The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other symmetries, such as 5-fold; these were not discovered until 1982, when a diffraction pattern out off a quasicrystal was first seen by the Israeli scientist Dan Shechtman, who won the 2011 Nobel Prize in Chemistry for his discovery.
Prior to the discovery of quasicrystals, crystals were modeled as discrete lattices, generated by a list of independent finite translations (Coxeter 1989). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group. The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups.
Read more about Crystallographic Restriction Theorem: Dimensions 2 and 3, Higher Dimensions, Formulation in Terms of Isometries
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