There are exactly 17 distinct symmetry groups for periodically tiling the plane

A wallpaper group (plane symmetry group) classifies the symmetries of a pattern that tiles the infinite plane periodically. There are exactly 17 such groups — proven complete. Each combines a subset of four symmetry operations: translation, rotation, reflection, and glide-reflection. Generative artists use this as a constraint: pick one of the 17 groups, generate a single tile, then let the group’s symmetry rule propagate it across the canvas, which is mathematically guaranteed to tessellate. A common misconception is that a more complex motif implies more groups — motif complexity and symmetry class are independent; the same 17 classes cover every periodic pattern regardless of how intricate the tile is.

Examples

Draw one asymmetric tile. Apply p4m symmetry (4-fold rotation + reflections) to fill the canvas; compare with p6m (hexagonal). Both tile perfectly because they belong to the 17 groups.

Assessment

Name three of the four symmetry operations that define wallpaper groups. Given a tiling, identify whether it uses rotation, reflection, or both. Produce a tiling in code using any one of the 17 groups.