A geometric pattern can be characterized by studying how it can be mapped into itself. For periodic patterns these mappings are translations. The set of these translations is the lattice. If the pattern can be described by a function defined on a geometric background, the Fourier series allows to break this function additively into elementary periodic functions like Sinus and Cosine.

A lattice Λ is carried into itself by its translational symmetry group of discrete
translations. Since the geometric correspondence between lattice points and
translational symmetry operations is one-to-one, we may denote the translational
symmetry group by the symbol Λ for the lattice. So far the translation or point
group was acting geometrically as a mapping of points into image points of space.
Given a real function f(x) defined on all points say of the plane, its functional
values can be viewed as a landscape over this plane. We can define an
action of a translation or rotation group element g on this function by a
map from f(x) to a new function (T_{g}f)(x) such that the landscape of all
functional values of f(x) is translated or rotated according to the chosen
group element g. The symbol T_{g} is the name of this new operation of
g G on a function. The formal prescription for this mapping of functions
is

| (1) |

Use of the inverse g^{-1} in this definition is necessary if we want successive
operations on a function to follow the group multiplication rule,

| (2) |

i. e. to give a homomorphism, see [47] pp. 102-11.

The systematic classification of lattices in E^{n} is the task of n-dimensional
crystallography, which in [7] and in [46] is developed from mathematical work by
Hermann [13] and Zassenhaus [48].

If eq. 1 is applied to functions on E^{1}, we can in particular characterize functions
f^{p}(x) periodic under Λ by the property

| (3) |

Any periodic real- or complex-valued functions on the real line E^{1} with primitive
period a, takes values on E^{1} which are completely determined by those on the
interval 0 ≤ x < a. This interval is called the fundamental domain of the
periodic function. The notion of a fundamental domain extends to lattices in the
space E^{n}.

Periodicity in coordinate space has significant consequences for the Fourier analysis. Any periodic function on the line can be developed into a Fourier series

| (4) |

with Fourier coefficients given by the integrals

| (5) |

over the fundamental domain. If the exponentials in eq. 4 are decomposed into
sines and cosines, the complex Fourier series becomes a series in these elementary
trigonometric functions. The points x = la,l Z form the lattice Λ for the
periodic function f^{p}(x). In crystallography it is customary to define on a real line
called k-space the reciprocal lattice Λ^{R} with points k^{R} Λ^{R} : k^{R} = l, l Z.
Then the Fourier coefficients eq. 5 can be assigned in k-space to the points of the
reciprocal lattice.

Similarly, n-fold periodic functions on E^{n} may be developed into an n-fold Fourier
series which then is interpreted as a function on a reciprocal lattice Λ^{R} on a
k-space of dimension n.

For a general function f(x), x E^{n} with reasonable properties there exists a
Fourier integral transform (k), which is a new function on k-space. The fact that
for periodic functions this Fourier integral transform reduces to the Fourier series
eq. 4, with coefficients attached to the discrete subset of points on the reciprocal
lattice Λ^{R}, is expressed by saying that the Fourier spectrum is pure
point.

These relations have observable consequences for the diffraction of waves by periodic distributions of matter: In the so-called Born approximation, the angular distribution of the scattered waves displays sharp diffraction peaks which are labelled by the points from the reciprocal lattice. The diffraction pattern can also be geometrically described in terms of Laue-Bragg reflections from the netplanes of the lattice. Crystallography as developed in the 19th and 20th century employs throughout this fundamental relation between periodic structures and diffraction. It offers experimental access to the periodic atomic structure postulated by Bravais.