Wednesday, October 30, 2013

On Reduction, 8 of 11

Projective Reduction

     Recall the “fish-tail” diagram:

           *     *
           |\   /|
           | \ / |
         x |  *  | y        z  =  x<+>y
           | /|\ |
           |/ |z\|
     If you flip this diagram left for right, then the lengths x, y and z remain the same; so x<+>y = y<+>x. Commutativity is illustrated by reflection.
     If you stretch it vertically then x, y and z become kx, ky and kz; so k(x<+>y)  =  kx <+> ky. So distribution is illustrated by stretching.
     So commutativity and distribution for reduction relate to the symmetries of the fish-tail diagram. The associative law is more complicated. It is illustrated by this diagram:

     AC, DE, FG, HI and JK are parallel;
     ABC, CIJ, BGJ, BEH, AEGIK and ADFHJ are collinear.

     If AC = x, JK = y and AB = z, then
           IH = x<+>y  ;  GF = y<+>z  ;  DE = (x<+>y) <+>z
           But (x<+>y)<+>z  = x<+>(y<+>z);
     So the AC and FG fishtail meets at E;
           Therefore C, E and F are collinear.
           If AC, DE, FG, IH and JK are parallel;
           and ABC, CIJ, BGJ, BEH, AEGIK and ADFHJ are collinear;
           then CEF are collinear.

     That theorem in affine geometry is equivalent to the associative law for reduction. I therefore call this the Affine Reductive Associative Theorem. Here is a projective version of the diagram, with D omitted, and including L as a point at infinity:

     Given these collinearities:
     Then CEF are collinear.

     This theorem is a consequence of Pappus’s Hexagon Theorem: for BCL are collinear points and FHJ are collinear points; therefore the opposite sides of the hexagon BHLFCJ meet in three collinear points: EGI. This proof from Pappus does not assume that the three lines BCL, FHJ and EGI meet at a point, as they do here.
     So the associativity of reduction is a consequence of a special case of Pappus’s Hexagon Theorem!

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