A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. There is 17 wallpaper groups. I made an example of each one with a domain coloring technique. I'd like to thank Frank A. Farris for his amazing book "Creating Symmetry".
I'll first show the 17 groups, then how we can play with them by morphing them or animating them.