Mirror, Rotate, Repeat

Symmetry means a shape or system looks the same after a transformation — reflection, rotation, or translation. Your face has approximate mirror symmetry. A snowflake has six-fold rotational symmetry. A brick wall has translational symmetry: shift it one brick over and the pattern is unchanged.

Mathematicians study symmetry through group theory, which classifies all the symmetries an object can have. For tiling a flat plane there are exactly 17 distinct patterns, and every wallpaper, fabric, and tile floor in history uses one of those 17 groups. The tilework of the Alhambra palace in Spain is a famous showcase of many of them.