  We show that for almost every (with respect to Masur-Veech measure) $\omega
\in \mathcal{H}(2)$, the set of angles $\theta \in [0, 2\pi)$ so that
$e^{i\theta}\omega$ has non-uniquely ergodic vertical foliation has Hausdorff
dimension (and codimension) $1/2$.
