The 17 wallpaper groups — all possible 2D crystal symmetries
There are exactly 17 distinct wallpaper groups — all possible symmetric tilings of the plane (proved by Fedorov 1891, Schoenflies 1891). Each group is characterized by its rotation orders, reflection axes, and glide reflections. 3D crystals have 230 space groups (Fedorov/Schoenflies 1891). In 2D, only rotation orders 1, 2, 3, 4, 6 are compatible with periodicity (crystallographic restriction).