# Zeno's spiral into madness

**Calculus**Level 3

Starting at \((0,0)\), Zeno goes up \(1\) meter and turns \(90^{\circ}\) clockwise. Each time, the length of the new step decreases to half of its previous step, followed by a turn \(90^{\circ}\) clockwise. So he progresses, \(1\) meter up, \(\frac{1}{2}\) meter right, \(\frac{1}{4}\) meter down...

At the very end, what is the magnitude of the displacement Zeno has travelled from the origin in meters? (Assume that the spiraling takes a finite amount of time.)