All that talk about numbering spots on the New York Thruway had a goal, that of establishing how we could set up a coordinate system for the points on a line. It turns out just as easy to do this for a curve, even one a little bit complicated like a branch of the Thruway. About the only constraints we said anything about were that we shouldn’t have branches. Lurking unstated was the idea that we didn’t have loops. For the Thruway that’s nothing exceptional; if we had a traffic circle in the middle of a high-speed limited-access highway we wouldn’t very long have a high-speed highway. Worse, we’d have some point — where the loop crosses itself — that would have two numbers describing its position. We don’t want to face that. But we’ve got this satisfying little system where we can assign unique numbers to all the points on a single line, or even a curve.
The natural follow-up idea is whether we can set up a system where we can describe a point on a surface or even in all of space using the same sort of coordinates scheme. And there’s the obvious answer of how to do it, using two numbers to describe where something is on a surface, since that’s a two-dimensional thing; or three numbers to describe where it is in space, since that’s a three-dimensional thing. So I’m not going to talk about that just now. I want to do something more fun, the kind of thing that could do nicely in late-night conversations in the dorm lounge if undergraduates still have late-night conversations in the dorm lounge.
If we have a long enough thread, or a strand of yarn, or whatever the quite correct term is, we know this can be set up with a coordinate system by marking off distance along that thread. We imagined doing that, more or less, with the numbering system on the Thruway and imagining the straightening out and curving and other moving around of the highway’s center line. As long as we didn’t stretch or compress the strand any, we could spread it out in any shape we liked, and have coordinates for whatever path the strand traces out.