Parallelepiped
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and space diagonals.  In 2009, dozens of perfect parallelepipeds were shown to exist,  answering an open question of Richard Guy. One example has edges 271, 106, and 103, minor face diagonals 101, 266, and 255, major face diagonals 183, 312, and 323, and space diagonals 374, 300, 278, and 272.