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Parallelepiped
Rhombohedron
Type Prism
Faces 6 parallelograms
Edges 12
Vertices 8
Symmetry group Ci
Properties convex

In geometry, a parallelepiped (now usually pronounced /ˌpærəlɛlɪˈpɪpɛd, ˌpærəlɛlɪˈpaɪpɛd, -pɪd/; traditionally /ˌpærəlɛlˈʔɛpɪpɛd/[1] in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes") is a three-dimensional figure formed by six parallelograms. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is This article deals with the four infinite series of Point groups in three dimensions ( n &ge1 with n -fold Rotational symmetry about one axis Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides Three equivalent definitions of parallelepiped are

The cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides See also Rhombohedral - Crystal system In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses

Parallelepipeds are a subclass of the prismatoids. A prismatoid is a Polyhedron where all vertices lie in two parallel planes

Contents

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). In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.

Since each face has point symmetry, a parallelepiped is a zonohedron. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is A zonohedron is a convex Polyhedron where every face is a Polygon with point Symmetry or equivalently symmetry under Rotations through Also the whole parallelepiped has point symmetry Ci (see also triclinic). In Crystallography, the triclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three basis vectors 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. In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to

A space-filling tessellation is possible with congruent copies of any parallelepiped. In Geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

Volume

Vectors defining a parallelepiped.
Vectors defining a parallelepiped.

The volume of a parallelepiped is the product of the area of its base A and its height h. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. 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 = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| = |\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})| = |\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})|

This is true because, if we choose b and c to represent the edges of the base, the area of the base is, by definition of the cross product (see geometric meaning of cross product),

A = |b| |c| sin θ = |b × c|,

where θ is the angle between b and c, and the height is

h = |a| cos α,

where α is the internal angle between a and h. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle

From the figure, we can deduce that the magnitude of α is limited to 0° ≤ α < 90°. On the contrary, the vector b × c may form with a an internal angle β larger than 90° (0° ≤ β ≤ 180°). Namely, since b × c is parallel to h, the value of β is either β = α or β = 180° − α. So

cos α = ±cos β = |cos β|,

and

h = |a| |cos β|.

We conclude that

V = Ah = |a| |b × c| |cos β|,

which is, by definition of the scalar product, equivalent to the absolute value of a · (b × c), Q.E.D.. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated"

The latter expression is also equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows (or columns):

V = \left| \det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \right| . In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

Special cases

For parallelepipeds with a symmetry plane there are two cases:

See also monoclinic. In Crystallography, the monoclinic Crystal system is one of the 7 lattice Point groups A crystal system is described by three vectors.

A cuboid, also called a rectangular parallelepiped, is a parallelepiped of which all faces are rectangular; a cube is a cuboid with square faces. In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex.

A rhombohedron is a parallelepiped with all rhombic faces; a trigonal trapezohedron is a rhombohedron with congruent rhombic faces. See also Rhombohedral - Crystal system In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses In Geometry, a rhombus (from Ancient Greek ῥόμβος - rrhombos “rhombus spinning top” (plural rhombi or rhombuses

Parallelotope

Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great

Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply n-parallelotope. Thus a parallelogram is a 2-parallelotope and a parallelepiped is a 3-parallelotope. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides

The diagonals of an n-parallelotope intersect at one point and are bisected by this point. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line Inversion in this point leaves the n-parallelotope unchanged. In Euclidean geometry, the inversion of a point X in respect to a point P is a point X * such that P is the midpoint of See also fixed points of isometry groups in Euclidean space. A fixed point of an isometry group is a point that is a fixed point for every Isometry in the group

The n-volume of an n-parallelotope embedded in \mathbb{R}^m where m \ge n can be computed by means of the Gram determinant. In Linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors v_1\dots v_n in an Inner product space is

Lexicography

The word appears as parallelipipedon in Sir Henry Billingsley's translation of Euclid's Elements, dated 1570. Sir Henry Billingsley (died November 22, 1606) was Lord Mayor of London and the first translator of Euclid into English Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek In the 1644 edition of his Cursus mathematicus, Pierre Hérigone used the spelling parallelepipedum. Pierre Hérigone ( Latinized as Petrus Herigonius) (1580-1643 was a French mathematician and astronomer. The OED cites the present-day parallelepiped as first appearing in Walter Charleton's Chorea gigantum (1663). Walter Charleton (February 1619-c April 1707 was an English writer educated at Magdalen Hall Oxford, who according to Jon Parkin was "the main conduit for the

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. Charles Hutton ( August 14, 1737 &ndash January 27, 1823) was an English Mathematician. Year 1795 ( MDCCXCV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar (or a Noah Webster (1806) includes the spelling parallelopiped. Noah Webster (October 16 1758 &ndash May 28 1843 was an American Lexicographer, textbook author Spelling reformer word enthusiast and editor Year 1806 ( MDCCCVI) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common 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. Year 1989 ( MCMLXXXIX) was a Common year starting on Sunday (link displays 1989 Gregorian calendar) The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English "MMIV" redirects here For the Modest Mouse album see " Baron von Bullshit Rides Again "

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

External links

Footnotes

  1. ^ Oxford English Dictionary 1904; Webster's Second International 1947

Dictionary

parallelepiped

-noun

  1. (geometry) Solid figure, having six faces, all parallelograms; all opposite faces being similar and parallel
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