alt Hacker NewsI think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".
There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.
Does it jump? I feel like it's a "fat perimeter".
I kind of ran into this when I was in high school and was introduced to limits.
For me the quandary was a "stair step" shape dividing a square with length of side "s" ("stairs" connecting two opposite diagonal corners). You could increase the number of steps—they get smaller—but the total rise + run of the stairs remains the same (2s). At infinity I reasoned you had a straight, diagonal line that should have been s√2 but was also still 2s in length.
At the very least you can say that the volume enclosed approached that of a right triangle (at infinity) but the perimeter stays stubbornly the same and not that of a right triangle at all.