关于晶体制约定理的一个严格证明（英文）

A rigorous proof on the crystallographic restriction theorem

关于晶体制约定理,有必要研究和提出更加严格和完美的证明。不存在C5轴等价于不能够用相互之间无任何空隙的五边形填充满所有的空间。以这一观点为基础,本文利用纯粹的数学方法严格地证明了不存在晶体的C5和Cn(n≥7)对称轴,而允许存在1,2,3,4以及6重转动对称轴,从而证明了晶体的转动对称轴只能够存在C1,C2,C3,C4和C6。

It is significant to find a more rigorous and satisfactory proof of the crystallographic restriction theorem . The inexistence of C 5 axis of symmetry is equivalent of that pentagons are impossible to fill all the space with a connected array of pentagons. On the basis of this viewpoint, using a purely mathematical approach the paper rigorously proves that C5 and Cn (n≥7) axes of symmetry can not exist , and one-, two-, three-, four-and six-fold axes of rotational symmetry are allowable, therefore, the axes of symmetry of the crystal can merely exist C1, C2, C3, C4 and C6.