FAMILIES OF NEARLY PERFECT PARALLELEPIPEDS

It is unknown whether there are perfect parallelepipeds, that is, parallelepipeds with integer-length edges, face diagonals and body diagonals. A stronger version of the problem also requires the coordinates to be integer. In that case, the vectors are integer-length integer vectors. We will show how to extend integer length integer vectors in any dimension to one higher dimension and utilize that construction to present three parametric families of parallelepipeds that are nearly perfect in the sense that only two conditions need be satisfied in order for the parallelepiped to be perfect. Computer searches show many examples where either, but not both, of those conditions may be satisfied.