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