Editing Parallelepiped (section)

== Parallelotope <!--'Parallelotope' redirects here--> ==

[[Coxeter]] called the generalization of a parallelepiped in higher dimensions a '''parallelotope'''<!--boldface per WP:R#PLA-->. In modern literature, the term parallelepiped is often used in higher (or arbitrary finite) dimensions as well.<ref>Morgan, C. L. (1974). Embedding metric spaces in Euclidean space. Journal of Geometry, 5(1), 101–107. https://doi.org/10.1007/bf01954540</ref>

Specifically in ''n''-dimensional space it is called ''n''-dimensional parallelotope, or simply {{mvar|n}}-parallelotope (or {{mvar|n}}-parallelepiped). Thus a [[parallelogram]] is a 2-parallelotope and a parallelepiped is a 3-parallelotope.

The [[diagonals]] of an ''n''-parallelotope intersect at one point and are bisected by this point. [[Inversion in a point|Inversion]] in this point leaves the ''n''-parallelotope unchanged. See also ''[[Fixed points of isometry groups in Euclidean space]]''.

The edges radiating from one vertex of a ''k''-parallelotope form a [[k-frame|''k''-frame]] <math>(v_1,\ldots, v_n)</math> 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 <math>\R^m</math> where <math>m \geq n</math> can be computed by means of the [[Gram determinant]]. Alternatively, the volume is the norm of the [[exterior product]] of the vectors:
<math display="block"> V = \left\| v_1 \wedge \cdots \wedge v_n \right\| .</math>

If {{math|1=''m'' = ''n''}}, this amounts to the absolute value of the determinant of [[Matrix (mathematics)|matrix]] formed by the components of the {{mvar|n}} vectors.

A formula to compute the volume of an {{mvar|n}}-parallelotope {{math|''P''}} in <math>\R^n</math>, whose {{nowrap|''n'' + 1}} vertices are <math>V_0,V_1, \ldots, V_n</math>, is 
<math display="block"> \mathrm{Vol}(P) = \left|\det \left(\left[V_0\ 1\right]^\mathsf{T}, \left[V_1\ 1\right]^\mathsf{T}, \ldots, \left[V_n\ 1\right]^\mathsf{T}\right)\right|,</math> 
where <math>[V_i\ 1]</math> is the row vector formed by the concatenation of the components of <math>V_i</math> and 1.       

Similarly, the volume of any ''n''-[[simplex]] that shares ''n'' converging edges of a parallelotope has a volume equal to one 1/[[factorial|''n''!]] of the volume of that parallelotope.