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  <gagster@flash.net> 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