From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Geometry - Line System Problem
Date: 26 Aug 1997 16:01:50 GMT
JamesBradleyHall wrote:
> This system of lines has the following properties. Draw the minimum
> number of lines in the system.
>
> 1. There exists a line and every line is point set.
> 2. If P and Q are two points, there is one and only one
> line containing them.
> 3. Every line contains exactly three points.
> 4. Every two lines has a point in common.
> 5. No line contains all points.
Gregory A Greenman wrote:
>I don't think this problem has a solution, transcendental or not.
>Two assumptions that most people will probably make that the rules don't
>actually impose are:
>
>1. All lines are straight.
>2. All points lie in a plane.
>
>Unfortunately, I don't see how removing either of these two assumptions
>changes the situation.
At the risk of completing the first poster's homework let me ask you to
consider this famous diagram:
o A
***
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* * *
* **** **** *
* **** * ****** *
* *** * ***
**** * ***
*** * **
*** * **
B o* * *o C
** * * * **
** * * * **
** * * * * *
* * * * * ** *
* * o D * *
* * * * * * *
* * * * * * *
* ** * * * * *
* ** * * * ** *
* ** * * * *
* * *** * ** * *
* * *** * *** * *
* * *** * *** * *
* * ***** * ****** * *
* * ******* * *
o*********************************o**********************************o
E F G
Here there are seven points A, ..., G and seven lines, where a "line" is the
set of three points in one of the seven "*" paths (the sides and bisectors of
the triangle as well as the circle).
Look up "incidence geometry", "projective plane", "finite geometry",
"Fano plane", etc.
dave