A mathematical bug shows us why the 3D universe carries the possibility of despair. Really.

Illustration for article titled A mathematical bug shows us why the 3D universe carries the possibility of despair. Really.

George Polya was a mathematician. Like most mathematicians, he was concerned with very strange concepts. One of them was the idea of "random walks," or the completely random path a strolling insect might take. He took this concept and expanded it until he could prove the chances of getting hopelessly, unendingly lost in the universe. Find out why.


Let's say that there is a universe that has nothing but space, time, and an immortal bug (hey, there are stranger ideas). This bug will always keep walking, no matter what. And since it has no higher purpose (and since any one place in universe it occupies is just as good as another), its walk is completely random. In 1921, in our universe — which has many more diversions than the bug universe — Hungarian math professor George Polya considered the odds that the beetle would ever make it back to the spot it originally inhabited.

In a one-dimensional universe (essentially a universe along a straight line) the bug would — with enough random walking and an infinite amount of time — make it back to its original starting point. It might make three steps forward and three steps back, and be done within a minute. Or it could wander for centuries. Eventually, though, it had to return home.

In a two-dimensional universe, a universe in which the beetle could take any path along a plane, the beetle would also always return home. This one might take a little longer, since the beetle had more freedom to wander, but the fact that is it would return back to the place it started.

A three-dimensional universe (like the one we live in) is the first universe in which the beetle might not make it home again. Polya crunched the numbers and came up with a disturbing conclusion. Assuming the bug could take any random walk through three-dimensional space, its chances of making it back home (after an infinite amount of time) are 0.34, or thirty-four percent. In our 3D universe, for the first time, there is such a thing as "never being able to go home again." No matter how long it wanders, the beetle has a two-thirds chance of being hopelessly lost.

It gets worse from there on up. For an n-dimensional universe, a random walk's chance of taking you back is (1/2n). The longer you walk, the less of a shot you have of going home, since you're likely to get more and more lost. So if you get dropped into a six-dimensional world, it's useful to pretend you just got lost in the woods. Don't try to find your way back. Just sit down, try to stay warm, and wait for the search-and-rescue to come find you. So yes, carry a whistle if you ever find yourself in six dimensions.

It's strange that the universe we occupy is the first one that carries the possibility of never getting the desired outcome. We occupy the world in which, even with an infinite amount of time, hopelessness is possible. On the other hand, it's all a matter of perspective. Plenty of people like the idea of permanently shaking a place that they don't like. Maybe we should look at George Polya's strange work as proof that we are in the first dimensional universe in which boredom is avoidable.


Image: Bohringer. Via AMT and The Math Book.




This seems kind of silly doesn't it? The bug has infinite energy and infinite time. There's no limiting action. The probability of it returning home is 1. If you agree it has to get home eventually in a 1D verse, then adding more dimensions merely means you have to get lucky and have all your 1Ds line up at the same time, but given infinite time and infinite tries, they will.

I could see how if you argue that he has to return on the same path the limit might approach a non-1 number (if you are 5 steps from home you have to hit the right vector 5 times in a row, and if you don't you're now 6 steps, and the odds decrease even more) but if we assume the bug can "recognize" home (and it must, otherwise how would the trial end) it's not necessary to walk back the way you came, any more than entering through the front door or the side door will still get me to my computer desktop.