When I arrived in Yerevan, Armenia, I was very impressed by how beautiful the sidewalk patterns were. I saw the pattern on the left-hand side just outside of my hostel, and I think that it was the first time that I had ever seen that pattern before. Then as I was walking around the city center, I saw all sorts of other patterns, as you can see above and below.

I immediately began to wonder what types of patterns are possible, and if there was any way to classify them. In the first pattern, for example, there is an interesting rotational symmetry: there are certain points where one could rotate the pattern 120 degrees (one third of a rotation) and still end up with the same pattern. This is quite different from the other patterns, many of which exhibit rotational symmetry of 90 degrees (one fourth of a rotation) or 180 degrees (one half of a rotation), but not 120 degrees.

Also, note that in the first pattern, there is no reflective symmetry (also known as mirror symmetry). The curvature of the edges is all oriented in a certain way, and if we were to look at a mirror image of this pattern, the orientation would change. This would not be the case if all of the curves were replaced with straight line segments; in that case, we would have reflective symmetry.

It became clear that different patterns had different types of symmetry, but I wondered how many different possibilities there could be. There is a finite amount of types of symmetries, so theoretically it should be possible to classify these patterns into groups based on their types of symmetries. So I did a quick internet search, and I learned about wallpaper groups. Apparently there are precisely 17 different groups of patterns, based on the types of symmetries they have, and they're called wallpaper groups because wallpaper makes use of these different patterns.

Read the Wikipedia article on wallpaper groups, focusing on the types of symmetries in each group and the examples. Use this knowledge to answer the following questions:

Sample Problems

1. Which type(s) of symmetry can be found in the left image above? Which wallpaper group does this pattern belong to?

2. Which type(s) of symmetry can be found in the middle image above? Which wallpaper group does this pattern belong to?

3. Which type(s) of symmetry can be found in the right image above? Which wallpaper group does this pattern belong to?

4. Which type(s) of symmetry can be found in the left image below? Which wallpaper group does this pattern belong to?

5. Which type(s) of symmetry can be found in the middle image below? Which wallpaper group does this pattern belong to?

6. Which type(s) of symmetry can be found in the right image below? Which wallpaper group does this pattern belong to?