A slanted box hides one of geometry's most stubborn unsolved puzzles
Take a cube and push it sideways so every face leans into a parallelogram. The result is a parallelepiped. In 2009 mathematicians found dozens whose edges and diagonals are all whole numbers, yet nobody knows whether a plain rectangular box can do the same.
A parallelepiped is to a parallelogram what a cube is to a square. It can be defined three equivalent ways: a six-faced solid with three pairs of parallel faces, a solid whose six faces are all parallelograms, or a prism standing on a parallelogram base. Its name comes from the Ancient Greek for a body having parallel planes, and it was traditionally stressed on the syllable ep, as in PARR-uh-lel-EP-ih-ped. The cube, the rectangular cuboid and the rhombohedron, with six rhombus faces, are all special cases.
Several neat properties follow. Its twelve edges fall into three groups of four parallel edges of equal length. Every parallelepiped is what you get by applying a linear transformation to a cube, which is why its volume equals the absolute value of the scalar triple product of three edge vectors, a quantity computed by a determinant. Any tetrahedron built on three edges meeting at one corner has exactly one sixth of the solid's volume. Each face is the mirror image of the one opposite, and identical copies of any parallelepiped can fill space with no gaps.
Symmetry sorts them into families. The most symmetric is the cube. Next come the square cuboid, with two square ends and four matching rectangles, and the trigonal trapezohedron, with six identical rhombi. Lower down are the rectangular cuboid, right and oblique rhombic prisms and the right parallelogrammic prism, each defined by which faces are rectangles, rhombi or general parallelograms.
The number puzzle concerns a perfect parallelepiped, where edges, face diagonals and space diagonals all have integer lengths. Richard Guy had asked whether any exist, and in 2009 dozens were found. One has edges of 271, 106 and 103, with space diagonals of 374, 300, 278 and 272. Some perfect examples even have two rectangular faces. But whether one exists with every face rectangular, a so-called perfect cuboid, remains open.
Source: Parallelepiped
