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, D_{n}, 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 ngon 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 n1 times. 
An ngon 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 ngon 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!

Example:

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^{1}F, 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: 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!