Point group symmetry

Initiation to the discovery of point group symmetry

Nicolas Schoeni and Gervais Chapuis
École Polytechnique Fédérale de Lausanne, Switzerland

The applet allows to study the symmetry operations of a small selection of polyhedra.

Every symmetry operation of an object (polyhedron, molecule, etc) which leaves at least one point invariant (i.e. unchanged) is described by its point group symmetry. If we limit ourselves to rotations of order 1, 2, 3, 4 or 6, we can find a total of 32 different groups, the so-called 32 point group symmetries.

The symbols of the 32 point groups in the international notation are given in the following table:

Systems point group
Cubic m3m, 43m, m3, 432, 23
Tetragonal 4/mmm, 42m, 4mm, 422, 4/m, 4, 4
Hexagonal 6/mmm, 62m, 6mm, 622, 6/m, 6, 6
Trigonal 3m, 3m, 32, 3, 3
Orthorhombic mmm, mm2, 222
Monoclinic 2/m, m, 2
Triclinic 1, 1