On 1/25/08, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:

I don't understand what the problem is. It's negative; the sheet is a three-dimensional object, even though one dimension is rather small, and you've reflected it across that small dimension. What's conundral about this? I'm obviously missing a subtlety...

Idealised, the sheet (together with its inscription) is a plane --- and so is the mirror. In real life, of course, an experimenter would have to separate them slightly to observe what's happening, if his imagination is unequal to the task of mental recreation. By the way, that inscription got a bit too clever for its own good: to avoid further cnfusion, maybe it should just have read "Happy New Year", or some such ... WFL

On 1/25/08, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:

You stipulated that the letters are different colours on the two sides. Therefore the sheet-plus-inscription is *not* a two-dimensional object in any relevant sense. (You might get away with "2+epsilon- -dimensional", but that's not enough for your argument.)

Replace "colour green" by vector normal to plane. Replace "sticky plastic letters" by three points in general position on plane. Now wriggle out of that! WFL

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

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

On 1/26/08, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:

On Friday 25 January 2008, Fred Lunnon wrote:

...

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.

OK. (I remain skeptical of the existence of a problem. But you knew that.)

...

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.

OK. (And the two are equivalent, if we adopt some universal convention like "the preferred side is the left side when you look along the preferred direction".)

That's exactly the point --- they are _not_ equivalent --- and if improper isometries act on the flat, they give different answers! It is a theorem that they coincide under proper isometries (at any rate, those that leave the locus fixed --- see my last posting), and doubtless under larger classes of proper transformations which I haven't at this stage considered.

...

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.

OK. (You've reversed the orientation of the line, while keeping the orientation of the ambient space.)

...

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!

(So now you've reversed the orientation of the ambient space, and with it you've exactly reversed whatever conventional relationship we might have established between your methods A and S.)

How are you defining "orientaion of the ambient space" ? Why should reversing this (whatever it may be) affect the relation between the components?

Why the "huh" and "urgh"?

Because I get different values for the orientation from my two components. So I no longer know unambiguously what the orientation of the transformed flat is.

... Or, less obliquely: There are many things you can do to a configuration of points and lines and vectors and so on. They have different effects. If your configuration includes some sort of orientation on some of the lines, then some of the things you do will preserve the orientations, some will reverse them, and some will have less consistent effects.

Not isometries.

In particular, as you observe, "orientation-preserving" isometries leave the relationship between your "method A" and "method S" unaltered, whereas "orientation-preserving" isometriesones flip it. Why should I regard this as problematic?

"Proper" isometries are defined by continuous connection with the identity; or alternatively as factoring into the composition of an even number of prime (hyperplane) reflections. "Improper" are the remainder. No mention of "orientation". Did you mean to say "whereas orientation-reversing isometries flip it" ?? If so, what do you mean by "orientation-reversing" ? If they are "orientation-reversing", why don't they reverse both U/D and L/R components of my belt-and-braces?

I'm having (like Joshua) some difficulty working out exactly what you mean by an "orientation".

I gave two (limited) definitions of what I meant by orientation earlier (A and S) --- I'd like to generalise those shortly, if we can sort out the terminological confusions ... Assuming you've got a competing definition, why not let's hear it?

Consider Experiment 7: rotate M through one *quarter* turn about the origin. You no longer have a name for the state it's in (according to either criterion A or criterion S).

Whatever notion of orientation you're using only makes sense in the context of applying some kind of transformation that preserves the objects you're putting orientations on.

Correct --- see my last posting. Keep it simple ...

When your configuration consists of a single line in the plane, of course things like "rotation through a half turn" and "reflection in M" can do that. But as soon as you have (almost) any more complicated configuration, this stops being true. So, e.g., you have an equilateral triangle, and you stick some arrowheads or whatever on its sides; now you can rotate the whole thing through 1/3 of a turn, preserving the *figure*, but not the *lines*, so what does it mean to ask what's happened to the orientation of a particular line?

Dunno. You tell me.

...

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.

Whereas, I take it, a parallel sort of guy would say that method S is unethical and should instantly be discarded by any straight-thinking mathematician. Apparently I am neither normal nor parallel, and I think you're erecting a straw man here. There's nothing terribly wrong with either method.

In your first posting, you completely ignored the (A) data --- from the legibility of the letters --- and focussed entirely on the (S) data --- from their colour. I hereby graciously accept your renouncing of the illiterate normal lifestyle.

It's just that you appear to have (what seem to me to be) peculiar expectations about what happens to the relationship between those methods when you apply various geometrical transformations.

If it is "peculiar" unthinkingly to assume the compatibility of two such apparently mundane concepts as Euclidean isometry and orientation of a flat, then I must plead guilty.

...

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?

Again: orientation of what?

Specifically, k-flats in n-space.

You can give an orientation to the whole space, and then a wide variety of transformations on the space are either orientation-preserving or orientation-reversing, and there's nothing incoherent going on there.

You can give an orientation to some particular configuration in the space; what happens then depends on what automorphisms (in some sense of that very flexible word) the configuration has. I don't see any incoherence there.

That's plainly true: but without concrete definitions to hand, it doesn't really seem to advance the discussion. For all I know, there may be an abstract theory applicable to this situation; apart from the discussion standard in topology texts, I don't know of one. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This seems an appropriate point to float the most general form of the two alternative definitions discussed earlier. I've tried to keep it brief. Criticism is invited. Any $k$-flat $F$ is (non-uniquely) the isometric transform of the intersection of $n-k$ (orthogonal, projective, homogeneous) coordinate primes. The same isometry transforms the intersection of the remaining $k+1$ coordinate primes into a $(n-1-k)$-flat we shall call the "perpendicular" $\per(F)$. Perpendiculars very nearly qualify as geometric duals, though the relation is not bijective. In particular, for any "finite" $k$-flat $F$ (not at infinity), $\per(F)$ is at infinity, and specifies the set of directions perpendicular to $F$: it corresponds to a system of finite $(n-k)$-flats meeting infinity in $\per(F)$. Initially a given $k$-flat $F$ is arbitrarily endowed with positive orientation; we then consider just those isometries $X$ which fix the locus of $F$ (that is, transform the set of points lying in $F$ into itself). We wish to investigate the various ways in which the image $G = X(F)$ of $F$ under the action of $X$ might be equipped with an "orientation", together with the relationships between them. R-orientation: by restriction. The orientation of $X(F)$ is positive, negative as the restriction of $X$ to $F$ is proper, improper resp. P-orientation: by perpendicular. The orientation of $X(F)$ is positive, negative as the restriction of $X$ to $\per(F)$ is proper, improper resp. As a nod to tradition, we could define also V-orientation: by vector. The orientation of the prime $X(F)$ is determined by the action of $X$ on a vector normal (perpendicular) to $F$; of a line $X(L)$ by the action of $X$ on a parallel vector. It's not very hard to see that V-orientation is a special case of P-orientation for primes, and of R-orientation for lines: an ambivalence which might perhaps give cause for trepidation, particularly for $n = 2$ where both overlap. In higher dimensions $n > 3$, the approach is plainly insufficient for any flats except primes and lines, without considerable modification. No problems would arise provided R-orientation and P-orientation were consistent: and the good news is, that for proper isometries $X$, they are indeed _always_ consistent. The bad news is that the proof relies on showing that they differ by factor $(-1)^l$, where $l$ denotes the parity of $X$; for improper $X$, by the same reasoning, they can _never_ be consistent! Fred Lunnon