parallelotope

It could be any shape, although the volume equals that of the parallelotope.

Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope.

An n-vector doesn't necessarily have a shape of a parallelotope - this is a convenient visualization.

Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply n-parallelotope.

The Gram determinant is the squared volume of the parallelotope with a, ..., a as edges.

Geometrically, the Gram determinant is the square of the volume of the parallelotope formed by the vectors.

(He defines parallelotope as a generalization of a parallelogram and parallelepiped in n-dimensions.)

The magnitude of the product should equal the volume of the parallelotope with the vectors as edges, which can be calculated using the Gram determinant.

A k-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant.

The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):

More generally a parallelotope, or voronoi parallelotope, has parallel and congruent opposite facets.

Similarly, the volume of any n-simplex that shares n converging edges of a parallelotope has a volume equal to one 1/n!

For a full lattice the square root of Determinant(L) is the volume of a fundamental parallelotope of the lattice.

Is there any known result on special polyedra like, e.g., a "parallelotope" (a lineare trasformation of the hypercube into a polyedron whose faces are pairwise parallel)?

This formula expresses the fact that the absolute value of the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.

An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x.

For instance, it is well known that the magnitude of the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix.

It is unimodular, meaning that it can be generated by the columns of an 8x8 matrix with determinant 1 (i.e. the volume of the fundamental parallelotope of the lattice is 1).

The translates of a hypercube (or of an affine transformation of it, a parallelotope) form a Helly family: every set of translates that have nonempty pairwise intersections has a nonempty intersection.

The n-dimensional parallelotope spanned by the rows of an nxn Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1.

Similar interpretations are true for any number of vectors spanning an n-dimensional parallelotope; the outer product of vectors a, a, ... a, that is , has a magnitude equal to the volume of the n-parallelotope.

These ideas can be extended not just to matrices but to linear transformations as well: the magnitude of the determinant of a linear transformation is the factor by which it scales the volume of any given reference parallelotope.

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.

Let v, ..., v, for n 2, be non-zero Euclidean vectors in n-dimensional space (ℝ) that are directed from a vertex of a parallelotope, forming the edges of the parallelotope.

The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.