[math-fun] more Reflections on Orientation
[With some trepidation, I'm risking re-opening the can of worms which got me (and the annelids) into hot water here a year ago ...] Summarising the situation as I currently view it, there are actually _two_ distinct kinds of orientation traditionally employed when discussing objects such as flats, spheres and more general manifolds under the action of spatial transformations such as isometries. Space for these purposes is also traditional: Euclidean, projective, inversive, or maybe elliptic, spherical, hyperbolic etc. The essential distinction is between orientation determined by the action of an isometry on the object itself, contrasted with determined by action on the complement. Into the first class fall cues such as the arrowhead on a line or "spear" in the plane, or the lettering on a banner in space; into the second, the left or right side of a line in the plane, or the colour scheme on one side of the banner. Why should the announcement of this notion cause such uncomprehending consternation among substantial numbers of my colleagues? The major reason is probably that for most purposes, it doesn't actually matter: if an object is subjected only to proper isometries [often referred to --- in a shorthand which might have been designed to perpetuate confusion over this matter --- "orientation preserving"] then cues of either type give the same result. It is only when one comes to consider the mysteries of improper ["orientation reversing"] isometries that an appalling anomaly becomes apparent: cues of different classes now (always) give inconsistent orientations! In attempting to put all this down coherently, I've run into another familiar elephant trap: nomenclature, over which I agonise endlessly in circles, and rarely with much success. I offer up the following alternatives for your consideration, and perhaps improvement: For the first class, "endogenous orientation" or "endo-sense" or "chirality"; for the second class, "exogenous orientation" or "exo-sense" or "chroma"; suggestions, anybody? Incidentally currently available coordinate systems might have been deliberately designed to exacerbate the situation (regarding user confusion, rather than etymological indecision). If the sign of the scale factor is interpreted as encoding orientation, rather than discarded as is traditional, then projective (homogeneous) matrices, as well as geometric (Clifford) algebras such as J.M.Selig's grade-4 algebra and my own DCQ algebra, all cope well with proper isometries, then lose it irreparably (in umpteen different ways) with improper. But I realised only recently that for Lie-sphere geometry there is light at the end of the Dupin cyclide. This is an inversive space, in which spheres (of codimension 1) are explicitly oriented via the sign of a radius component. In 3-space for instance, an appropriate grade-6 algebra has generaters x,y,z (axes), u,v (inversions) and r (radius), where x^2 = y^2 = z^2 = u^2 = +1, v^2 = r^2 = -1. And --- hallelujah --- under _all_ Lie transformations (including Euclidean, Moebius and Laguerre), exo-sense/chroma is transformed correctly! At last, a sign of divine favour? But soft ... why should either class be preferred to the the other? After all, they could be viewed (in a slightly nightmare scenario, it is true) as duals of one another. Well now --- there is in practice very little to choose between the algebra above and the alternative x^2 = y^2 = z^2 = u^2 = -1, v^2 = r^2 = +1 [it's the difference between Selig's algebra and DCQ's]. And lo' and behold: when the squared signs are reversed, the conserved orientation changes from exo-sense/chroma to endo-sense/chirality. Fred Lunnon
On 2/2/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... For the first class, "endogenous orientation" or "endo-sense" or "chirality"; for the second class, "exogenous orientation" or "exo-sense" or "chroma"; suggestions, anybody?
Well, no budding lexicographer out there has offered any opinions / suggestions. After much vacillation, I seem to be coming down on the side of "chirality" for orientation determined by left vs. right-handedness, arrowheads on lines, legibility of text etc; and "chroma" based on left vs right half-spaces, colour of painted surface, rotation sense etc. [But I did toy briefly with "incense" and "essence" ...] And now for the visit of our typo fairy, without which no peroration (perorientation) would be complete ... [stretching the terminology somewhat --- in this instance, thumping-great-goof-giant might be more accurate.]
... Lie-sphere geometry ... is an inversive space, in which spheres (of codimension 1) are explicitly oriented via the sign of a radius component. In 3-space for instance, an appropriate grade-6 algebra has generaters x,y,z (axes), u,v (inversions) and r (radius), where x^2 = y^2 = z^2 = u^2 = +1, v^2 = r^2 = -1. And ... chroma is transformed correctly!
... there is in practice very little to choose between the algebra above and the alternative x^2 = y^2 = z^2 = u^2 = -1, v^2 = r^2 = +1 [it's the difference between Selig's algebra and DCQ's]. And lo' and behold: when the squared signs are reversed, the conserved orientation changes from chroma to chirality.
Utter twaddle. Both algebras behave exactly the same in respect of isometries etc of subspaces (oriented spheres, circles, etc). What makes the difference is whether the mapping from algebra to geometry represents intersection resp. union by the wedge product. [This is not an option for Euclidean algebra. If you try to choose union --- sadly, everybody does --- the metricals go pear-shaped.] Fred Lunnon
On 2/6/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Utter twaddle. Both algebras behave exactly the same in respect of isometries etc of subspaces (oriented spheres, circles, etc). What makes the difference is whether the mapping from algebra to geometry represents intersection resp. union by the wedge product. [This is not an option for Euclidean algebra. If you try to choose union --- sadly, everybody does --- the metricals go pear-shaped.]
And now for the correction to the correction, a few minutes of simple algebra having demonstrated that my intuition was well down to its customary misleading standard. There are three different ways the Clifford algebra representation for Lie-sphere geometry might be tweaked: (1) use Cl(n+1,2) instead of Cl(2,n+1); (2) represent linear complexes by grade 1 (vectors), rather than cycles (spheres); (3) use the dual product for wedging (meet, join etc), rather than straight Clifford product. It doesn't apparently matter what options are taken --- the orientation is always exogenous, a chroma: or to put it another way, the orientated sphere behaves like a half-space, rather than a billboard, under improper transformations. I do not know whether the same is true of every order-(n+3) matrix representation of the symmetry group: notice there is no room for manoeuvre along the lines of the notorious Atiyah-Bott-Shapiro "twist", since these coordinate systems are purely homogeneous: scalar factors are irrelevant, including their sign. The upshot seems to be the algebra suggests there is something inherently more natural about chromatic / exogenous orientation, as opposed to chiral / endogenous. It's all rather strange, seeing that the two are duals of one another. I sincerely hope I can shut up about this for a while now! Fred Lunnon
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Fred lunnon