Parallelepipedin

Parallelepiped, or parallelepipedin in some languages, is a six-faced polyhedron in which every face is a parallelogram. Opposite faces are parallel and congruent, and adjacent faces meet along edges. It can be viewed as a prism with a parallelogram base or as the Minkowski sum of a parallelogram and a line segment.

If the three edges meeting at a vertex are represented by vectors a, b, and c, then the opposite faces are the parallelograms spanned by pairs of these vectors. The volume of a parallelepiped is V = |a · (b × c)|, equivalently base area times height. In coordinates, if a, b, c are the column vectors of a 3×3 matrix, the volume is the absolute value of its determinant.

Special cases include the rectangular parallelepiped, where all dihedral angles are right angles; the cube, which is a rectangular parallelepiped with equal edge lengths; and the rhombohedron, where all edges have equal length but angles are not necessarily right. An oblique parallelepiped has no right angles.

Diagonals: there are four space diagonals connecting opposite vertices; they intersect at the center of the solid and are generally of different lengths. Each face has diagonals that bisect each other.

A parallelepiped is defined by three edge vectors and can be parameterized as { s a + t b + u c | 0 ≤ s, t, u ≤ 1 }. In linear algebra, its volume equals the determinant of the matrix with a, b, c as columns, and the Gram determinant relates to V^2. Parallelepipeds are fundamental in vector geometry and 3D modeling.