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

In geometry, a parallelepiped (now usually pronounced /ˌpærəˌlɛləˈpaɪpɪd/, traditionally^[1] /ˌpærəlɛlˈɛpɪpɛd/ in accordance with its etymology in Greek παραλληλ-επίπεδον, a body "having parallel planes") is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. Three equivalent definitions of parallelepiped are Image File history File links A parallelepiped, by Ulrik Sverdrup File links The following pages link to this file: Parallelepiped User:Sverdrup/Images ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... A parallelogram. ... The symmetry group of an object (e. ... This article deals with the four infinite series of point groups in three dimensions (n≥1) with n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and no other rotational symmetry (n=1 covers the cases of no rotational symmetry... Table of Geometry, from the 1728 Cyclopaedia. ... This chart shows concisely the most common way in which the International Phonetic Alphabet (IPA) is applied to represent the English language. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... A parallelogram. ...

• a prism of which the base is a parallelogram,
• a hexahedron of which each face is a parallelogram,
• a hexahedron with three pairs of parallel faces.

Parallelepipeds are a subclass of the prismatoids. In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... A parallelogram. ... A hexahedron is a polyhedron with 6 faces. ... A prismatoid is a polyhedron where all vertices lie in two parallel planes. ...

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 transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... A cube[1] 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. Also the whole parallelepiped has point symmetry C_i (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. The symmetry group of an object (e. ... A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic... In crystallography, the triclinic crystal system is one of the 7 lattice point groups. ... 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 its mirror image by rotations and translations alone. ...

A space-filling tessellation is possible with congruent copies of any parallelepiped. In geometry, a honeycomb is a name for a space-filling tessellation, just as a tiling is a tessellation of a plane or 2-dimensional surface. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...

Volume

The volume of a parallelepiped is the product of the area of the base b and the height h. Here, the base is any of the six parallelograms that make up the parallelepiped. The height is the perpendicular distance between the base and the top face. The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

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). In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). ...

This is true because the base parallelogram has two edges as the vectors b and c, which have an internal angle of θ; the area of this parallelogram is |b| |c| sin θ = |b × c|. The reason for this is that a parallelogram can be considered as two similar triangles - one of them having edges b and c which means the area of one of these triangles is ½|b| |c| sin θ (formula for area of a triangle).

From the diagram, the height is perpendicular to b and equal to |a| cos α where α is the angle between a and (b × c). So base × height = |a| |b × c| cos α, which is the scalar product of a and (b × c).

This is equivalent to the absolute value of the determinant In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

.

Special cases

For parallelepipeds with a symmetry plane there are two cases:

• it has four rectangular faces
• it has two rhombic faces, while of the other faces, two adjacent ones are equal and the other two also (the two pairs are each other's mirror image).

See also monoclinic. In crystallography, the monoclinic crystal system is one of the 7 lattice point groups. ...

A cuboid 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. ... A cube[1] 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 hexahedral trapezohedron is a rhombohedron with congruent rhombic faces. The n-sided trapezohedron or deltohedron is the dual polyhedron of a regular n-sided antiprism. ... For other uses of the word rhombus, see Rhombus (disambiguation) This shape is a rhombus In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i. ... The trapezohedron is the dual polyhedron of the corresponding antiprism. ... For other uses of the word rhombus, see Rhombus (disambiguation) This shape is a rhombus In geometry, a rhombus (or rhomb; plural rhombi) is a quadrilateral in which all of the sides are of equal length, i. ...

Arbitrary dimensions

The word parallelepiped is also sometimes used for the higher-dimensional analogues.

A parallelepiped in 3-space is often called just a parallelepiped. In n-dim space it is called n-dimensional parallelepiped, or simply n-parallelepiped. In 1D it is an interval, in 2D a parallelogram. Though actual perceptible space-time is a 4-dimensional Minkowski space (see special relativity), human beings usually perceive space as a three-dimensional space as long they dont notice anything with high relative velocity. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... A parallelogram. ...

The diagonals of an n-parallelepiped intersect at one point and are bisected by this point. Inversion in this point leaves the n-parallelepiped unchanged. See also fixed points of isometry groups in Euclidean space. In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... 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 the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from... A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. ...

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. Sir Henry Billingsley (d. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... Events January 23 - The assassination of regent James Stewart, Earl of Moray throws Scotland into civil war February 25 - Pope Pius V excommunicates Queen Elizabeth I of England with the bull Regnans in Excelsis May 20 - Abraham Ortelius issues the first modern atlas. ... // Events February to August - Explorer Abel Tasmans second expedition for the Dutch East India Company maps the north coast of Australia. ... Pierre Hérigone (Latinized as Petrus Herigonius) (1580-1643) was a French mathematician and astronomer. ...

Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon. Charles Hutton (August 14, 1737 - January 27, 1823) was an English mathematician. ... 1795 was a common year starting on Thursday (see link for calendar). ...

Noah Webster (1806) includes the spelling parallelopiped. Noah Webster Noah Webster (October 16, 1758 – April 28, 1843) was an American lexicographer, textbook author, spelling reformer, political writer, and editor. ... 1806 was a common year starting on Wednesday (see link for calendar). ...

The 1989 edition of the Oxford English Dictionary describes parallelipiped and parallelopiped explicitly as incorrect forms, but these are listed without comment in the 2004 edition. Pronunciation has the emphasis consistently on the fifth syllable pi (/paɪ/). 1989 (MCMLXXXIX) was a common year starting on Sunday of the Gregorian calendar. ... The Oxford English Dictionary print set The Oxford English Dictionary (OED) is a dictionary published by the Oxford University Press (OUP), and is generally regarded as the most comprehensive and scholarly dictionary of the English language. ... 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ...

The OED also cites the present-day parallelepiped as first appearing in Walter Charleton's Chorea gigantum (1663). Walter Charleton (1619 - 1707), miscellaneous writer, educated at Oxford, was titular physician to Charles I. He was a copious writer on theology, natural history, and antiquities, and published Chorea Gigantum (1663) to prove that Stonehenge was built by the Danes. ... // Events Prix de Rome scholarship established for students of the arts. ...

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

External links

• Eric W. Weisstein, Parallelepiped at MathWorld.

MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

Footnotes

1. ^ e.g. Oxford English Dictionary, 1904