Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning).
[a] Three equivalent definitions of parallelepiped are The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all special cases of parallelepiped.
"Parallelepiped" is now usually pronounced /ˌpærəˌlɛlɪˈpɪpɪd/ or /ˌpærəˌlɛlɪˈpaɪpɪd/;[1] traditionally it was /ˌpærəlɛlˈɛpɪpɛd/ PARR-ə-lel-EP-ih-ped[2] because of its etymology in Greek παραλληλεπίπεδον parallelepipedon (with short -i-), a body "having parallel planes".
Any of the three pairs of parallel faces can be viewed as the base planes of the prism.
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).
A space-filling tessellation is possible with congruent copies of any parallelepiped.
of a parallelepiped is the product of the base area
the volume is: Another way to prove (V1) is to use the scalar component in the direction of
An alternative representation of the volume uses geometric properties (angles and edge lengths) only: where
The proof of (V2) uses properties of a determinant and the geometric interpretation of the dot product: Let
, ...) The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof).
In 2009, dozens of perfect parallelepipeds were shown to exist,[3] answering an open question of Richard Guy.
But it is not known whether there exist any with all faces rectangular; such a case would be called a perfect cuboid.
Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope.
In modern literature, the term parallelepiped is often used in higher (or arbitrary finite) dimensions as well.
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
Alternatively, the volume is the norm of the exterior product of the vectors:
If m = n, this amounts to the absolute value of the determinant of matrix formed by the components of the n vectors.
is the row vector formed by the concatenation of the components of
The term parallelepiped stems from Ancient Greek παραλληλεπίπεδον (parallēlepípedon, "body with parallel plane surfaces"), from parallēl ("parallel") + epípedon ("plane surface"), from epí- ("on") + pedon ("ground").
[5][6] 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.
[5] 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 (/paɪ/) are given.
