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in crystallography, listing of the ways in which the orientation of a crystal can be changed without seeming to change the positions of its atoms. These changes of orientation must involve just the point operations of rotation about an axis, reflection in a plane, inversion about a centre, or sequential rotation and inversion. Only 32 distinct combinations of these point operations are possible, as demonstrated by a German mineralogist, Johann F.C. Hessel, in 1830. Each possible combination is called a point group, or crystal class. A crystal can be assigned to one of these point groups on the basis of its external shape, or morphology. The addition of translational changes will yield a total of 230 possible combinations; these are called space groups.