The symmetry found in regular polygons has been used to created interesting patterns throughout history. The mathematical study of such symmetry patterns is at the heart of group theory, a major focus in abstract algebra. Each uniform tessellation of the plane by undecorated tiles has an underlying wallpaper or planar symmetry group. I am seeking funding to characterize the point symmetry group properties of patterns obtainable using 1-uniform and 2-uniform tessellations. This work includes identifying the possible geometrically unique symmetry subgroups admissible for each tessellation. Another important issue is determining the symmetry types of the polygons comprising the tessellations. I will create algorithms and new software to implement these appropriate symmetry groups for each tessellation. Using this new software, I will create a catalog of example patterns for each possible point symmetry group for each tessellation.