´ÓÎÒ¸öÈËÀí½â,Õâ¸öpage²ûÊöÁËNiggliÔÚthree-dimensional arithmetic crystal classes (symmorphic space groups)×ö³öÁ˺ܴóµÄ¹±Ï×. Èç¹ûÄã²éÒ»ÏÂʲôÊÇsymmorphic space groups, Äã»áµÃµ½ÏÂÃæµÄÐÅÏ¢.
Arithmetic crystal classes (73 in three dimensions). These are determined by the point group together with the action of the point group on the subgroup of translations. In other words the arithmetic crystal classes correspond to conjugacy classes of finite subgroup of the general linear group GLn(Z) over the integers. A space group is called symmorphic (or split) if there is a point such that all symmetries are the product of a symmetry fixing this point and a translation. Equivalently, a space group is symmorphic if it is a semidirect product of its point group with its translation subgroup. There are 73 symmorphic space groups, with exactly one in each arithmetic crystal class. There are also 157 nonsymmorphic space group types with varying numbers in the arithmetic crystal classes.