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 $Λ_{d,K}$, 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.