Newsgroups: sci.math
From: boshuck@triples.math.mcgill.ca (William Boshuck)
Subject: Re: Your favourite proofs
Date: Thu, 14 Jul 1994 20:10:28 GMT
Hi,
I just remembered a really cute proof of the following fact: Any
finite collection L of points in the plane satisfying
(*) a line passing through any two points in L passes
through a third point in L
is collinear.
Proof: Suppose not. Among all triples (p,q,r) of non-collinear points
in L, pick one which minimizes the distance between r and the line pq,
and let p' be a point witnessing (*) for the line pq. Without loss, p'
lies between p and q on pq. Then the distance between p' and (at
least) one of the lines rp, rq is smaller than the distance between r
and pq. Contradiction.
cheers,
==============================================================================
Newsgroups: sci.math
From: stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith)
Subject: Re: Your favourite proofs
Date: 15 Jul 94 05:16:02 GMT
In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (Wi
lliam Boshuck) writes:
>Hi,
>I just remembered a really cute proof of the following fact: Any
>finite collection L of points in the plane satisfying
> (*) a line passing through any two points in L passes
> through a third point in L
>is collinear.
>Proof: [ the short proof is ommitted]
Apparently this was an unsolved problem for quite a while.
Stephen
==============================================================================
Newsgroups: sci.math
Subject: Re: Your favourite proofs
From: kubo@zariski.harvard.edu (Tal Kubo)
Date: 15 Jul 94 03:32:25 EDT
In article
stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes:
>boshuck@triples.math.mcgill.ca (William Boshuck) writes:
>
>>[...] Any finite collection L of points in the plane satisfying
>
>> (*) a line passing through any two points in L passes
>> through a third point in L
>>
>>is collinear.
>
>>Proof: [ the short proof is ommitted]
>
>Apparently this was an unsolved problem for quite a while.
Both this problem and the previous one (N red, N blue points) are
projectively invariant assertions but the short proofs use
metric concepts. Maybe "purity of method" was the stumbling block?
(Speaking of which, are there projective geometry solutions to either
problem? The one above even has a *combinatorial* solution but it's
not simple.)
=============================================================================
Newsgroups: sci.math
From: edgar@math.ohio-state.edu (Gerald Edgar)
Subject: Re: Your favourite proofs
Date: 15 Jul 1994 09:21:58 -0400
In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (Wi
lliam Boshuck) wrote:
>Hi,
>
>I just remembered a really cute proof of the following fact: Any
>finite collection L of points in the plane satisfying
>
> (*) a line passing through any two points in L passes
> through a third point in L
>
>is collinear.
>
>Proof: Suppose not. Among all triples (p,q,r) of non-collinear points
>in L, pick one which minimizes the distance between r and the line pq,
>and let p' be a point witnessing (*) for the line pq. Without loss, p'
>lies between p and q on pq. Then the distance between p' and (at
>least) one of the lines rp, rq is smaller than the distance between r
>and pq. Contradiction.
>
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?
--
Gerald A. Edgar Internet: edgar@math.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
==============================================================================
Newsgroups: sci.math
Subject: Re: Your favourite proofs
From: kubo@germain.harvard.edu (Tal Kubo)
Date: 15 Jul 94 17:48:30 EDT
In article <3062hm$bpb@math.mps.ohio-state.edu>
edgar@math.ohio-state.edu (Gerald 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?
It's true in all of the above. The result follows from a purely
combinatorial theorem of Haim Hanani stating that if the N points
are not collinear then they determine at least N distinct "lines".
This doesn't rely on any geometry whatsoever, just pure incidence
relations.
==============================================================================
Newsgroups: sci.math
From: geoff@math.ucla.edu (Geoffrey Mess)
Subject: Re: Your favourite proofs
Date: Sat, 16 Jul 94 02:01:04 GMT
In article <1994Jul15.033225.33540@hulaw1.harvard.edu> writes:
> In article
> stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes:
> >boshuck@triples.math.mcgill.ca (William Boshuck) writes:
> >
> >>[...] Any finite collection L of points in the plane satisfying
> >
> >> (*) a line passing through any two points in L passes
> >> through a third point in L
> >>is collinear.
> >
> >>Proof: [ the short proof is ommitted]
Tal Kubo:
> (Speaking of which, are there projective geometry solutions to either
> problem? The one above even has a *combinatorial* solution but it's
> not simple.)
???
Gerald Edgar:
>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?
It's false in CP^2: consider the 9 inflection points of a nonsingular
cubic. They lie on 12 lines each containing 3 points. It's also false
in many planes over finite fields: if the characteristic is not 3
and the field F_q contains enough roots of unity then the Hessian configuration
of 9 points and 12 lines reduces mod p to embed in the projective plane
over F_q. Here q is a power of p. I think its enough that the field
contains a cube root of unity regardless of the characteristic, but
I'd have to think some more in characteristic 2. The Hessian example is
valuable as motivation for the "extraneous" approach to the problem
in the real plane.
--
Geoffrey Mess
Department of Mathematics, UCLA. geoff@math.ucla.edu
==============================================================================
Newsgroups: sci.math
From: hrubin@snap.stat.purdue.edu (Herman Rubin)
Subject: Re: Your favourite proofs
Date: Sat, 16 Jul 1994 20:23:37 GMT
In article <3062hm$bpb@math.mps.ohio-state.edu> edgar@math.ohio-state.edu (Geral
d Edgar) writes:
>In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (W
illiam Boshuck) wrote:
>>Hi,
>>
>>I just remembered a really cute proof of the following fact: Any
>>finite collection L of points in the plane satisfying
>>
>> (*) a line passing through any two points in L passes
>> through a third point in L
>>
>>is collinear.
How about planes over finite fields?
Since a plane over a finite field (affine or projective) is finite,
the theorem is obviously false for those planes.
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
==============================================================================
Newsgroups: sci.math
From: edgar@math.ohio-state.edu (Gerald Edgar)
Subject: Re: Your favourite proofs
Date: 16 Jul 1994 17:14:32 -0400
>>>I just remembered a really cute proof of the following fact: Any
>>>finite collection L of points in the plane satisfying
>>>
>>> (*) a line passing through any two points in L passes
>>> through a third point in L
>>>
>>>is collinear.
>
> How about planes over finite fields?
>
>Since a plane over a finite field (affine or projective) is finite,
>the theorem is obviously false for those planes.
Well...
Oh, yeah.. I should have said fields of finite characteristic. That't it.
--
Gerald A. Edgar Internet: edgar@math.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
==============================================================================