Everything about Parallelepiped totally explained
In
geometry, a
parallelepiped (now usually ; traditionally /ˌpærəlɛlˈʔɛp
ɪpɛd/ in accordance with its etymology in
Greek παραλληλ-επίπεδον, a body "having parallel planes") is a three-dimensional figure formed by six
parallelograms. Three equivalent definitions of
parallelepiped are
Parallelepipeds are a subclass of the
prismatoids.
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
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):
» where
can be computed by means of the
Gram determinant.
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
OED 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 (/paɪ/) 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. (This is the same
epi- used when we say a mapping is an epimorphism/surjection/onto.)
Sources
Earliest Known Uses of Some of the Words of Mathematics
Further Information
Get more info on 'Parallelepiped'.
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