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