Crystallographic Symmetry
point groups · space groups · Bravais lattices
Physics: Crystallography: 7 crystal systems, 14 Bravais lattices, 32 point groups, 230 space groups in 3D; 17 wallpaper groups in 2D. Symmetry operations: rotation Cₙ (n=1,2,3,4,6 only — crystallographic restriction theorem), reflection σ, inversion i, screw axis, glide plane. Neumann's principle: any physical property must have at least the symmetry of the point group. Pyroelectric (10 polar groups), piezoelectric (20 non-centrosymmetric groups), ferroelectric (requires switchable polarization). Space groups determine systematic absences in X-ray diffraction patterns.