The proof of the existance of crystals was furnished by Max von Laue and colleagues in 1912. They discovered the x-ray diffraction by crystals and the crystal-dependent, regular pattern of points resulting from it. This discovery and its theoretical substantiation won von Laue the Nobel Prize in Physics in 1914. Bragg had shown that x-rays could be used to exactly define the position of atoms within a crystal, thus identifying it's three dimensional structure. They were awarded the Nobel Prize in Physics in 1915.
Steno, Rome, and Hauy discovered that crystals consist of periodically repeating basic units having the same shape as the crystal itself. For this purpose we are looking at ideal crystals that are infinite and have translational symmetry. The atomic positions are merged into one another by a translational movement within space so that the whole crystal is formed by the translations of one basic unit.
Of course, ideal crystals do not exist. A real crystal is always finite and has defects, which are deviations from regularity. Still for many purposes, these idealized crystal are still important to study.
Each translation within the three-dimensional Euclidean space is defined by a translation vector that can be described as a combination of the multiples of three independent basic vectors, and transform the ideal crystal into itself, have three linearly independent basic vectors so that each of these translations can be described as a combination of integer multiples of the basic vectors. All translation vectors translate the crystal into itself form a three-dimensional lattice within the space, the crystal lattice.
The choice of the the basis of a lattice is not unambiguous. Therefore, one tried to find a basis where the unit cell reflects the same of the crystal as exactly as possible. This is not always possible, but it is important to be easily view the crystal's symmetry, one waives the requirement that the vectors forming the unit cell be a lattice basis and uses three vectors forming larger (non-primitive) unit cells instead. In this case, the translations of the crystal lattice can be described as a combination of rational multiples of these vectors.
Auguste Bravais classified the different possible crystal lattices by indicated both primitive and non-primitive unit cells. They are named Bravais lattices afterhim. In three dimensions there are exactly 14 Bravais lattices (i.e. exactly 14 translation groups of all possible ideal crystals).
Crystals are classified according to their symmetric proeprties.
The isometry group of a crystal lattice is called the crystallographic group or space group. The translations of the lattice belong to the crystallographic group. They describe the long-range order of the crystal. However, there also isometries of the crystal lattice with one fixed point
This group described the symmetry of the basic unit and thus of the crystal itself. It is called the crystal's point group. There are exactly 32 of such crystallographic point groups, also called crystal classes. Being abstract groups, these groups are of mathematical interest.
There are 32 crystallographic point groups (crystal classes) besides the 14 translation groups (crystal lattices). There are 230 different crystallographic groups in total (i.e. isometry groups of crystal lattices in three dimensions.)
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The seven crystal systems was introduced by Weiss. Crystal systems constitute a symmetry related classification of crystal by means of crystallographic axes of coordinates.
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Hexagonal
Cubic
[1] Gert-Martin Greuel. "Crystals and Mathematics. A historical outline of the interaction between two disciplines" http://www.mathematik.uni-kl.de/~greuel/Paper/GeneralType/14Crystals-and-math.pdf