Today, we are going to start to look at symmetries of objects by looking at some of the common symmetrical objects people look at, that is, regular polygons. We are going to need some polygons to play with.
In order to make our polygons, we have a few options. To begin with, I have already mentioned straight-edge and compass constructions in Comparing fractions with Pythagoras. We can use such techniques in order to construct a number of regular polygons. Doing this can be a lot of fun in class as it gives students things to work with as they follow along. If doing this in class, I would suggest projecting your work on a screen using either Geogebra or Geometer’s Sketchpad. Both provide computer simulated compass and straight-edge and allow for all the geometric constructions. While I like the interface of Geometer’s Sketchpad slightly better, Geogebra is free to use and doesn’t require a download, which is helpful when working in different classrooms.
Instead of using straight-edge and compass, I am going to work with paper folding techniques. I like this technique because I often find myself without a compass, but I am always able to find paper (especially scrap paper) to work with. Furthermore, depending on the maturity of the students, it may not be a great idea to hand them compasses.
Making a triangle
We begin with constructing an equilateral triangle.
- Start off by folding a piece of paper in half the long way as shown below.
- In order to help see the fold, I would draw the line where the fold is. Then fold the paper so that the corner of one side is on the previously drawn line. Mark the point of intersection.
- Then fold the paper so that you create a crease through the corner and the point you just labeled. Repeat for the other corner.
- We then can draw in these lines so that we can see them.
- At this point I would then cut out the triangle I have drawn. In addition, I will want another triangle to use, so I will cut that one out by tracing the triangle I just constructed.
- In order to continue with the project label the corners of both of your triangles.
- One of these will remain stationary, but we will be moving the other one around, so we also want to label the corners on the back. We should do this in such a way that each corner is labeled the same on each side. In the picture below, you should be able to see the numbers that are labeled on the other side as the ink bled through.
Symmetries of an equilateral triangle
The goal now is to get your students to find all the symmetries of an equilateral triangle. I would begin by posing the question to them, “Line up the two triangles so that they are on top of each other. Now, if we leave the bottom triangle stationary, what can we do to the top triangle so that, while it may move, the corners will still line up with the bottom triangle?”
When I work on this project in class, I will have my students working together in groups. I will leave the question above open to interpretation to see what they come up with and will work my way around the classroom. Some students will jump in and get the point, while others may not grasp the goal. Therefore, I try to give them hints one by one until they catch on. The hints I pass on, if needed are,
- If I do nothing, do the two triangles line up? What would you say you’ve done here.
- Suppose I rotate the triangle a small angle. Do the corners again line up? In this case they won’t so this would be an example of a movement that does not create a symmetry.
- Keep rotating the triangle about the center until you have rotated 120 degrees. Note then that your corners will now be lined up again, so you will have another symmetry.
- If they still need help show rotations of 240 degrees, then relate the do nothing symmetry to a rotation of 0 degrees.
- Show that you can also get the corners to line up by flipping the triangle over. In particular, you can flip over the altitude through each of the vertices.
- Note that the purpose of numbering the corners is so that you can see each of these symmetries is distinct. That is, each symmetry will send one corner somewhere different than the other symmetries. In this way, let them continue until they realize these are all the symmetries of the equilateral triangle.
- Provide names for all the symmetries; R0 is the rotation of 0 degrees, R120 is a rotation of 120 degrees, R240 is a rotation of 240 degrees, F1 is a flip over the altitude through the corner labeled 1, F2 is a flip over the altitude through the corner labeled 2, and F3 is a flip over the altitude through the corner labeled 3.
Making a square
This is actually much easier than constructing the triangle. Here we only need one fold and one cut.
- Fold your piece of paper so that the bottom edge lines up with the side edge.
- Cut off the excess and unfold the square. Label the corners so that each corner is labeled the same on each side.
I would now follow the same set up as last time for finding the symmetries of the square. Here, however, we will have the symmetries; R0 is the rotation of 0 degrees, R90 is a rotation of 90 degrees, R180 is a rotation of 180 degrees, R270 is a rotation of 270 degrees, H is a flip over the horizontal line through the bisectors of sides 1 4 and 2 3, V is the vertical flip over the line through the bisector of the sides 1 2 and 3 4, / is the flip over the corresponding diagonal and \ is the flip over this diagonal.
At this point the students have a square and a triangle in front of them. I would start with the subjective questions, “Which of the two shapes do you like more?” or “Which of the shapes do you find more aesthetically pleasing?” Then ask them why they like that shape. Whichever choice people make, have them explain why they like that one more? After they have provided their answers, state that the square is more symmetrical than the equilateral triangle. We can see this by noting that there are more symmetries of a square than a triangle. If they had liked the square more, make the connection to symmetry and beauty. If they had chosen the triangle, then work with the fact that beauty is subjective and not everyone sees the same thing.
If you would like to generalize these findings to a general regular n-gon you can now do this. While you can use paper folding techniques to make a regular pentagon and some more regular polygons, I would instead suggest that you use images that are constructed ahead of time. The construction process can take up a significant amount of time when you get more edges involved. However, you should be able to get your students to come to the conclusion that, if you have a regular n-gon, you will have n rotations with angles being the multiples of 360/n. You will also have flips, but these will break into two cases. In the odd cases, the flips will be over the line through a vertex and perpendicular to the opposite side. In the even case, you will have flips over lines through the two opposite vertices and flips over the lines that bisect two opposite sides. In each case we will have 2n symmetries of each regular n-gon. To go a step further, you can also find the symmetries of the circle, that is reflection over any line through the center or a rotation of any angle.
Now that you’ve been able to classify the symmetries of the regular polygons and the circle, you can make a comparison to how symmetric each shape is. Clearly the more sides or points, the more symmetries there will be. Noting that shapes with more symmetry are more aesthetically pleasing in general is also worth pointing out to your class. I would point out that increased symmetry leading to a higher class is the basis of class structure used in Flatland as a metaphor for the class structure in Victorian society.
If you go through this project on your own or in class, let me know how it goes and how your students like it. I always enjoy working with this project in my classes, and I hope you have the same experience with this.
If you liked this post, let me know by liking it below or leaving a comment. Next time in the symmetry series we will continue with this topic. In particular, we will look at what happens when we take an object and perform two or more symmetries in a row. What will this composition of symmetries look like and, furthermore, what will the set of all symmetries look like under this composition?