Friday, February 26, 2010

Q: Why does string get tangled up in knots?

A: Because it can.

This is not a joke answer. The question is very nearly tautological. A knot is, by definition, a conformation of string which resists untangling. If you perturb a length of string randomly, then at any given time it may either become tangled in a knot or remain free. If it is in a knot, it will resist becoming a non-knot; if it is not a knot, it is free to change. If a knot is possible, one will eventually emerge. The only way that a knot could fail to emerge is if it were completely impossible for an unknotted string to tangle into a knot.

A very similar question would be why a shaken ratchet eventually turns forward. Of course, nobody would ask that question. The lower dimensionality means that the answer is transparent.

There is a similar result in evolutionary population dynamics which states that as time goes to infinity, given a fixed population cap and a randomized chance to reproduce, every species goes extinct. Intuitively, if you roll k dice infinity times, then eventually they all come up sixes.

The many, many applications of this principle are left as an exercise for the reader.

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