## The Group of Symmetries of the Square

The square has eight symmetries -
four rotations, two mirror images, and two diagonal flips:

These eight form a group under composition (do one, then another). Let's
give each one a color:

## The Multiplication Table of D4 With Color

You can rotate the square in the eight possible ways
by clicking on successive buttons to see
which element results. Note: this group is not abelian, clicking is
not commutative. To use the multiplication table the first operation
is the column index and the second the row. Thus R1 followed by M1 is
D2, while M1 followed by R1 is D1.

*Warning.
*
When using Internet Explorer sometimes the first click after loading
the applet does nothing. Behavior is fine with firefox.

The colorful design and original applet code is the work of
Erin Carmody at the MSRI Mathematical Graphics Workshop for
undergraduate math majors held at Reed College in 2005. Visit her at
www.erincarmody.org/.
You can see her version of the applet at merganser.math.gvsu.edu/david/reed05/projects/carmody/html/presentation.html.
Ethan Bolker rewrote the applet in
June 2007 for Yu Zhang to use in her thesis. The source code for this
version is in file RotateSquare.java. To run the applet as an
application, or to modify it, you will need the jar file
rotate.jar.

I think it would make an interesting exercise to generalize this
application so that it calculated in an arbitrary
dihedral
group.
Some students at the Reed summer institute started the job two years
earlier, but without Carmody's sense of color. You can see their work at
http://merganser.math.gvsu.edu/david/reed03/projects/ettingerGuy/.