This is maybe ambitious, for the details are obviously not completely accessible to the high school level; but the beauty of the ideas is, and this video is really a superb example of divulgation. [Smale's theoem on the eversion of the 2-sphere and Thrston's construction][1].
[1]:http://www.youtube.com/watch?v=BVVfs4zKrgk&feature=player_embedded