Symmetry Groups in R^2

What comes to mind when we think of symmetry? In nature, we often think of a butterfly’s wings or a snowflake. We can also find symmetry in the faces of our peers. These examples are perhaps the most basic notion of symmetry we have. Our goal however, is to study symmetry groups of the plane. We define a symmetry group of a figure in the plane to be the set of all isometries that carry the figure onto itself, or leave it unchanged. These symmetry groups can be classified into two cases: finite and infinite symmetry groups, and both will be examined. When we open our minds to the possibility of an infinite symmetry group, we can analyze the unique designs within wallpaper and jewelry that result from periodic—or repeating—designs in the plane. Specifically, we concern ourselves with two types of infinite symmetry groups known as the frieze and crystallographic groups. We pay particular attention to the crystallographic groups, sometimes called the wallpaper groups, first studied by 19th-century crystallographers who were studying the arrangement of atoms in crystalline solids. We discover that no wallpaper group can contain 5-fold rotational symmetry—or, a rotation of order 5. In fact, this result is a critical component of the fact that a wallpaper group can only contain rotational symmetries of order 1, 2, 3, 4, and 6—a fact known as the crystallographic restriction.