This applet dynamically demonstrates a
mathematical transformation called conformal mapping. Simply put, this
takes a point in the complex plane and moves ("maps") it to a
different place. This has all sorts of interesting applications, from
aerodynamics to fractals. I think they do
interesting things dynamically (some of which you can try here).
How it works: drag the mouse to move the source location, which is a grid of points two units square. This is transformed (using the equation shown at upper right) into the green points displayed on the graph. You can optionally turn off the axis, shown in black, which displays the unit circle and the real and imaginary axes. You can optionally display the source region, shown as a red square. The slider controls the zoom factor: sliding it to the right zooms in. The "Next Map" button cycles through several available transformations; the last one is the identity [f(z) = z] that does not change the source.
For more about the mathematics behind this, there's a good overview at the Mathworld site: http://mathworld.wolfram.com/ConformalMapping.html