  The concept of midpoint percolation has recently been applied to characterize
the double percolation transitions in negatively curved structures. Regular
$d$-dimensional hypercubic lattices are in the present work investigated using
the same concept. Specifically, the site-percolation transitions at the
critical thresholds are investigated for dimensions up to $d=10$ by means of
the Leath algorithm. It is shown that the explicit inclusion of the boundaries
provides a straightforward way to obtain critical indices, both for the bulk
and surface parts. At and above the critical dimension $d=6$, it is found that
the percolation cluster contains only a finite number of surface points in the
infinite-size limit. This is in accordance with the expectation from studies of
lattices with negative curvature. It is also found that the number of surface
points, reached by the percolation cluster in the infinite limit, approaches 2d
for large dimensions $d$. We also note that the size dependence in
proliferation of percolating clusters for $d\ge 7$ can be obtained by solely
counting surface points of the midpoint cluster.
