> but no matter how small you make the steps, the area never changes

Sorry, this is a bit off-topic, but I have to call this out.

The area absolutely does change, you can see this in the trivial example from the first to second step in https://yagmin.com/blog/content/images/2026/02/blocks_cuttin...

The corners are literally cut away.

What doesn't change is the length of the edges, which is a kind of manhattan distance.

The length of the edge has a limit of the straight line, but does not actually approach the limit.

The area however absolutely does approach the limit, as in fact you remove half the "remaining" area each iteration.