Dihedral group visualizer

How to use the visualizer

With this tool you can visualize and interact with the dihedral group \(D_{2n}\) acting on a regular \(n\)-gon.

The elements \(r_0, r_1, r_2, \ldots\) represent rotations, while \(s_0, s_1, s_2, \ldots\) are the reflections. (\(r_0\) is the identity element, as a rotation by \(0\) has no effect.)

The visualizer will show in the top-left corner the result of the current computation. For example, one can start with the equilateral triangle, apply \(s_1\) (reflection across the axis \(y=\sqrt{3} x\)) and then \(r_1\) (rotation by \(\frac{2 \pi}{3}\) radians). The result of this will be \(s_2\), in other words, composing these two actions is the same as a reflection across the axis \(y=-\sqrt{3}\). In cycle notation, the operation \(r_1 s_1 = s_2\) can be written as \((123)(12) = (23)\).

In this way, one can use the visualizer as a sort of calculator.

One can also see fixed points very easily: all of the reflections in \(D_6\) fix one point, while none of the rotations have fixed points. On the other hand, in \(D_8\), the even reflections fix two points, while the odd reflections have no fixed points.

Rotations

\(r_k\) corresponds to a counterclockwise rotation by \(\frac{2\pi k}{n}\) radians.

Reflections

\(s_k\) corresponds to a reflection across the axis making an angle of \(\frac{\pi k}{n}\) radians from the origin.

Source

Source code is available here and licensed under GPLv2.