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. 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. 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. 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! Extending this argument, it becomes evident that under proper isometries the L/R and U/D components of the state always agree about the orientation; however, under improper isometries they always disagree! 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. Unfortunately, adopting this attitude then leaves him unable to assign an orientation to subspaces in 3-space, where (apparently) only method A is available for lines, and method B for planes. In the Happy New Year problem, orientation via lettering is (very indirectly) equivalent to method A, orientation via colour to method B. Again under reflection in the mirror, they both fail to agree [well now, one of them could hardly fail by itself, could it?]. I ought to add that these considerations do no more than lift the lid from an extremely vigorous can of worms, which I have at this stage only partially excavated. 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? Fred Lunnon On 1/25/08, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
So now your configuration consists of a plane-with-normal-vector, and again it's obviously changed by reflection in that plane, and again I simply don't understand what's supposed to be difficult here. I think maybe we just have different intuitions about ... well, I don't know, but something.
Maybe it would help if you'd say more explicitly what your argument is for why the answer to your original question might be, or at least might feel like it ought to be, "positive" (for either the sheet with letters or the plane with points and normal vector), because I'm just not seeing it.
-- g