In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a parallelepiped is a
three-dimensional figure formed by six
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
s (the term ''
rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
just as a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
relates to a
square. In
Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general
affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are
*a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all o ...
with six faces (
hexahedron), each of which is a parallelogram,
*a hexahedron with three pairs of parallel faces, and
*a
prism of which the base is a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
.
The rectangular
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...
(six
rectangular faces),
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
(six
square faces), and the
rhombohedron (six
rhombus faces) are all specific cases of parallelepiped.
"Parallelepiped" is now usually pronounced or ; traditionally it was in accordance with its etymology in
Greek παραλληλεπίπεδον ''parallelepipedon'', a body "having parallel planes".
Parallelepipeds are a subclass of the
prismatoid
In geometry, a prismatoid is a polyhedron whose vertices all lie in two parallel planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces are either parallelograms or ...
s.
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
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
(for the non-degenerate cases: the bijective linear transformations).
Since each face has
point symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is inv ...
, a parallelepiped is a
zonohedron. Also the whole parallelepiped has point symmetry (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
A parallelepiped can be considered as an
oblique prism with a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
as base.
Hence the volume
of a parallelepiped is the product of the base area
and the height
(see diagram). With
*
(where
is the angle between vectors
and
), and
*
(where
is the angle between vector
and the
normal to the base), one gets:
The mixed product of three vectors is called
triple product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
. It can be described by a
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
. Hence for
the volume is:
Another way to prove () is to use the scalar component in the direction of
of vector
:
The result follows.
An alternative representation of the volume uses geometric properties (angles and edge lengths) only:
where
,
,
, and
are the edge lengths.
;Corresponding tetrahedron
The volume of any
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...
that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see
proof).
Surface area
The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms:
(For labeling: see previous section.)
Special cases by symmetry
*The parallelepiped with O
h symmetry is known as a
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only ...
, which has six congruent square faces.
*The parallelepiped with D
4h symmetry is known as a square cuboid, which has two square faces and four congruent rectangular faces.
*The parallelepiped with D
3d symmetry is known as a
trigonal trapezohedron
In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rh ...
, which has six congruent
rhombic faces (also called an isohedral rhombohedron).
*For parallelepipeds with D
2h symmetry, there are two cases:
**
Rectangular cuboid: it has six rectangular faces (also called a rectangular parallelepiped, or sometimes simply a ''cuboid'').
**Right rhombic prism: it has two rhombic faces and four congruent rectangular faces.
**:Note: the fully rhombic special case, with two rhombic faces and four congruent square faces
, has the same name, and the same symmetry group (D
2h , order 8).
*For parallelepipeds with C
2h symmetry, there are two cases:
**Right parallelogrammic prism: it has four rectangular faces and two parallelogrammic faces.
**Oblique rhombic prism: 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).
Perfect parallelepiped
A ''perfect parallelepiped'' is a parallelepiped with integer-length edges, face diagonals, and
space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist, answering an open question of
Richard Guy. One example has edges 271, 106, and 103, minor face diagonals 101, 266, and 255, major face diagonals 183, 312, and 323, and space diagonals 374, 300, 278, and 272.
Some perfect parallelepipeds having two rectangular faces are known. But it is not known whether there exist any with all faces rectangular; such a case would be called a perfect
cuboid
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a c ...
.
Parallelotope
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.
Specifically in ''n''-dimensional space it is called ''n''-dimensional parallelotope, or simply -parallelotope (or -parallelepiped). Thus a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
is a 2-parallelotope and a parallelepiped is a 3-parallelotope.
More generally a parallelotope, or ''voronoi parallelotope'', has parallel and congruent opposite facets. So a 2-parallelotope is a
parallelogon which can also include certain hexagons, and a 3-parallelotope is a
parallelohedron, including 5 types of polyhedra.
The
diagonals
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
of an ''n''-parallelotope intersect at one point and are bisected by this point.
Inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
in this point leaves the ''n''-parallelotope unchanged. 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. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more ...
.
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