  By examining the 3 surface angles which exist at any of the 8 vertices of a
Diophantine parallelepiped, and classifying them by the appearance of a right
angle, it is discovered that 5 unique classes of Diophantine parallelepipeds
exist. It is proposed to name these classes: acute (triclinic), obtuse
(triclinic), 1-ortho (biclinic), 2-ortho (monoclinic), and rectangular,
according to the count of rights angles which may exist. A Diophantine analysis
of the 83 possible rational components of the piped reveals that only 27
rationality checks need to be made when examining for rationality; such as skew
triangles, body or face parallelograms, face diagonals, body diagonals, and
volume. A computer search of 1,981,336,681 tetrahedrons with 6 rational face
diagonals uncovers interesting examples of pipeds, including the perfect
parallelepiped of Sawyer-Reiter (and 5 others), and the rectangular integer
cuboid. Other interesting pipeds were also discovered in the 115 unique
categories which the computer searches revealed. Some questions, conjectures
and possible studies are provided at the conclusion.
