SMARANDACHE COUNTER-PROJECTIVE GEOMETRY
                                  
                                by         
                            
        Sandy P. Chimienti                       Mihaly Bencze     
        Mathematics and Science Department       6, Hatmanului Street
        University of New Mexico                 2212 Sacele 3
        Gallup, NM  87301, USA                   Jud. Brasov, Romania


Abstract:
  All three axioms of the projective geometry are denied in this 
new geometry. 

Key Words: Projective Geometry, Smarandache Geometries, Geometrical Model

Introduction:
  This type of geometry has been constructed by F. Smarandache[4] in 1969.

  Let P, L be two sets, and r a relation included in PxL.  The elements of   
P are called points, and those of L lines.  When (p, l) belongs to r, we
say that the line l contains the point p.
For these, one imposes the following COUNTER-AXIOMS:

   (I) Not always two distinct points determine a line.

  (II) Let p1, p2, p3 be three non-collinear points, and q1, q2 two 
       distinct points.  Suppose that {p1, q1, p3} and {p2, q2, p3} are
       collinear triples.  Then the line containing p1, p2, and the line
       containing q1, q2 do not always intersect.

 (III) Not every line contains at least three distinct points.

We consider that in a discontinuous space one can construct a model  
to this geometry.


References:
  [1] Ashbacher, Charles, "Smarandache Geometries", <Smarandache Notions
      Journal>, Vol. 8, No. 1-2-3, 212-215, 1997.
  [2] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Counter-Projective
      Geometry", <Bulletin of Pure and Applied Sciences>, Delhi, India, 
      Vol. 17E, No. 1, pp. 117-118, 1998.
  [3] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Counter-Projective
      Geometry", <Smarandache Notions Journal>, Vol. 9, No. 1-2, 47-48, 
      1998.
  [4] Brown, Jerry L., "The Smarandache Counter-Projective Geometry",
      <Abstracts of Papers Presented to the American Mathematical Society
      Meetings>, Vol. 17, No. 3, Issue 105, 595, 1996.
  [5] Smarandache, Florentin, "Collected Papers" (Vol. II), University of
      Kishinev Press, Kishinev, 5-28, 1997.
  [6] Smarandache, Florentin, "Paradoxist Mathematics" (lecture), Bloomsburg
      University, Mathematics Department, PA, USA, November 1985.