Editing Reciprocal lattice (section)

===Higher dimensions===
The formula for <math>n</math> dimensions can be derived assuming an <math>n</math>-[[dimension (vector space)|dimensional]] [[real number|real]] vector space <math>V</math> with a [[basis (linear algebra)|basis]] <math>(\mathbf{a}_1,\ldots,\mathbf{a}_n)</math> and an inner product <math>g\colon V\times V\to\mathbf{R}</math>. The reciprocal lattice vectors are uniquely determined by the formula <math>g(\mathbf{a}_i,\mathbf{b}_j)=2\pi\delta_{ij}</math>. Using the [[Permutation#Two-line notation|permutation]]
: <math>\sigma = \begin{pmatrix}
  1 & 2 & \cdots &n\\
  2 & 3 & \cdots &1
\end{pmatrix},</math>
they can be determined with the following formula:
:<math>
     \mathbf{b}_i = 2\pi\frac{\varepsilon_{\sigma^1i\ldots\sigma^ni}}{\omega(\mathbf{a}_1,\ldots,\mathbf{a}_n)}g^{-1}(\mathbf{a}_{\sigma^{n-1}i}\,\lrcorner\ldots\mathbf{a}_{\sigma^1i}\,\lrcorner\,\omega)\in V
</math>
Here, <math>\omega\colon V^n \to \mathbf{R}</math> is the [[volume form]], <math>g^{-1}</math> is the inverse of the vector space isomorphism <math>\hat{g}\colon V \to V^*</math> defined by <math>\hat{g}(v)(w) = g(v,w)</math> and <math>\lrcorner</math> denotes the [[Interior product|inner multiplication]].

One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions, <math>\omega(u,v,w) = g(u \times v, w)</math> and in two dimensions, <math>\omega(v,w) = g(Rv,w)</math>, where <math>R \in \text{SO}(2) \subset L(V,V)</math> is the [[rotation]] by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation<ref>{{cite book | last=Audin|first=Michèle | title=Geometry | publisher=Springer | year=2003|page=69}}</ref>).