On Friday 25 January 2008, Fred Lunnon wrote:
Let's examine a simpler example, where the problem is even more starkly illustrated, but at the same time so familiar that it seems somehow easier to sweep under the carpet.
OK. (I remain skeptical of the existence of a problem. But you knew that.)
Staying in 2-space now, we have only points and lines to play with. There are two intuitively obvious ways in which a line might oriented:
Method A --- slap an arrowhead on it; or if you prefer, associate with it a vector parallel to the line.
Method S --- choose one side of it; or if you prefer, associate with it a vector normal to the line.
OK. (And the two are equivalent, if we adopt some universal convention like "the preferred side is the left side when you look along the preferred direction".)
Now let M denote the line x = 0, N the line y = 0. Just to be on the safe side, I shall orient M using both methods. Initially it will point upwards, with its right side distinguished: denote this state by the abbreviation UR.
Experiment 1: rotate M through one half turn about the origin. It is now in state DL. Both components indicate that the orientation has been reversed --- belt and braces cooperating magnificently.
OK. (You've reversed the orientation of the line, while keeping the orientation of the ambient space.)
Experiment 2: reflect M in N. It is now in state DR. Huh?
Experiment 3: reflect M in M. It is now in state UL. Urgh!
(So now you've reversed the orientation of the ambient space, and with it you've exactly reversed whatever conventional relationship we might have established between your methods A and S.) Why the "huh" and "urgh"? Experiment 4: rub out the arrowhead and redraw it so it points the other way. Hnhh? Experiment 5: subject the plane to a nonlinear transformation that acts like a shear in each of the two halves into which M divides it, in opposite directions, so that it distorts the arrowhead and makes it point the other way. Eh? Experiment 6: replace the line with a little smiley face. WTF? Or, less obliquely: There are many things you can do to a configuration of points and lines and vectors and so on. They have different effects. If your configuration includes some sort of orientation on some of the lines, then some of the things you do will preserve the orientations, some will reverse them, and some will have less consistent effects. In particular, as you observe, "orientation-preserving" isometries leave the relationship between your "method A" and "method S" unaltered, whereas "orientation-reversing" ones flip it. Why should I regard this as problematic? I'm having (like Joshua) some difficulty working out exactly what you mean by an "orientation". Consider Experiment 7: rotate M through one *quarter* turn about the origin. You no longer have a name for the state it's in (according to either criterion A or criterion S). Whatever notion of orientation you're using only makes sense in the context of applying some kind of transformation that preserves the objects you're putting orientations on. When your configuration consists of a single line in the plane, of course things like "rotation through a half turn" and "reflection in M" can do that. But as soon as you have (almost) any more complicated configuration, this stops being true. So, e.g., you have an equilateral triangle, and you stick some arrowheads or whatever on its sides; now you can rotate the whole thing through 1/3 of a turn, preserving the *figure*, but not the *lines*, so what does it mean to ask what's happened to the orientation of a particular line?
At this point, a normal sort of a guy might object that method A is patently unethical and should instantly be discarded by any right-thinking mathematician.
Whereas, I take it, a parallel sort of guy would say that method S is unethical and should instantly be discarded by any straight-thinking mathematician. Apparently I am neither normal nor parallel, and I think you're erecting a straw man here. There's nothing terribly wrong with either method. It's just that you appear to have (what seem to me to be) peculiar expectations about what happens to the relationship between those methods when you apply various geometrical transformations.
The simple fact that orientation as a coherent concept is viable only under proper transformations has come as a considerable (and most unwelcome) surprise to me; and perhaps also to a few other people?
Again: orientation of what? You can give an orientation to the whole space, and then a wide variety of transformations on the space are either orientation-preserving or orientation-reversing, and there's nothing incoherent going on there. You can give an orientation to some particular configuration in the space; what happens then depends on what automorphisms (in some sense of that very flexible word) the configuration has. I don't see any incoherence there. -- g