Smarandache geometries

An axiom is said smarandachely denied (S-denied) if in the same space the axiom behaves differently (i.e., validated and invalided; or only invalidated but in at least two distinct ways).

A Smarandache geometry (SG) is a geometryMathworldPlanetmath which has at least one smarandachely denied axiom (1969).

Thus, as a particular case, EuclideanPlanetmathPlanetmath, Lobachevsky-Bolyai-Gauss, and Riemannian geometries may be united altogether, in the same space, by some SGs. These last geometries can be partially Euclidean and partially non-Euclidean.

The novelty of the SG is the fact that they introduce for the first time the degree of negationMathworldPlanetmath in geometry, similarly to the degree of falsehood in fuzzy or neutrosophic logic. For example an axiom can be denied in percentage of 30
Also SG are defined on multispaces, i.e. unions of Euclidean and non-Euclidean subspacesMathworldPlanetmathPlanetmath, or unions of distinct non-Euclidean spaces.

As an example of S-denying, a propositionPlanetmathPlanetmath ϕ, which is the conjunctionMathworldPlanetmath of a set ϕi of propositions, can be invalidated in many ways if it is minimally unsatisfiable, that is, such that the conjunction of any proper subsetMathworldPlanetmathPlanetmath of the ϕi is satisfied in a structureMathworldPlanetmath, but ϕ itself is not.

Many axioms have one of the following forms in the sequential logic:
Let n,m1 be two integers, also A=A1×A2××An and B=B1×B2××Bm.
Let x=(x1,x2,,xn) and y=(y1,y2,,ym), also 𝑃(x) and 𝑄(x) be relationships on x and 𝑅(x,y) a relationship between x and y. Then:
a1 xA, and is unique yB, such that 𝑅(x,y).
 a2 If xA such that 𝑃(x), then 𝑄(x).
a3 xA one has 𝑃(x).

One can invalidate in many ways the first class of axioms a1 as follows:
1.1. xA and finitely many (at least two) yB such that 𝑅(x,y).
1.2. xA and infinitely many yB such that 𝑅(x,y).
1.3. xA such that yB for which 𝑅(x,y).

One can invalidate in many ways the second class of axioms a2 as follows:
2.1. a unique xA such that 𝑃(x) but ¬𝑄(x).
2.2. finitely many (more than one) xA such that 𝑃(x) but ¬𝑄(x).
2.3. infinitely many xA such that 𝑃(x) but ¬𝑄(x).
2.4. xA such that 𝑃(x) one has ¬𝑄(x).

One can invalidate in many ways the third class of axioms a3 as follows:
3.1. xA and is unique such that ¬𝑃(x).
3.2. finitely many xA such that ¬𝑃(x).
3.3. infinitely many xA such that ¬𝑃(x).

Not all axioms can be smarandachely denied.

Here it is an example of what it means for an axiom to be invalidated in multiple ways [2]: As a particular axiom let’s take Euclid’s Fifth Postulate. In Euclidean or parabolic geometry a line has one parallelMathworldPlanetmathPlanetmath only through a given point. In Lobacevskian or hyperbolic geometry a line has at least two parallels through a given point. In Riemannian or elliptic geometry a line has no parallel through a given point. Whereas in Smarandache geometries there are lines which have no parallels through a given point and other lines which have one or more parallels through a given point (the fifth postulate is invalidated in many ways).

Therefore, the Euclid’s Fifth Postulate (which asserts that there is only one parallel passing through an exteriorMathworldPlanetmath point to a given line) can be invalidated in many ways, i.e. Smarandachely denied, as follows:

- first invalidation: there is no parallel passing through an exterior point to a given line;

- second invalidation: there is a finite number of parallels passing through an exterior point to a given line;

- third invalidation: there are infinitely many parallels passing through an exterior point to a given line.


  • 1 S. Chimienti, M. Bencze, Smarandache Paradoxist Geometry, Bulletin of Pure and Applied Sciences, Vol. 17E, No. 1, 123-1124, 1998.
  • 2 H. Iseri, Smarandache Manifolds, Am. Res. Press, 2002. smarandache/Iseri-book.pdfThe book is also online..
  • 3 L. Kuciuk, M. Antholy An Introduction to Smarandache Geometries, JP Journal of Geometry and Topology, Vol. 5, No. 1, 77-82, 2005.
    Presented at New Zealand Mathematics Colloquium, Massey University, Palmerston North, New Zealand, December 3-6, 2001.
  • 4 Linfan Mao, An introduction to Smarandache geometries on maps, 2005 International Conference on Graph TheoryMathworldPlanetmath and Combinatorics, Zhejiang Normal University, Jinhua, Zhejiang, P. R. China, June 25-30, 2005.
  • 5 Linfan Mao, Automorphism GroupsMathworldPlanetmath of Maps, Surfaces and Smarandache Geometries, partially post-doctoral research for the Chinese Academy of Science, Beijing, 2005. smarandache/Linfan.pdfSecond book which is online..
  • 6 F. Smarandache, Paradoxist Mathematics, in Collected Papers (Vol. II), Kishinev University Press, Kishinev, 5-28, 1997. smarandache/CP2.pdfThird book which is online..
Title Smarandache geometries
Canonical name SmarandacheGeometries
Date of creation 2013-03-22 15:28:24
Last modified on 2013-03-22 15:28:24
Owner jonnathan (5141)
Last modified by jonnathan (5141)
Numerical id 7
Author jonnathan (5141)
Entry type Definition
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Related topic FlorentinSmarandache