<< You might have hoped you'd heard the last of this topic. However, a discussion with David Gale has inspired the following vicious conundrum. Consider a transparent sheet of overhead projector film, onto which has been glued the legend "P A B L / N H Ø O" in plastic symbols, coloured green on the front and red on the back [from a numerate classicist toddler's alphabet: there would have been 10 symbols, but the swastika's been eaten]. Flatten it out on the table [the sheet, not the toddler], with the lettering legible, and coloured green. Let's call this positively oriented. Now turn it over, as a page from a book, and slide it back again. It occupies the same space as before; and the legend is illegible and coloured red. The orientation of the sheet is now negative. Then begin again, but instead lay it against a mirror. The image again occupies the same space (more or less) as the original. The legend in the reflected image is legible and coloured red [or if you foozled it, illegible and coloured green --- whichever]. Question: is the reflected orientation positive, or is it negative? If anyone comes across a previous reference to something related, I should like to hear about it. [Martin Gardner's "The Ambidextrous Universe" might perhaps be a good place to start looking.]
I've been rading this thread with the hopes of jumping at some point, since I feel orientation is a concept I have a good handle on. But I can't, since at no point do I feel that any clearly defined mathematics question has been asked. I will say this: * There is no such thing as a "positive" or "negative" orientation. Orientations of the same manifold can be compared with each other and declared "same" or different". But they cannot be labeled "positive" or "negative" in any consistent way. * An orientation on a Euclidean space R^n does not determine an orientation on a lower-dimensional subspace thereof. * There is, however, a way for an orientation on R^n to determine an orientation on an affine R^(n-1) that does not pass through the origin of R^n, and vice versa: such a subspace can determine an orientation on the R^n in which it lies. Since I don't really understand the question(s) under discussion, I don't know if these facts will be of any help. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele