These pages contains the derivation all possible symmetry patterns of wallpapers that has rotational and or translational symmetries. (Reflection or glide reflection are not included here.) Technically, it is a derivation of all discontinuous groups of rotation and translation in the plane. A appendix contains illustrations of all the 17 wallpaper groups.

This exposition is written by a math lover for math lovers. The topic is developed from a pure math point of view. It began when I was learning wallpaper groups and wanted a rudimental understanding. The materials are originally based on chapter 2 of David Hilbert and S. Cohn-Vossen's *Geometry and the Imagination*, 1932. (English translation published by Chelsea Publishing Company, 1952.) Later additions and revisions gather information from many sources. (see the Reference Section)

This exposition is divided into the following chapters:

- Introduction. This is the page you are reading.
- Some Theorems on Rotation and Translation. Theorems that will be used throughout later chapters.
- The Discontinuous Groups of Rotation and Translation in the Plane. A setting of the groups to be discussed.
- Derivation and Classification of Groups. The actual derivation, using theorems in previous chapters.
- Appendix: The 17 Wallpaper Groups with a wallpaper gallery.
- References and Related Web Sites.

This exposition would be of interest to anyone who wants a introduction to wallpaper groups. For example, students who have just learned the concept of groups, scientists and engineers with a casual interest in the mathematics of crystallography, or any general math lovers.

A knowledge of highschool level geometry is needed for reading the chapter: *Some theorems of rotation and translation*. For later chapters, you should know vectors and understand the concept of a group, usualy taught in 3rd year college. If you are a highschool student and find this writing too difficult, visit some of the sites in the section Related Web Sites.

I was born in 1968. I studied mathematics in a two-year community college (in California, US) and spend several years studying math on my own. To me, mathematics is very important, to the point that everything else in life seems trivial. As of 1998, I do not have a degree. My lifelong goal is to be a contributing mathematician. (I'm very disatisfied with my experience at school. I felt that the education system has become a bureaucratic rigmarole and people forget the goal of education. School system is now a pestering fashionable business full-fleged with political trendies.)

Please send me (xah@xahlee.org) any comments, suggestions, or corrections you have on this exposition.

In our context, symmetry will mean rotation or translation but excluding reflection and glide reflection. Similarly, a transformation will mean rotation, translation, or identity only. Sometimes we will use the word “motion”; to mean transformation. Exceptions should be obvious.

Let r[{x,y},α] denote a rotation of α around fixed point with coordinates {x,y}, and t[{x,y}] denote a translation with vector {x,y}. Capital letters will usually denote a point in the plane. We'll use the notation r[C,α][P] to mean a point obtained by applying the rotation r[C,α] to P. Similarly, t[V][P] means a point obtained by applying the translation t[V] to P. A double equal sign will mean a true statement of equality, ⁖ 2+2==4. A colon-equal sign will mean a definition or assignment, ⁖ a:=7, Q:=t[V][P]. The distinction is for clarity of meaning but you could just use a single equation sign as in conventional text. For brevity, we'll use r1[C,α] to mean r1:=r[C,α]. Similarly, t1[V] is a short hand for the assignment t1:=t[V]. After such assignments, we can string a series of symbols to indicate a succesion of transformations, ⁖ r1*r2*t1*r2^-1 means rotation r1 followed by rotation r2, then translation t1, then the inverse of rotation r2. If m is a motion, then m^-1 will denote it's inverse. For example, the inverse of r[C,α] is r[C,-α] and can be written as (r[C,α])^-1. Similarly, (t[V])^-1==t[-V].

A icon indicates a Mathematica notebook. A icon indicates GeoGebra file. These are supplementary files that help understanding and exploration.

Most graphics in this work are generated by Mathematica by the package PlaneTiling.m. The package is available at Plane Tiling Mathematica Package.

There are a number of softwares that specializes in 2D tiling and symmetry. I highly recommend trying them out. Please see the tiling section at my other page: Great Math Software.

1997