For centuries, a crystal was considered a solid body with an external shape resembling a polyhedron. Classic examples included gemstones like diamonds and amethysts, as well as cooking salt and sugar crystals. Kepler and Aai hypothesized that the external shape was a result of the internal structure, which was an ordered arrangement of identical basic units. At the beginning of the 20th century, Friedrich, Knipping, and von Laue proved this hypothesis through the first X-ray crystallography experiments. Since then, it has been agreed that a crystal is a periodic arrangement of atoms.
The formal definition of a periodic crystal structure is based on a Bravais lattice and is as follows: an n-dimensional lattice is a group of points where the vectors are linearly independent. The parallelepiped spanned by the vectors is called a unit cell, or “fundamental domain” in mathematical terms. From this definition, it is clear that the unit cell repeats periodically throughout the entire space. A motif (sometimes also called a “basis”) is a pattern that fills the unit cell. In nature, this is an arrangement of atoms. A crystal structure is a combination of a Bravais lattice and a motif, resulting in a periodic (and therefore infinite) arrangement of atoms.