Rotational Symmetry - Formal Treatment - Rotational Symmetry With Translational Symmetry

Rotational Symmetry With Translational Symmetry

2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. There are two rotocenters per primitive cell.

Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell:

• p2 (2222): 4×2-fold; rotation group of a parallelogrammic, rectangular, and rhombic lattice.
• p3 (333): 3×3-fold; not the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored.
• p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice.
• p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice.

• 2-fold rotocenters (including possible 4-fold and 6-fold), if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not apply.
• 3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor

• 4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor
• 6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.

Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.

3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is 2√3 times their distance.