a crystal is a shape which tessellates the plane (or space) in a periodicway, so that the pattern repeats at regular intervals in all directions.the group of symmetries (translational, rotational, reflectional) of sucha tessellation is called a crystallographic group. in two dimensions thereare exactly 17 different kinds of symmetry, the so-called "wallpapergroups", which i'll describe. i'll also talk about what happens in threedimensions, in hyperbolic space, and how you can make "quasiperiodic"tessellations (penrose tilings) with five-fold symmetry.