Wallpaper Group - Dependence of Wallpaper Groups On Transformations

Dependence of Wallpaper Groups On Transformations

• The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).
• Translational symmetry is preserved under arbitrary bijective affine transformations.
• Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.
• Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.

Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.

Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.

If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.