Orbits in space group theory

Space group theory is developed by the decoration of the fourteen possible ways of arranging points in regular 3D arrays, in space, the Bravais lattices of Crystallography. In Crystallography, the decoration about a lattice point is called the basis or, less commonly. 

that since five-fold rotational symmetry cannot be propagated on alattice, there are only 32 Crystallographic point groups, since the icosahedral groups are excluded. 

The stereograms include symbols to identify the locations of the symmetry elements of the structures with respect to the regular orbit points on the unit sphere. These are shown normally as filled polygons for proper rotational axes and empty polygons for improper rotations, which give rise to actions across the hemispherical plane, while binary rotations are shown as ellipses, either filled or empty, but mirror planes, the improper axes of binary rotation, are distinguished on the stereograms as solid lines. 

Individual orbits, as sets of coordinate points, which are decorations of the lattice points by the basis elements of structure are listed using Wyckoff numbers for individual unit cells or conveniently grouped sets of unit cells in the International Tables for Crystallography published. Two examples of these listings are reproduced in Tables 2.3 and 2.4. 

---