First off, the penny has finally dropped over what Gareth means by an "orientation-preserving / reversing" isometry. This evidently refers to the special case where k = n, when any improper isometry must reverse what I called R-orientation (generalising A-orientation), because the perpendicular flat is empty. Enantiomorphs are always exchanged --- e.g. in 3-space a left-handed glove becomes right-handed --- and the ambiguity I've been exploring still miraculously fails to materialise. It only rears it head once flat dimension = k < n = space dimension. On 1/26/08, Dan Asimov <dasimov@earthlink.net> wrote:
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.
Indeed: the initial point of the original problem is quite simply that it's unanswerable; having grasped that, we might perhaps be stimulated to investigate why. The original problem allowed the observer to employ two distinct methods of defining the orientation of the sheet: the colour of the letters, and their legibility. As long as the sheet is waved around (properly) before being returned to its original locus, both methods give the same answer. Once it has been reflected in the mirror (improperly), they give different answers --- leaving us without any way to define its orientation, even in this restricted setting, without making some arbitrary choice between definitions.
* 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.
Agreed --- up to a point, anyway. It's to avoid this difficulty that for the present I only consider isometries fixing the locus of a given k-flat.
* An orientation on a Euclidean space R^n does not determine an orientation on a lower-dimensional subspace thereof.
Partly bearing on the "orientation-preserving / reversing" confusion above ...
* 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.
Presumably this is related to using the sign of the (homogeneous) equation of a prime to specify its orientation. This is a case of what might be called an "algebraic" definition, derived from the sign of the flat's representation in some chosen coordinate system. There are a few juicy worms tucked away under this stone as well, which I was saving for later: one being the gruesome Atiyah-Bott-Shapiro "twisted representation", which can be regarded as patching up the way geometric (Clifford) algebra versors and (indefinite) orthogonal matrices do inconsistent things with this sign (so complicating further its employment for orientation).
Since I don't really understand the question(s) under discussion, I don't know if these facts will be of any help.
Seems to me you understand it just about as well as anybody else does. Join the club! Fred Lunnon