Wallpaper Groups
Wallpaper patterns result when symmetrical repetition takes place in two directions, thus forming a two-dimensional pattern that covers the plane with no gaps or overlaps. Although it seems that there would be a limitless variety of repeating patterns, it turns out that there are only 17 systems which mathematicians classify as symmetry groups. E.S. Fedorov published his proof of only 17 possibilities in 1891, but it appeared only in Russia. In 1924, George Polya published his paper, "Uber die Analogie der Kristallsymmetrie in der Ebene," translated "On the analogy of crystal symmetry in the plane." This is important to note because it reflects that fact that wallpaper patterns are related to the crystals studied by chemists. His paper gave an example of each wallpaper pattern and these examples were an inspiration for the Dutch artist M.C. Escher, famous for his intriguing drawings that are examples of these symmetries.
Each symmetry involves a combination of transformations on the plane. To determine how to categorize or organize these different wallpaper patterns we can ask a series of questions about the isometries. The answers to these questions will generate a way to think about the organization of the patterns.
First we begin with the question "What order of rotations are there?" To answer this, first we must recognize that there can be rotations of orders 1, 2, 3, 4, and 6 in the wallpaper groups, and these are the only possibilities. This is called the "crytallographic restriction". Next, we ask are there any reflections or glide reflections? If so, is the lattice rectangular or rhombic (with respect to the axis of reflection)? If rectangular, are there reflections (as opposed to glide reflections?) If so, is there an axis of reflection through a point of maximal rotation? Finally, if the maximal order of rotation is three, you have to look carefully: there are two choices, reflection through the midpoints or reflection through the axes. The following table shows the number of symmetry types having some of these properties.
Order of rotation |
1 |
2 |
3 |
4 |
6 |
Reflections - no |
1 |
1 |
1 |
1 |
1 |
Reflection - yes |
3 |
4 |
2 |
2 |
1 |
Rhombic |
1 |
1 |
|||
Rectangular |
2 |
3 |
Although to teach this curriculum unit, this level of detail is beyond the scope of the students' learning, it is a desirable foundation for the instructor to have for preparing activities and relaying the concept properly and accurately.
Comments: