Parallelepiped

Under a parallelepiped (from Greek επίπεδον epipedon " surface "; Synonyms: Spat, parallelepiped, parallelotope ) refers to a geometric body bounded past six pairs of congruent ( congruent ) in parallel planes parallelogram. The name derives from the calcite spar ( calcite, chemical: CaCO3), a result whose crystals have the shape of a parallelepiped.

A parallelepiped has twelve edges, each of which four are parallel to each other and are of equal length, and eight corners where converge these edges in a maximum of three different angles to each other. By taking these three coincident at a vertex edges as vectors represent, so the volume of the parallelepiped obtained from the amount of Spatproduktes (mixed scalar and cross product )

Volume

The volume V is the product of the base G and the parallelepiped height h, with ( with θ as an angle between and ) and the height. Here, α is the angle between and the height h

The volume can be regarded as a determinant of a 3 × 3 matrix, which is also called scalar triple product.

There are, the angles between the edges. Then the volume:

The area of ​​the surface resulting from the sum of the individual Parallelogrammflächen

Comments

• Cuboid (all angles 90 °) and rhombohedral (all edges the same length ) are special forms of the parallelepiped. The cube combines the two special forms in a figure.
• The parallelepiped is a special ( crooked ) prism with a parallelogram as base.
• Each parallelepiped is a space filler, that is, the space can be shifted with parallel copies of P cover so that any two have in common at most boundary points among them.

Generalization to the n-dimensional space ( n> 1)

The generalization of the parallelepiped in n-dimensional space is called for parallelotope or n - parallelotope. The two-dimensional analogue of the parallelepiped is the parallelogram.

Definition

An n- parallelotope P is the image of the unit cube E under an affine transformation. The unit cube is a set of points whose coordinates have a value between 0 and 1, ie

The parallelotope P is a convex polytope with vertices. For m

Volume

An affine transformation can be written as, where the projection matrix and the shift. The volume of the unit cube is one. To determine the volume of Parallelotops P, so it must be investigated how much the affine transformation changes the volume. As a volume is independent of the displacement, this value is only inserted in the picture matrix. By calculating the determinant of this matrix, we obtain also the factor by which the volume changes. The dashes indicate the amount here. Multiplying this factor by the volume of the unit cube, so holds trivially, therefore applies

Wherein the picture matrix of the affine transformation is that defines the parallelotope.