Cubic and ultimate groups of complete symmetry

Summary

It is shown that the systems of definite actions described by polar and axial tensors of the second rank and their combinations during the superposition of their elements of complete symmetry with the elements of complete symmetry of the “grey” cube, result in 11 cubic crystallographical groups of complete symmetry. There are 35 ultimate groups (i.e., the groups having the axes of symmetry of infinite order) in complete symmetry of finite figures. 14 out of these groups are ultimate groups of symmetry of polar and axial tensors of the second rank and 24 are new groups. All these 24 ultimate groups areconventional groups since they cannot be presented by certain finite figures possessing the axes of symmetry$\bar \infty , \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\bar \infty } ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\infty }$ . Geometrical interpretation for some of the groups of complete symmetry is given. The connection between complete symmetry and physical properties of the crystals (electrical, magnetic and optical) is shown.