This really is the week for mathematical bugs. First, one beetle showed us why we live in a universe of despair. Now, another shows us that whatever can go wrong, will go wrong...and the more that things can go wrong, the more things will go wrong. See how a bug proves a knotty conjecture.
Welcome to December, the season of Christmas lights. If you annually deck your halls, chances are your lights are tangled in a hideous knobby green wad at the bottom of a box somewhere. They were probably in an organized loop when you put them there. But now, they look like the Christmas Cat coughed up a hairball. Why do they do that?
Because they're subject to Murphy's Law, an informal law that says that anything that can go wrong will go wrong. When it comes to long strands of string, from proteins in a person's cells to the rigging in a ship, this means spontaneous knotting. People have written papers about how string knots up the minute it's given a chance to jiggle around. In 1988, two mathematicians — De Witt Sumners and Stuart Whittington — informally proved that knotting had to occur under these circumstances, and they did it using the mathematician's favorite creature, the random-walk-taking bug.
A random walk is a walk in which each step can be in any direction, regardless of where the previous step was. It can occur in a one-dimensional, two-dimensional, or multi-dimensional setting. Sumners and Whittington's bug goes for a three-dimensional walk but with a twist. The bug cannot re-cross over a spot it has previously occupied. This aspect of the "walk" makes it analogous to a piece of string. A string can wind around in any direction, but two parts of it can't occupy the same place. The bug is basically tracing out the "string" with its walk.
The two mathematicians found that any of the bug's sufficiently long walks had to contain one knot. The longer the walk was, the more knots there were. This showed that if you put a string in an area and let it move randomly, it will almost always work itself into a knot. The longer it is, or the more strings are there, the more knots you can expect to untangle. Anything that can leave you on your knees in a cold garage, cursing and trying to untangle blinking strings of lights will leave you on your knees in a cold garage, cursing and trying to untangle blinking strings of lights.
But this does lead to some nice things as well, such as knots in human proteins that stabilize their structure. You're stuck with the lights, though.
Image: Tony Wills. Via Maa and UCSD.