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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.


The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and we can treat them together.


A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original.


The four translation vectors (three of which are given by r, and one which connects A' and B' given by r') form a parallelogram. Therefore, the length of r' is also given by:





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