Weekly 2: Wallpaper groups (Revised)

One interesting connection between geometry and algebra is the idea of wallpaper groups. To explain what it is, let's first discuss symmetry groups. Remember in MTH 410 (Modern Algebra II) where we first started discussing groups. A group is a set combined with one operation that satisfy the four group axioms; let me explain.

A group is a set G on which we define one binary operation "*" that satisfy the following axioms:
1. For all a,b in G, a*b is also in G (G is closed under its operation).
2. For all a,b,c in G, (a*b)*c = a*(b*c), (* is associative in G).
3. There exists an element i in G such that for all a in G, a*i = a = i*a (G has an identity element).
4. For all a in G, there exists b in G such that a*b = i = b*a (each element in G has an inverse).

Some examples of a group are: the set of integers over addition, the set of non-zero rational numbers under multiplication, and the set of all n by n matrices with real entries under standard addition. For a symmetry group, the set is a collection of symmetries (rotations, translations, reflections, and glide-reflections) and the operation is function composition. For example, to perform a horizontal reflection composed with a 180 degree rotation on a rectangle, you would first rotate the rectangle 180 degrees and then reflect it about a horizontal line through its center. Some other examples are the group of symmetries of a square, triangle, or hexagon. So then a wallpaper group is a 2 dimensional symmetric group that is a repetitive pattern (like a wallpaper) that is categorized and classified by its symmetries. There are only 17 different types of wallpaper groups. A reason for this is because though many patterns and tessellations may look completely different, they would still fall into the same wallpaper group (essentially they have the same symmetries). The first proof that there are only 17 was actually written pretty recently by Evgraf Fedorov in 1891. Though, knowledge of some of these different groups were already known throughout history. Some of the works by M.C. Escher actually belong in one of the 17 groups. Now let's explore some of the 17 wallpaper groups. An example of the wallpaper group "p1" is

http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Wallpaper_group-p1-3.jpg/91px-Wallpaper_group-p1-3.jpg

This group has 2 translational symmetries. For this particular tessellation, it can be translated in two ways (up/down and left/right). An example of the wallpaper group "p2" is

http://en.wikipedia.org/wiki/File:Wallpaper_group-p2-2.jpg

This group has translations and rotations only. The tessellation above can be translated  to the left or right and rotated 180 degrees. An example of the wallpaper group "pm" is

http://en.wikipedia.org/wiki/File:Wallpaper_group-pm-3.jpg

This group has a reflection symmetry and a translational symmetry. This tessellation can be reflected about a vertical line through the center and be shifted to the left/right and still keep its design. An example of the wallpaper group "p4" is

http://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/A_tri-colored_Pythagorean_tiling_View_4.svg/120px-A_tri-colored_Pythagorean_tiling_View_4.svg.png

This group has translations and 90 degree rotations. This particular tessellation can translated at about a 150 degree angle and be rotated 90 degrees and still keep its shape. Things get difficult when certain groups start having glide reflections (combination of a reflection and translation). The groups I mentioned above are the more simpler ones. Wallpaper groups are a great way to connect geometry to algebra (or more generally, symmetry groups). Take MTH 410 if you would like to learn more about symmetry groups as a whole. In the future, I would like to explore why there are only 17 wallpaper groups and the proof behind it. I tried to look online and I couldn't find a real good proof. So the best option would probably be to get a book about symmetry groups on amazon.

References:

http://en.wikipedia.org/wiki/Wallpaper_group#The_seventeen_groups

A. Nelson, H. Newman, M. Shipley: 17 PLANE SYMMETRY GROUPS. wordpress.com

     http://caicedoteaching.files.wordpress.com/2012/05/nelson-newman-shipley.pdf