Whether there exists a parallelepiped with edges, face diagonals, and main diagonals all of integer length is an open question. This is equivalent to thirteen linked quadratic diophantine equations. We look at the basic diophantine equation: the structure of integer length vectors in dimensions two, three, and four and give matrix generators for producing all the 3-dimensional integer length integer vectors. Parametric families of parallelepipeds that have good properties and the results of computer searches for perfect parallelepipeds are also described.

perfect parallelepiped, perfect cuboid.