  We study the set of visible lattice points in multidimensional hypercubes.
The problems we investigate mix together geometric, probabilistic and number
theoretic tones. For example, we prove that almost all self-visible triangles
with vertices in the lattice of points with integer coordinates in $\mathcal
W=[0,N]^d$ are almost equilateral having all sides almost equal to
$\sqrt{d}N/\sqrt{6}$, and the sine of the typical angle between rays from the
visual spectra from the origin of $\mathcal W$ is, in the limit, equal to
$\sqrt{7}/4$, as $d$ and $N/d$ tend to infinity. We also show that there exists
an interesting number theoretic constant $\Lambda_{d,K}$, which is the limit
probability of the chance that a $K$-polytope with vertices in the lattice
$\mathcal W$ has all vertices visible from each other.
