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As a math major, I took a class called Geometry, where we studied visual symmetry through the lens of abstract algebra.
In it, we learned about the 17 "wallpaper groups" and that there are provably only 17 different ways to tile a plane, i.e. there are 17 different ways in which you can arrange identical shapes to completely cover a plane while maintaining perfect symmetry. You can achieve these patterns by applying a combination of translations, rotations, and reflections to a single "motif".
I found this fun page that interactively illustrates each wallpaper pattern, and I wanted to see what happens when you apply different wallpaper patterns to a dynamic motif.
The motif is a simple square that moves in a random snake-like path. The transformations are applied to the square as specified, then tessellated across the plane.