I want to show some interesting quite simple application coded in Swing. Here you can see the result:
Source code you can find here: https://github.com/TomaszWaszczyk/Conwey ,the application shows how small change in the system can have a big influence on the future.
More reference: https://en.wikipedia.org/wiki/Conway's_Game_of_Life or https://de.wikipedia.org/wiki/Conways_Spiel_des_Lebens (in German)
I want to also recommend some great application for making gif files: http://www.cockos.com/licecap/