Title: Parallelepiped Content: In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square or as a cuboid to a rectangle. In Euclidean geometry, its definition encompasses all four concepts (i.e., parallelepiped, parallelogram, cube, and square). In this context of affine geometry, in which angles are not differentiated, its definition admits only parallelograms and parallelepipeds. Three equivalent definitions of parallelepiped are *a polyhedron with six faces (hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. ==Properties== Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length. Parallelepipeds result from linear transformations of a cube (for the non-degenerate cases: the bijective linear transformations). Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not. A space-filling tessellation is possible with congruent copies of any parallelepiped. ==Volume== right|thumb|240px|Vectors defining a parallelepiped. The volume of a parallelepiped is the product of the area of its base A and its height h. The base is any of the six faces of the parallelepiped. The height is the perpendicular distance between the base and the opposite face. An alternative method defines the vectors a = (a1, a2, a3), b = (b1, b2, b3) and c = (c1, c2, c3) to represent three edges that meet at one vertex. The volume of the parallelepiped then equals the absolute value of the scalar triple product a · (b × c): : V = \left| \mathbf)|, where [V_i\ 1] is the row vector formed by the concatenation of V_i and 1. Indeed, the determinant is unchanged if [V_0\ 1] is subtracted from [V_i\ 1] , and placing [V_0\ 1] in the last position only changes its sign. Similarly, the volume of any n-simplex that shares n converging edges of a parallelotope has a volume equal to one 1/n! of the volume of that parallelotope. == Lexicography == The word appears as parallelipipedon in Sir Henry Billingsley's translation of Euclid's Elements, dated 1570. In the 1644 edition of his Cursus mathematicus, Pierre Hérigone used the spelling parallelepipedum. The Oxford English Dictionary cites the present-day parallelepiped as first appearing in Walter Charleton's Chorea gigantum (1663). Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon, showing the influence of the combining form parallelo-, as if the second element were pipedon rather than epipedon. Noah Webster (1806) includes the spelling parallelopiped. The 1989 edition of the Oxford English Dictionary describes parallelopiped (and parallelipiped) explicitly as incorrect forms, but these are listed without comment in the 2004 edition, and only pronunciations with the emphasis on the fifth syllable pi  are given. A change away from the traditional pronunciation has hidden the different partition suggested by the Greek roots, with epi- ("on") and pedon ("ground") combining to give epiped, a flat "plane". Thus the faces of a parallelepiped are planar, with opposite faces being parallel.