If all shaded blocks are squares derived from the largest one with edge length 1, does the area converge?
2026-02-22 21:04:20.1771794260
does the area converge?
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The horizontal distance from the midpoint of the largest square to the leftmost point can be computed as $$\frac12+\frac12+\left(\frac{1}{\sqrt2}\right)^2+\left(\frac{1}{\sqrt2}\right)^2+\left(\frac{1}{\sqrt2}\right)^4+\left(\frac{1}{\sqrt2}\right)^4+\cdots\\=1+2\left(\frac12+\frac14+\frac18+\cdots\right)=3$$ So the total width of the system is $6$. The total height is $$1+1+\left(\frac{1}{\sqrt2}\right)^2+\left(\frac{1}{\sqrt2}\right)^2+\left(\frac{1}{\sqrt2}\right)^4+\left(\frac{1}{\sqrt2}\right)^4+\cdots=4$$ So, as the wiki article says, it is bounded in a $6\times 4$ box, and so is finite in area.