Wallis Sieve

A compact set $W_\infty$ with Area

$$\mu(W_\infty)={8\over 9}\, {24\over 25} \,{48\over 49} \cdots = {\pi\over 4}$$

created by punching a square hole of length $1/3$ in the center of a square. In each of the eight squares remaining, punch out another hole of length $1/(3\cdot 5)$, and so on.