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", , Vol. 8, No. 1-2-3, 212-215, 1997. [2] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Counter-Projective Geometry", , Delhi, India, Vol. 17E, No. 1, pp. 117-118, 1998. [3] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Counter-Projective Geometry", , Vol. 9, No. 1-2, 47-48, 1998. [4] Brown, Jerry L., "The Smarandache Counter-Projective Geometry", , 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.