From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: Desargues configuration
Date: Wed, 28 Jul 1999 08:34:49 GMT
Newsgroups: sci.math.research
Keywords: Desargues' theorem, Pappus' theorem, coordinatization

In article <7niv4s$lkj$1@news1.tc.umn.edu>,
  hardy@calhoun.stat.umn.edu (Michael Hardy) wrote:
>      On page 145 of _Indiscrete_Thoughts_ Gian-Carlo Rota writes
> about the "Desargues configuration."  He refers to "a figure
> displaying incident straight lines, more than 50 straight lines
> if I remember correctly."  This left me with the impression that
> that is what the Desargues configuration is.
>
>      But running "Desargues configuration" on a search engine and
> glancing at the results leaves me with the impression that the term
> refers to a set of ten points and ten lines, so arranged that each
> of the ten points is on exactly three of the ten lines, and each
> of the ten lines passes through exactly three of the ten points.
> This configuration always spans a projective subspace of dimension
> 2 or 3.  This is the picture that would normally accompany an account
> of "Desargues' theorem", which says that in a projective space over
> a field (or any associative division ring, I think, but not over
> non-associative creatures like octonions) two triangles are
> "co-axial" (definitions below) if and only if they are "co-polar."

That is what I understand by the Desargues configuration

>      Can anyone explain what figure with more than 50 lines Rota
> is talking about?

I can't.

>      Then he goes on to refer to "the fundamental role of Desargues'
> theorem in the coordinatization of synthetic projective geometry."
> What is that about?  "Coordinatization of projective geometry" makes
> me think of "homogeneous coordinates", whereby two (n+1)-tuples of
> scalars represent the same point in n-dimensional projective space
> iff one is a scalar multiple of the other, and "synthetic" often
> means "without coordinates."

It's the converse to the result you quoted above: that Desargues'
theorem is true in a projective plane coordinatized using a division
ring.

If we try to axiomatize projective planes we start with the obvious
incidence axioms (two points determine a line, two lines meet in
a point) then throw in some non-triviality conditions (each line has
at least three points, there are three non-collinear points etc.)
we find that we get some wacky examples of projective planes which
bear scant resemblance to those constructed by homogeneous coordinates
over division rings. However in these planes. the Desargues theorem
is false. If we impose the Desargues theorem as an extra axiom then
one can prove that the plane can be coordinatized by a division ring.
Thus the Desargues' theorem is the missing link from a "synthetic"
(axiomatic, coordinate-free) approach to a coordinate approach.
[However we don't require Desargues to axiomatize 3 or higher
dimensional prjective space.]

The other theorem similar to Desargues' theorem is the theorem
of Pappus. This states: let A_1, A_2 and A_3 be points on a line l_1
and B_1, B_2 and B_3 be points on a line l_2. Let C_1 be the
intersection of lines A_2B_3 and A_3B_2, C_2 be the intersection
of lines A_3B_1 and A_1B_3 and C_3 be the intersection of lines
A_1B_2 and A_2B_1. Then C_1, C_2 and C_3 are collinear.
The theorem is that Pappus is true in a projective plane iff
it can be coordinatized by a field (commutative division ring).
Thus Pappus implies Desargues. For a finite projective plane
Desargues implies Pappus since finite division rings are commutative
by Wedderburn's theorem. AFAIK no-one has yet found a synthetic
(coordinate-free) proof of this.


Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "They did not have proper palms at home in Exeter."
    Peter Carey, _Oscar and Lucinda_


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