# Paradoxism in geometry: smarandache geometries (sg) and the degree of negation in geometries.

We present a general class of geometries resulted from paradoxism.

Definition:

An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space (i.e., validated and invalided, or only invalidated but in multiple distinct ways).

A Smarandache Geometry is a geometry which has at least one Smarandachely denied axiom (1969).

Notations:

Let's note any point, line, plane, space, triangle, etc. in a smarandacheian geometry by s-point, s-line, s-plane, s-space, s-triangle respectively in order to distinguish them from other geometries.

Applications:

Why these hybrid geometries? Because in reality there does not exist isolated homogeneous spaces, but a mixture of them, interconnected, and each having a different structure.

The Smarandache geometries (SG) are becoming very important now since they combine many spaces into one, because our world is not formed by perfect homogeneous spaces as in pure mathematics, but by non-homogeneous spaces. Also, SG introduce the degree of negation in geometry for the first time [for example an axiom (or theorem, or lemma, or proposition) is denied in 40% of the space and accepted in 60% of the space], that's why they can become revolutionary in science and this thanks to the idea of partially denying and partially accepting of axioms/ theorems/ lemmas/ propositions in a space (making multi-spaces, i.e. a space formed by combination of many different other spaces), similarly as in fuzzy logic (or in neutrosophic logic--the last one is a generalization of the fuzzy logic) the 'degree of truth' (i. e. for example 40% false and 60% true).

Smarandache geometries are starting to have applications in physics and engineering because of dealing with nonhomogeneous spaces.

In the Euclidean geometry, also called parabolic geometry, the fifth Euclidean postulate that there is only one parallel to a given line passing through an exterior point, is kept or validated.

In the Lobachevsky-Bolyai-Gauss geometry, called hyperbolic geometry, this fifth Euclidean postulate is invalidated in the following way: there are infinitely many lines parallels to a given line passing through an exterior point.

While in the Riemannian geometry, called elliptic geometry, the fifth Euclidean postulate is also invalidated as follows: there is no parallel to a given line passing through an exterior point.

Thus, as a particular case, Euclidean, Lobachevsky-BolyaiGauss, and Riemannian geometries may be united altogether, in the same space, by some Smarandache geometries. These last geometries can be partially Euclidean and partially NonEuclidean. Howard Iseri [3] constructed a model for this particular Smarandache geometry, where the Euclidean fifth postulate is replaced by different statements within the same space, i.e. one parallel, no parallel, infinitely many parallels but all lines passing through the given point, all lines passing through the given point are parallel.

Linfan Mao [4, 5] showed that SG are generalizations of Pseudo-Manifold Geometries, which in their turn are generalizations of Finsler Geometry, and which in its turn is a generalization of Riemann Geometry.

Let's consider Hilbert's 21 axioms of Euclidean geometry. If we Smarandachely deny one, two, three, and so on, up to 21 axioms respectively, then one gets:

[sub.21][C.sub.1] + [sub.21][C.sub.2] + [sub.21][C.sub.3] + ... [sub.21][C.sub.21] = [2.sup.21] - 1 = 2,097,151

Smarandache geometries, however the number is much higher because one axiom can be Smarandachely denied in multiple ways.

Similarly, if one Smarandachely denies the axioms of Projective Geometry, etc.

It seems that Smarandache Geometries are connected with the Theory of Relativity (because they include the Riemannian geometry in a subspace) and with the Parallel Universes (because they combine separate spaces into one space only) too. A Smarandache manifold is an n-D manifold that supports a smarandacheian geometry.

Examples:

As a particular case one mentions Howard's Models [3] where a Smarandache manifold is a 2-D manifold formed by equilateral triangles such that around a vertex there are 5 (for elliptic), 6 (for Euclidean), and 7 (for hyperbolic) triangles, two by two having in common a side. Or, more general, an n-D manifold constructed from n-D submanifolds (which have in common two by two at most one m-D frontier, where m<n) that supports a Smarandache geometry.

A Mode for a particular Smarandache Geometry:

Let's consider an Euclidean plane ([for all]) and three noncollinear given points A, B, and C in it. We define as s-points all usual Euclidean points and s-lines any Euclidean line that passes through one and only one of the points A, B, or C. Thus the geometry formed is smarandacheian because two axioms are Smarandachely denied:

a) The axiom that through a point exterior to a given line there is only one parallel passing through it is now replaced by two statements: one parallel, and no parallel. Examples:

Let's take the Euclidean line AB (which is not an s-line according to the definition because passes through two among the three given points A, B, C), and an s-line noted (c) that passes through s-point C and is parallel in the Euclidean sense to AB:

--through any s-point not lying on AB there is one s-parallel to (c).

--through any other s-point lying on the Euclidean line AB, there is no s-parallel to (c).

b) And the axiom that through any two distinct points there exist one line passing through them is now replaced by: one s-line, and no s-line.

Examples:

Using the same notations:

--through any two distinct s-points not lying on Euclidean lines AB, BC, CA, there is one s-line passing through them;

--through any two distinct s-points lying on AB there is no s-line passing through them.

Miscellanea:

First International Conference on Smarandache Geometries will be held, between May 3-5, 2003, at the Griffith University, Queensland, Australia, organized by Dr. Jack Allen. Conference's page is at: http://at.yorku.ca/cgi-bin/amcacalendar/public/display/conference info/fabz54.

And it is announced at http://www.ams.org/mathcal/ info/2003 may3-5 goldcoast.html as well.

There is a club too on "Smarandache Geometries" at http://clubs.yahoo.com/clubs/smarandachegeometries and everybody is welcome.

Question:

Is there a general model for all Smarandache Geometries in such a way that replacing some parameters one gets any of the desired particular SG?

References:

(1.) Ashbacher, C.,"Smarandache Geometries", Smarandache Notions Journal, Vol. 8, 212-215, No. 1-2-3, 1997.

(2.) Chimienti, S. and Bencze, M., "Smarandache Paradoxist Geometry", Bulletin of Pure and Applied Sciences, Delhi, India, Vol. 17E, No. 1, 123-1124, 1998; http://www.gallup. unm.edu/ ~smarandache/prd-geo1.txt.

(3.) Kuciuk, L., Antholy M., "An Introduction to Smarandache Geometries", Mathematics Magazine, Aurora, Canada, Vol. 12, 2003, and online: http://www. mathematicsmagazine.com/1-2004/Sm_Geom_1_2004.htm; also presented at New Zealand Mathematics Colloquium, Massey University, Palmerston North, New Zealand, December 3-6, 2001, http://atlas-conferences.com/c/a/h/f/09. htm; also presented at the International Congress of Mathematicians (ICM2002), Beijing, China, 20-28 August 2002, http://www.icm2002.org.cn/B/Schedule Section04.htm and in 'Abstracts of Short Communications to the International Congress of Mathematicians', International Congress of Mathematicians, 20-28 August 2002, Beijing, China, Higher Education Press, 2002; and in JP Journal of Geometry and Topology, Allahabad, India, Vol. 5, No. 1, 77-82, 2005.

(4.) Mao, Linfan, "An introduction to Smarandache geometries on maps", 2005 International Conference on Graph Theory and Combinatorics, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China, June 25-30, 2005.

(5.) Mao, Linfan, "Automorphism Groups of Maps, Surfaces and Smarandache Geometries", partially post-doctoral research for the Chinese Academy of Science, Am. Res. Press, Rehoboth, 2005.

(6.) Iseri, H., "Partially Paradoxist Smarandache Geometries", http://www.gallup.unm.edu/~smarandache/ Howard--Iseri-paper.htm.

(7.) Iseri, H., "Smarandache Manifolds", Am. Res. Press, 2002, http://www.gallup.unm.edu/~smarandache/Iseri-book1. pdf

(8.) PlanetMath, "Smarandache Geometries", http:// planetmath. org/encyclopedia/SmarandacheGeometries.html.

(9.) Perez, M., "Scientific Sites", in 'Journal of Recreational Mathematics', Amityville, NY, USA, Vol. 31, No. 1, 86, 20022003.

(10.) Smarandache, F., "Paradoxist Mathematics", in Collected Papers (Vol. II), Kishinev University Press, Kishinev, 5-28, 1997.

L. KUCIUK (USA), M. ANTHOLY (Canada)