Newsgroups: sci.math
From: mark@oracorp.com (Mark Bickford)
Subject: Re: Your favourite proofs
Date: Mon, 18 Jul 1994 15:52:13 GMT

In article <3062hm$bpb@math.mps.ohio-state.edu> edgar@math.ohio-state.edu (Geral
d Edgar) writes:
>As Stephen M-S notes, this proof uses apparenly extraneous
>metric properties.  Is the theorem true in other geometries?
>Say the two-dimensional rational plane?  [Yes, embed it in the
>real plane...another extraneous construction.]  How about
>the two-complex-dimensional "plane", where line means of course
>one-complex-dimensional line??  How about planes over finite fields?

In "Introduction to Geometry" by Coxeter, Coxeter gives a proof of the
theorem:
If n points are not all collinear, there is at least one line containing
exactly two of them.

The proof uses only the basic axioms of ordered geometry.
In ordered geometry the only primitive concepts are points A,B,...
and a ternary relation called "intermediacy" written [ABC] which
intuitively means "B is strictly between A and C".
In terms of this, the notions of segment, ray and line can be defined.

Other than some obvious axioms about intermediacy, the only non obvious
axiom used in the proof of the theorem is:

Axiom: If A,B,C are not collinear and points D and E satisfy
  [BCD] and [CEA], then there exists a point F on the line DE
for which [AFB].

In the introduction to chapter 12, Ordered Geometry, Coxeter
defines "absolute geometry" to be geometry without the parallel postulate,
so theorems of absolute geometry are theorem of both Euclidean and
of hyperbolic geometry.
He also defines "affine geometry" as the part of euclidean geometry that
is preserved by parallel projection from one plane to another.
Theorems of affine geometry are theorems of both Euclidean geometry
and of Minkowski's geometry of time and space (I'm quoting Coxeter).

Ordered geometry lies in the theory that is common to both
absolute geometry and affine geometry.
So, since the theorem in question (Coxeter calls it Sylvester's problem)
is a theorem of ordered geometry, it should hold in most of the
"planes" that you mentioned.

  ----Mark Bickford