Re: [math-fun] Reflections on Orientation
<< 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
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
On Saturday 26 January 2008, Dan Asimov 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.
Definitely a handle rather than a cross-cap :-). -- g
Dan gave several definitions of "orientation", all confined to the case k = n in the way that everybody else seems to have interpreted the word, and all mentioning 2 connected components. It's now evident that a major source of confusion was that [coming from a computational geometry direction --- orientation?] I'm concerned rather with orientation of a subspace within the whole space. If we consider more general geometries --- Moebius / conformal / Poincar\'e, Lie sphere / physicist's conformal, and so on --- the Lie group corresponding to Euclidean isometries has 4 connected components rather than 2. On the other hand, orientation of subspaces does not appear to be any more difficult than before. So the idea occurs to me that perhaps the solution to my problem is to consider instead a 4-valued "sub-orientation"? Why should this be justified? WFL
participants (3)
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Dan Asimov -
Fred lunnon -
Gareth McCaughan