For decades scientists have accepted the premise that solid matter can only order in two ways: amorphous (or glassy) like window glass or crystalline with atoms arranged according to translational symmetry. The science of crystallography, now two centuries old, was able to relate in a simple and efficient way all atomic positions within a crystal to a frame of reference in which a single unit cell was defined. Positions within the crystal could all be deduced from the restricted number of positions in the unit cell by translations along vectors formed by a combination of integer numbers of unit vectors of the reference frame. Of course disorder, which is always present in solids, could be understood as some form of disturbance with respect to this rule of construction. Also amorphous solids were naturally referred to as a full breakdown of translational symmetry yet preserving most of the short-range order around atoms. Incommensurate structures, or more simply modulated crystals, could be understood as the overlap of various ordering potentials not necessarily with commensurate periodicities.
For so many years, no exception to the canonical rule of crystallography was discovered. Any crystal could be completely described using one unit cell and its set of three basis vectors. In 1848 the French crystallographer Bravais demonstrated that only 14 different ways of arranging atoms exist in three-dimensional space according to translational symmetry. This led to the well-known cubic, hexagonal, tetragonal, and associated structures. Furthermore the dihedral angle between pairs of faces of the unit cell cannot assume just any number since an integer number of unit cells must completely fill space around an edge.