Clifford A. Reiter and Jordan O. Tirrell, Pursuing the Perfect Parallelepiped, JP Journal of Algebra, Number Theory and Applications, 6 2 (2006) 279-294.

Whether there exists a parallelepiped with edges, face diagonals, and main diagonals all of integer length is an open question. We look at the structure of integer length vectors in dimensions two, three and four and develop an algebraic view of the structure of those vectors in three dimensions. Namely, we 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 described.

See Also: · Preprint of this manuscript. · Auxiliary Materials for this manscript. · [Abstract][Preprint] for related manuscript "Matrix generation of the Diophantine Solution of 3 ≤ n ≤ 9 Squares that are Square". · [Abstract][Preprint] for related manuscript "Families of Nearly Perfect Parallelepipeds". · [Abstract][Preprint] for related manuscript "Generalized Perfect Parallelograms and Their Matrix Generators". · [Materials] for related manuscript "Perfect Parallelepipeds Exist".