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.