On 1/25/08, Joshua Zucker <joshua.zucker@gmail.com> wrote:
On Jan 25, 2008 11:35 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
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.
But I would call this the same orientation. Maybe I'm thinking wrongly about what you mean by this word.
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!
I would say both of these states are signs that the orientation has been reversed.
Well, I suppose that you are at liberty to define "orientation" how you wish, given that my topic is whether such a notion actually exists at all [in the context of oriented subspaces of euclidean space]. But this seems a very strange way to try to do it. Going to 3-space for a minute [I hope that's not too confusing], I can imagine being very concerned about the S-orientation of my (quite flat) umbrella, given that it's otherwise likely to let the rain in. And (assuming it decorated with the red/green Happy New Year) I can imagine being equally concerned about the A-orientation, otherwise I may be unable to read the caption. Adopting your convention, I should apparently be unconcerned about both being reversed simultaneously. Do you know of an application where such a convention might be useful? If so, I'm quite prepared to consider add it to my list of alternatives (8 currently to hand). But I have to point out that, under most isometries it would be appear to be inconsistent with all the others --- including proper, where they all agree --- hardly a promising start!
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!
I think I'm confused about whether we're talking about isometries of the line or of the plane or of the way the line is embedded in the plane?
Apologies --- I should have made clear that I'm limiting consideration to isometries of n-space [n = 2 for the lines, n = 3 for the Happy New Year sheet/brolly] fixing the locus of a given k-flat [k = 1, 2 resp.] --- that is fixing the set of points lying in a k-dimensional subspace. The reason for doing this is to isolate the relationship between the various definitions, without complicating the issue further by attempting to quantify orientation at varying loci. Fred Lunnon