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