clever and super interesting! Although the 'sierpenski' levels aren't really sierpenski...that's the Menger Sponge.
hmm. It would appear so; if you think of it as "take any point, is there a hole there". However... If you think about it, each iteration, you remove 1/9 of the area, and 1/9 is less every time, which seems like it would approach a finite value. But another way to think of it is that the total carpet area at iteration n is 8/9 (n-1). This is always less and less continually. And sure enough, a sequence calculator says it approaches 0.
confusing...
Wikipedia has the last word:
The area of the carpet is zero (in standard Lebesgue measure).
Proof: Denote as ai the area of iteration i. Then ai + 1 = 8/9ai. So ai = (8/9)i, which tends to 0 as i goes to infinity.
I guess the only way to really tell is to make one, or steal Sierpenski's and weigh it. :)
Now we're entering the fundamentals of fractal math :)
Weighing it is a good call; you could derive the concept of "fractal dimensions" by scaling it and weighing it repeatedly.
For those who are curious, this video by 3Blue1Brown does a much better job at explaining it than I can ever do.
Actually... Perhaps I can somehow make fractal dimensions a mechanic in the full game? We'll see...