Then read through the following definitions, using Kali to explore the terms defined. We generated several plane patterns when we experimented with three mirrors perpendicular to the desk. Figure 8 A wallpaper pattern has reflective symmetry if there is a reflection that transforms one half of the pattern into the other half. If you set a mirror down on a line of mirror symmetry of such a pattern, you will see the same pattern in the mirror that you would if you replaced the mirror with a piece of glass. A pattern is said to have translational symmetry if there is some translation of the pattern that takes each image of the motif to some other image.
All wallpaper patterns have translational symmetries in at least two different directions.
One family of wallpaper patterns has only translational symmetry. A pattern has rotational symmetry if some rotation of the pattern takes each image in the pattern to some other image. A pattern has glide reflective symmetry if there is some glide reflection that transforms the pattern into itself. This is the rule rather than an exception. The fact that a wallpaper pattern is generated by repeatedly applying some set of transformations to a finite motif guarantees that this will happen for all the wallpaper groups.