parallelogon which can also include certain hexagons, and a 3-parallelotope is a parallelohedron, including 5 types of polyhedra. The diagonals of an n-parallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the n-parallelotope unchanged. See also fixed points of isometry groups in Euclidean space. The edges radiating from one vertex of a k-parallelotope form a k-frame of the vector space, and the parallelotope can be recovered from these vectors, by taking linear combinations of the vectors, with weights between 0 and 1. The n-volume of an n-parallelotope embedded in where can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors: If , this amounts to the absolute value of the determinant of the vectors. Another formula to compute the volume of an -parallelotope in , whose vertices are , is where is the row vector formed by the concatenation of and 1. Indeed, the determinant is unchanged if is subtracted from (), and placing 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. Etymology The term parallelepiped stems from Ancient Greek (parallēlepípedon, "body with parallel plane surfaces"), from parallēl ("parallel") + epípedon ("plane surface"), from epí- ("on") + pedon ("ground"). Thus the faces of a parallelepiped are planar, with opposite faces being parallel. In English, the term parallelipipedon is attested in a 1570 translation of Euclid's Elements by Henry Billingsley. The spelling parallelepipedum is used in the 1644 edition of Pierre Hérigone's Cursus mathematicus. In 1663, the present-day parallelepiped is attested in Walter Charleton's Chorea gigantum. 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. See also Lists of shapes Notes References Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 122, 1973. (He defines parallelotope as a generalization of a parallelogram and parallelepiped in n-dimensions.) External links