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Crystallographic restriction theorem
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 diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman.〔Shechtman et al (1982)〕
Crystals are modeled as discrete lattices, generated by a list of independent finite translations . 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.
==Dimensions 2 and 3==
The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we can treat them together.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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