The Dihedral Group

by: Bree Ettinger and G. Michael Guy

Presented in this page is an exploration of the Dihedral group, the symmetries it represents and various other algebraic and geometric properties of the Dihedral group. The Dihedral group, Dn, is the symmetry group of a regular n sided polygon. The group is generated by a rotation by 360/n degrees and a flip across an axis through the center and a vertex. As we will see, only these two operations are needed to generate the entire Dihedral group.

An n-gon is symmetric with respect to a rotation by T=360/n degrees. Click on the applet to see a rotation. You may change the number of sides of your polygon using the control at the top of the box. Note that after rotating n times you come back to the original position. A rotation by T degrees is represented by R, and thus, symbolically we have R n = e , where e is the identity. One should also note that all rotations can be acquired with only using one rotation button. For example, pressing the "Rotate -" button once is the same as pressing the "Rotate +" button n-1 times.

An n-gon is also symmetric with respect to a flip across an axis through the center and a vertex. Click on the applet to see a flip. Note that after flipping twice the polygon returns to the original position. A flip is represented by an F. Thus, symbolically we have F 2 = e .

You may have noticed that in the applet above, you are only able to flip across one axis. Yet, an n-gon is symmetric with respect to any flip along an axis through the center and a vertex. One can accomplish these other flips by first rotating the desired vertex to the flipping position, flipping, then followed by rotating back to the original position. This is called conjugation by a rotation. In the applet below, you may both rotate and flip a pentagon. Give it a try!

  • Click the "Rotate +" button 2 times to get the (4) at the top.
  • Click the "Flip" button once.
  • Click the "Rotate -" button 2 times to get the (4) back where into its initial position.
Ta-dah! You have flipped about vertex (4). The (4) stayed fixed and the (3) and the (5) vertices switched as well as the (1) and the (2) vertices. Symbolically, we represent the above combination of operations by R2 F R-2. Experiment with the applet after hitting the reset button to see if you can discover how to flip about the other axes. You did it? Congratulations! You are a conjugating maniac! Don't forget this skill on your resume.

At this point, we make another observation about how the flip interacts with the rotation. A reasonable question to ask is whether flipping then rotating is the same as first rotating and then flipping, i.e. is the Dihedral group abelian? Using the applet above, try this and see if you can predict the answer. Did you try it yet, or are you just continuing to read to find the answer? Try it, damn it! Okay, great. So did you get it? You did?! Excellent! So you discovered that F R =R-1F, which is also the same as saying that F R F -1=R-1.

Combining what you have learned above, we can now define the Dihedral group using only two generating elements, R and F. We have that the Dihedral group is given by:

Dn={R iF j: R n=e, F 2=e, F R F -1= R -1}
The fact that this definition gives all symmetries of the regular n-gon follows easily from a counting argument.

Now, by using the applet, we can see all the symmetries of a regular polygon by using only a rotation and a flip.

Oops!! there is one element in the Dihedral group which we forgot to demonstrate. Here it is!

This project was completed while attending MSRI's Mathematical Graphics Program July 14-25, 2003 at Reed College in Portland, OR.