Möbius Metaphor
/A couple of hours before class last Thursday, I got a text from Eric asking if I could talk about the Möbius strip. He had this idea, not completely worked out at the time (seriously, like two hours before class), that the structure of Aronofsky's \(\Pi\): Faith in Chaos could be modeled by a Möbius strip in some way. OK, I said, and quickly made one out of a strip of paper right before I left for class (second week in a row that I couldn't get a spot in my "secret" parking lot; I guess it's not so secret anymore).
The film is jarring in many ways, one of which is the repetition of Max's routines. When he feels a headache coming on his thumb twitches and he begins to panic and then he pops some pills and maybe takes an intravenous injection of some medication; all of this is edited together in rapid succession, heightening the tension. The background score throbs, making the viewer edgier still. Then come the hallucinations (a brain in the sink with ants crawling on it--ewww) until we get a bright flash and then Max wakes up on the floor with a bloody nose. Add the physical troubles to his relentless drive to find a pattern in the stock market and it's no wonder he's starting to lose grip of his sanity.
This repetition is what led Eric to think of the Möbius strip as a metaphor for the structure, but it's not quite clear at first that it's the right one. In case you don't remember, the Möbius strip is the simplest example of a nonorientable surface--it has only one side. You can make one yourself by taking a strip of paper, giving one end a half-twist and then taping the ends together. Here's a picture:
from a cylinder to a Möbius strip to a twisted cylinder--two sides to one and back to two. image from https://plus.maths.org/issue26/features/mathart/Twist.gif.
If you look closely at the arrows (on the orange side), we see that in the beginning the cylinder has two sides. By cutting it apart, adding a half-twist, and taping it back together, we see that if we begin at a point on the cut line and move along a horizontal curve through the middle, then when we get back to the point (remember, this is a two-dimensional object; it has no thickness) where we started, the arrows point in the "wrong" direction. This is the essence of nonorientability: choose an outward pointing normal vector and follow it along a closed loop; if you always get back to arrows pointing in the same direction the surface is orientable, but if not the surface is nonorientable. If we go around again, then we are truly back where we started with everything pointing in the right direction. Note also that if we put another twist in the strip, we get something orientable--the arrows line up and it's two-sided again.
How is this idea manifested in \(\Pi\)? Well, one of our brilliant students had an idea: In the beginning of the film, Max knows nothing (well, that's not exactly true, but let's go with it). As we move along in time, he discovers a lot--a mystical \(216\)-digit number which the Hasidic Jews in the film believe is the true name of God; he can make predictions about stock prices (or can he?). This knowledge drives him mad, however. His headaches get worse until finally he decides not to take the medication and uses a drill to take out the portion of his brain that is torturing him (again, ewww). He then is back where he started--he knows nothing. See? Möbius strip!
Well, maybe it's a bit of a stretch. In any case, I asked the question: Is this movie even about mathematics? I'm not convinced. It's a device, certainly, but it's really about unknowability and the madness that can cause. More than anything, the film is about obsession and the idea that if you believe something is important you'll see it everywhere (Max's former Ph.D. advisor, Sol, tells him as much). Numerology plays a big role here, and in the end that's what Max's work devolves into.
Serious mathematicians have fallen into this trap. In the late 1990s we got The Bible Code, in which we are told that God encrypted lots of messages into the Torah via skip codes. The biggest, most prophetic example in it is that Yitzhak Rabin's name is crossed by the phrase "assassin that will assassinate;" this did come to pass, of course, so voila, God must be trying to tell us something. But you can play all kinds of games like this. Consider the following passage from the Declaration of Independence (H/T to Pat Ballew's blog for this):
“When in the course of human events,
it becomes necessary for one people to
dissolve the political bands which have
connected them with another, and to
assume among the powers of the Earth,
the separate and equal station to which
the Laws of Nature and
of Nature’s God entitle them
”
(Not sure why the quotation marks don't line up properly, but let's forge ahead.) Begin in the first row. Choose any word you like. Say you choose "course." That word has six letters, so count to the sixth word following it; you land on "necessary." This has nine letters, so count off nine words to get to "which" in the third line. Lather, rinse, repeat. Where do you land when you can't continue this process? In this case you land on "God" in the last line. Go ahead and try a few others. I bet you always land on "God." So, if I wanted to interpret this as proof that the Founders intended the United States to be a Christian nation, I could certainly do so. I mean, this can't just be a coincidence, right?
Well, yes it can. And the Bible Code is just a coincidence, too. In fact, many mathematicians wrote solid refutations of the Bible Code. For example, you can take Moby-Dick and do the same thing; you get lots of interesting "prophetic" sentences. It's all a consequence of something called the Kruskal count, discovered by the physicist Martin Kruskal in the mid-70s. The link takes you to a discussion of a really good card trick based on the idea. The point is that if you begin at some point and then have some algorithm for generating a sequence in your set, then no matter where you start, the sequences all coincide after a while (with high probability). So it shouldn't be at all surprising that we can find "hidden messages" in texts, just as Max should have known that the "patterns" he was seeing were likely coincidental. Just now, as I'm writing this in a coffeehouse, Teenage Lobotomy is playing over the speakers. Coincidence, or is God telling me something? I mean, I'm writing about a movie in which the main character lobotomizes himself and this song comes on. That can't be a coincidence.
But this is what we do as humans. We can't deal with randomness so we look for patterns or assign divine causes to random events. The truth of course is that the universe is a random place. God really does play dice.
One final note about the film. The title is \(\Pi\): Faith in Chaos. I asked the question: does Max have faith in chaos, or is he looking for faith in chaos? I don't know. Talk amongst yourselves.