**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:

ABCL, FGL,
HIL, JKL, CIJ, BGJ, BEH, AEGIK, AFHJ

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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