of the conditions are fulfilled, the twelfth-degree polynomial is irreducible over the ring of integers . Almost-perfect cuboids An almost-perfect cuboid has 6 out of the 7 lengths as rational. Such cuboids can be sorted into three types, called body, edge, and face cuboids. In the case of the body cuboid, the body (space) diagonal is irrational. For the edge cuboid, one of the edges is irrational. The face cuboid has one of the face diagonals irrational. The body cuboid is commonly referred to as the Euler cuboid in honor of Leonhard Euler, who discussed this type of cuboid. He was also aware of face cuboids, and provided the (104, 153, 672) example. The three integer cuboid edge lengths and three integer diagonal lengths of a face cuboid can also be interpreted as the edge lengths of a Heronian tetrahedron that is also a Schläfli orthoscheme. There are infinitely many face cuboids, and infinitely many Heronian orthoschemes. The smallest solutions for each type of almost-perfect cuboids, given as edges, face diagonals and the space diagonal , are as follows: Body cuboid: Edge cuboid: Face cuboid: , there are 167,043 found cuboids with the smallest integer edge less than 200,000,000,027: 61,042 are Euler (body) cuboids, 16,612 are edge cuboids with a complex number edge length, 32,286 were edge cuboids, and 57,103 were face cuboids. , an exhaustive search counted all edge and face cuboids with the smallest integer space diagonal less than 1,125,899,906,842,624: 194,652 were edge cuboids, 350,778 were face cuboids. Perfect parallelepiped A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, dozens of perfect parallelepipeds were shown to exist, answering an open question of Richard Guy. Some of these perfect parallelepipeds have two rectangular faces. The smallest perfect parallelepiped has edges 271, 106, and 103; short face diagonals 101, 266, and 255; long face diagonals 183, 312, and 323; and body diagonals 374, 300, 278, and 272. See also Pythagorean quadruple Notes References Arithmetic problems of solid geometry Diophantine equations Pythagorean theorem Unsolved problems in number theory Cuboids