  For a graph $G$, its \emph{cubicity} $cub(G)$ is the minimum dimension $k$
such that $G$ is representable as the intersection graph of (axis--parallel)
cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a
$d$--dimensional hypercube $H_d$, $\frac{d-1}{\log d} \le cub(H_d) \le 2d$. In
this paper, we show that $cub(H_d) = \Theta(\frac{d}{\log d})$.The parameter
\emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is
defined as the minimum dimension $k$ such that $G$ is representable as the
intersection graph of axis parallel boxes in $k$ dimensional space. Since
$box(G) \le cub(G)$ for any graph $G$, our result implies that $box(H_d) =
O(\frac{d}{\log d})$. The problem of determining a non-trivial lower bound for
$box(H_d)$ is left open.
