  We consider the question of how approximations satisfying Dirichlet's theorem
spiral around vectors in $\mathbb{R}^d$. We give pointwise almost everywhere
results (using only the Birkhoff ergodic theorem on the space of lattices). In
addition, we show that for $\textit{every}$ unimodular lattice, on average, the
directions of approximates spiral in a uniformly distributed fashion on the
$d-1$ dimensional unit sphere. For this second result, we adapt a very recent
proof of Marklof and Str\"ombergsson \cite{MS3} to show a spherical average
result for Siegel transforms on
$\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$. Our
techniques are elementary. Results like this date back to the work of
Eskin-Margulis-Mozes \cite{EMM} and Kleinbock-Margulis \cite{KM} and have
wide-ranging applications. We also explicitly construct examples in which the
directions are not uniformly distributed.
