is it fair to say that this is more a consequence of the fact that we never fully fill a region than anything specific to properties or circles?

@Apass.Jack Yep, I’m familiar with the Cantor set and the proof that it’s uncountable (it consists of all real numbers in the interval $[0, 1]$ whose ternary representation uses just 0s and 2s, which can be bijected to the set of all infinite binary sequences).

@XanderHenderson I generated the first with a program I wrote and drew the second myself.

You’ve raised an interesting point I hadn’t considered. I had assumed a point was “covered” if it was either inside a circle or on the boundary of a circle. Does that change the result?

@Nezuko The second step in your derivation is not possible with the grammar because you cannot replace $S$ with $aA$.