Symmetry principles in old and new physics

This is the authors' abstract. We don't add key points for this paper.

Symmetry and invariance considerations have long played important roles in physics. The 32 crystal classes—that is, groups of rotations in three-dimensional space all the elements of which are of the order 2, 3, 4 or 6—were determined 137 years ago, in the same year in which group theory was born. The determination of the 230 space groups, by Schonflies and by Fedorov (these are the discrete subgroups of the Euclidean group which contain three noncoplanar translations) was a masterpiece of analysis and so was the determination by Groth of the possible properties of crystals with the symmetries of these space-groups.