In Chapter 28 of Contemporary Abstract Algebra, Joseph A. Gallian investigates the two types of infinite symmetry groups determined by periodic designs in a plane, frieze groups and crystallographic groups. This paper is intended to provide an overview of these groups with special concentration on the crystallographic groups, also known as wallpaper groups. There are exactly 17 of these groups. I would like to prove there are only 17 groups and discuss methods to determine which group a particular pattern belongs to. As an aid in this examination, I will be using resources provided by John H. Conway, a contemporary mathematician from Liverpool, England who spent many years on the faculty at Cambridge and Princeton University. Conway has made significant contributions to group theory and has done a great deal of important work with patterns. As a visual aid, I will be incorporating the artwork of M.C. Escher. Escher was very fond of using mathematical ideas as a basis for his abstract space-filling works. These artistic pieces, such as Study of Regular Division of the Plane with Human Figures and Study of Regular Division of the Plane with Fish and Birds, are frequently used to illustrate the crystallographic groups. First, I would like to provide some background on the material I will be discussing. We can identify a planar symmetry as a transformation of a plane that moves a pattern so that it falls back on itself. The only transformations that we'll consider are those that preserve distance, called isometries. There are four kinds of planar isometries: translations, rotations, reflections, and glide reflections. Gallian defines a translation as a function that carries all points the same distance in the same direction. In other words, if you have a given repeating pattern, you can slide it along a certain direction a certain distance and it will fall back upon itself with all the patterns exactly matching.
