Winding Number and Topological Explanations22 Mar 1995 11:54
Take a rubber band. Wrap it around a cylinder. Run your finger around the band. Add up the number of times you circle the cylinder counter-clockwise, and subtract the number of times you went round clockwise. That's the winding number. You can twist and pluck and pull at the band all you wish without changing it; to do that, you have to take it off the cylinder (possibly by cutting and splicing). As they say in the trade, "winding number is invariant under continuous deformations." There are fancier definitions of winding number, with integrals and stuff, and related concepts for other sorts of shapes, which are also constant under continuous deformations. All this falls under topology.
Toss a stone in a still lake. It makes waves. Even without friction, the waves spread out and die away. In an infinite lake, they would eventually die away to zero; the wave disperses. Under the right conditions (like in canals), you can get other kinds of waves, called solitions, which don't disperse, but travel, pulse-like, indefinitely. Things like solitons show up in lots of physical theories, called various solitions, kinks, hedgehogs, monopoles and lumps. (I like "lumps.") For many lumps, you can prove they won't disperse by using things like the winding number. A flat lake, after a wave has dispersed, has a winding number of zero. (It's actually one of the winding number's cousins; let it go.) Lumps have non-zero winding number. Therefore you can only get from a lump to a flat lake by a discontinuous transformation.
Nature doesn't like discontinuities.
Winding number also shows up in pattern formation. If you put the right chemicals together --- a mix called the Belusov-Zhabotinskii reaction is particularly famous --- you get a cyclic reaction. If conditions are just right, different parts of the solution are at different phases in the cycle, and the mixture spontaneously forms spiral waves. (I may have to break down and get a picture for this. It's pretty.) Your initial even mix of chemicals has winding number zero; spiral waves don't. (And these are the real true original honest-to-Gauss winding numbers, too.) So how can spiral waves form, if Nature dislikes discontinuities so much? The trick is that spirals can point in two senses, clockwise and counter-clockwise, and that these two senses give winding numbers of opposite sign. Winding number will be conserved if spiral waves always emerge in pairs, rotating in opposite directions. (Since one goes round clockwise, and the other anti-clockwise, the total winding number is +1 -1 = 0.) And in fact this is what we always see; you even see it in computer simulations of the reaction.
Now at first glance this kind of topological argument may look like that much-fabled and long-sought beast, a non-reductionist explanation of an emergent property. I don't think it's anything of the kind. The conservation of winding number shows up, after all, in computer models where we know only local, micro-level rules are involved. What this looks like to me is that the topological reasoning is a hack: a clever shortcut which lets us see consquences of the micro-rules without elaborate calculations. If you want to explain the emergence of any particular pair of spiral waves microscopically, you can do so in detail by following the application of the rules over time: and does it get more reductionist than that?
- Sidney Coleman, "Some Classical Lumps and their Quantum Descendants" in his Aspects of Symmetry [Excellent on lumps and kinks and so forth in theoretical physics; unfortunately you have to know field theory to follow him. I'm looking for an explanation for the laity.]
- David Griffeath, Primordial Soup Kitchen [I wish he'd put his lecture notes on-line as well...]
- A. T. Winfree, Where Time Breaks Down [More on spiral waves, self-organization and topology. Math-free.]