SMARANDACHE PARADOXIST 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:
  This new geometry is important because it generalizes and
unites in the same time all together: Euclid, Lobachevsky/Bolyai/Gauss, 
and Riemann geometries.  And separates them as well!
It is based on the first four Euclid's postulates, but the fifth one is 
replaced so that there exist various straight lines and points exterior to 
them in such a way that none, one, more, and infinitely many parallels
can be drawn through the points in this mixted smarandacheian space.       

Key Words: Non-Euclidean Geometry, Euclidean Geometry, Lobacevskyian
            Geometry, Riemannian Geometry, Smarandache Geometries, 
            Geometrical Model

Introduction:
  A new type of geometry has been constructed by F.Smarandache[5] in 1969
simultaneously in a partial euclidean and partial non-euclidean space by a
replacement of the Euclid's fifth postulate (axiom of parallels)
with the following five-statement proposition:

     a) there are at least a straight line and a point exterior
        to it in this space for which only one line passes through 
        the point and does not intersect the initial line;
          [1 parallel]
     b) there are at least a straight line and a point exterior
        to it in this space for which only a finite number of 
        lines l , ..., l  (k >= 2) pass through the point and do not 
               1        k
        intersect the initial line;
          [2 or more (in a finite number) parallels]
     c) there are at least a straight line and a point exterior
        to it in this space for which any line that passes through 
        the point intersects the initial line;
          [0  parallels]
     d) there are at least a straight line and a point exterior
        to it in this space for which an infinite number of lines 
        that pass through the point (but not all of them) do not 
        intersect the initial line;
          [an infinite number of parallels, but not all lines passing
             throught]
     e) there are at least a straight line and a point exterior
        to it in this space for which any line that passes through 
        the point does not intersect the initial line;
          [an infinite number of parallels, all lines passing throught
             the point]

  I have found a partial geometrical model, different from Popescu's [1], by 
putting together the Riemann sphere (Ellyptic geometry), tangent to the 
Beltrami disk (Hyperbolic geometry), which is tangent to a plane (Euclidean 
geometry).  But is it any better one?  
(because this doesn't satisfy all the above required axioms).


References:
  [1] Charles Ashbacher, "Smarandache Geometries", <Smarandache Notions
      Journal>, Vol. 8, No. 1-2-3, Fall 1997, pp. 212-215.
  [2] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Paradoxist 
      Geometry", <Bulletin of Pure and Applied Sciences>, Delhi, India,
      Vol. 17E, No. 1, 123-1124, 1998.
  [3] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Paradoxist 
      Geometry", <Smarandache Notions Journal>, AZ, USA, Vol. 9, No. 
      1-2, 43-44, 1998.
  [4] Mike Mudge, "A Paradoxist Mathematician, His Function, Paradoxist
      Geometry, and Class of Paradoxes", <Smarandache Notions Journal>,
      Vol. 7, No. 1-2-3, August 1996, pp. 127-129.
      reviewed by David E. Zitarelli, <Historia Mathematica>, Vol. 24,
      No. 1, p. 114, #24.1.119, 1997.
  [5] Marian Popescu, "A Model for the Smarandache Paradoxist Geometry",
      <Abstracts of Papers Presented to the American Mathematical Society
      Meetings>, Vol. 17, No. 1, Issue 103, 1996, p. 265.
  [6] Florentin Smarandache, "Collected Papers" (Vol. II), University of
      Kishinev Press, Kishinev, pp. 5-28, 1997.
  [7] Florentin Smarandache, "Paradoxist Mathematics" (lecture), Bloomsburg
      University, Mathematics Department, PA, USA, November 1985.