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", , Vol. 8, No. 1-2-3, Fall 1997, pp. 212-215.
[2] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Paradoxist
Geometry", , Delhi, India,
Vol. 17E, No. 1, 123-1124, 1998.
[3] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Paradoxist
Geometry", , AZ, USA, Vol. 9, No.
1-2, 43-44, 1998.
[4] Mike Mudge, "A Paradoxist Mathematician, His Function, Paradoxist
Geometry, and Class of Paradoxes", ,
Vol. 7, No. 1-2-3, August 1996, pp. 127-129.
reviewed by David E. Zitarelli, , Vol. 24,
No. 1, p. 114, #24.1.119, 1997.
[5] Marian Popescu, "A Model for the Smarandache Paradoxist Geometry",
, 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.