32 crystallographic point groups — stereographic projections and symmetry operations
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Hexagonal
Cubic
Select a group
Click any point group to see its stereographic projection and symmetry elements.
Crystallographic Point Groups: The 32 crystallographic point groups enumerate all ways a finite 3D object can be symmetric while being compatible with a periodic lattice (the crystallographic restriction: only 2-, 3-, 4-, 6-fold rotations allowed). Each group is characterized by rotation axes (Cₙ), mirror planes (σ), inversion center (i), and improper rotations (Sₙ). Stereographic projection maps 3D symmetry elements onto a 2D disk: poles of rotation axes appear as points, mirror planes as lines. The 32 point groups distribute across 7 crystal systems. Groups with inversion symmetry (centrosymmetric) cannot be piezoelectric. Groups with polar axis allow spontaneous polarization (ferroelectrics).