a crystal is a shape which tessellates the plane (or space) in a periodic way, so that the pattern repeats at regular intervals in all directions. the group of symmetries (translational, rotational, reflectional) of such a tessellation is called a crystallographic group. in two dimensions there are exactly 17 different kinds of symmetry, the so-called "wallpaper groups", which i'll describe. i'll also mention what happens in three dimensions, in hyperbolic space, and how you can make "quasiperiodic" tessellations (penrose tilings) with five-fold symmetry.