Wallpaper groups and the crystallographic restriction theorem for the plane are a wonderful application/example of finite group theory and group actions.

This is a really good relevant clip: http://www.youtube.com/watch?v=7zLi47yYlcc#t=7m43s (queued up at the relevant point, whence came the #t=7m34s).

which continues: http://www.youtube.com/watch?v=xP52g6eQRmY&feature=related

Also, Bronowski was a mathematician.