Fixed Point Theorem

(This entry originally written April 16, but it seemed sort of silly and I postponed it. It still seems silly but here it is anyway, because analogy-making is fun even if the analogy is vague. )

Current Music: Lesson #8, Sunday in the Park with George, by Sondheim

The Fixed Point Theorem just came to mind as I pondered a favorite situation: say I'm sitting with a girl watching my favorite movie and I notice that the plot of the movie seems to be following the plot of my own life. A moment inevitably comes, then, when the scene in the movie is nearly identical to the scene in real life as we watch the movie. If it were an exact mapping from the movie to real life (or is that vice versa?) then that moment would be like the moment when you stand in front of a bathroom sink with a mirrored medicine cabinet and notice that another mirror stands just opposite to the medicine cabinet, so that closing the mirrored cabinet door causes the start of an infinite regress. You start to see many copies of yourself and the medicine cabinet with it's nearly-closed, just slightly angled door. At some point it seems an infinity must happen as the two mirrors become parallel. (You never can see the infinite regress, though, as your eyes get in the way. It makes one wonder: are there poor, unfortunate light particles trapped forever, bouncing back and forth between the two mirrors? Of course not, but it's amusing.)

It's cathartic when the movie scene and real-life scene match up. It almost feels inevitable, a result just like the closing-of-the-cabinet in the parallel mirror situation. It also is reminiscent of the Fixed Point Theorem from analysis, or better, the 2-D map fixed-point theorem from dynamic systems. It seems inevitable that if one watches the entire movie, then there will exist a point p, that moment that is parallel in both the movie and real life. Of course, none of this is actually guaranteed and the analogy is very vague.