Smarandache Geometries

An axiom is said smarandachely 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). Therefore, we say that an axiom is partially negated, or there is a degree of negation of an axiom.

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

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

(i)  For the first case, when the axiom is partially validated (true) and partially invalidated (false), the Smarandache Geometries are particular cases of the NeutroGeometry.

(ii) For the second case, when the axiom is only invalidated (100% false), then the Smarandache Geometry are particular cases of the AntiGeometry.

The most important contribution of Smarandache Geometries was the introduction of the degree of negation of an axiom (and more general the degree of negation of a theorem, lemma, property, algorithm, scientific or humanistic proposition, etc.) which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or more general like the negation in neutrosophic logic (with a degree of truth, a degree of falsehood, and a degree of neutrality (neither true nor false, but unknown, ambiguous, indeterminate) [not only Euclid geometrical axioms, but any scientific or humanistic proposition in any field] or partial negation of an axiom (and, in general, partial negation of a scientific or humanistic proposition in any field).

These geometries connect many geometrical spaces with different structures into a heterogeneous multispace with multistructure.

In general, a rule R  R  in a system (Σ; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Σ, i.e. validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache system (Σ; R) is a system which has at least one Smarandachely denied rule in R.

In particular, a Smarandache geometry is such a geometry in which there is at least one Smarandachely denied rule, and a Smarandache manifold (M;A) is an n-dimensional manifold M that supports a Smarandache geometry.

In a Smarandache geometry, the points, lines, planes, spaces, triangles, ... are respectively called s-pointss-liness-planess-spacess-triangles, ... in order to distinguish them from those in classical geometry.

Howard Iseri constructed the Smarandache 2-manifolds by using equilateral triangular disks on Euclidean plane R2. Such manifold came true by paper models in Rfor elliptic, Euclidean and hyperbolic cases. It should be noted that a more general Smarandache n-manifold, i.e. combinatorial manifold and a differential theory on such manifold were constructed by Linfan Mao.

A curve and a surface in a Smarandache Geometry are called a Smarandache Curve and Smarandache Surface respectively. They actually are curves and surfaces of hybrid geometrical structures.

Nearly all geometries, such as pseudo-manifold geometries, Finsler geometry, combinatorial Finsler geometries, Riemann geometry, combinatorial Riemannian geometries, Weyl geometry, Kahler geometry are particular cases of Smarandache geometries.

[Prof. Dr. Linfan Mao, Chinese Academy of Sciences, Beijing, P. R. China, 2005-2023]

Particular cases of Smarandache Geometries:

Books:

Linfan Mao (2017). (让我们插上翅膀飞翔 -- 数学组合与Smarandache重叠空间). Chinese Branch Xiquan House, 352 p.

Hu Chang-Wei (2012). Vacuum, Space-Time, Matter and the Models of Smarandache Geometry (真空、时空、物质和). Educational Publishers, 112 p.

Linfan Mao (2011). Education Publisher, 484 p.

Yanpei Liu (2010). Introductory Map Theory. Kapa & Omega, 502 p.

Yuhua Fu, Linfan Mao, Mihaly Bencze (ed) (2007). Scientific Elements - Applications to Mathematics, Physics, and Other Sciences. International book series, Vol. 1. ProQuest Information & Learning, 200 p.

Linfan Mao (2006). Chinese Branch Xiquan House, 200 p.

Linfan Mao (2006). Smarandache Multi-Space Theory. Hexis, 274 p.

Linfan Mao (2005). American Research Press, 114 p.

Howard Iseri (2002). American Research Press, 96 p.

Articles:

Ion Patrascu (2023): Smarandache Geometries (or Hybrids)Octogon Mathematical Journal 31(2), 966-969.

F. Smarandache (2011). Degree of Negation of Euclid's Fifth Postulate. University of New Mexico, 6 p.

L. Kuciuk, M. Antholy (2005). An Introduction to the Smarandache Geometries. JP Journal of Geometry & Topology, 5(1), 77-81.

S. Bhattacharya (2005). A Model to a Smarandache Geometry. Alaska Pacific University, presentation.

Ovidiu Sandru (2004). Un model simplu de geometrie Smarandache construit exclusiv cu elemente de geometrie euclidiana. Universitatea Politehnica Bucharest, Romania, 3 p.

Howard Iseri (2003). A Classification of s-Lines in a Closed s-Manifold. Mansfield University, 3 p.

Howard Iseri (2003). Partially Paradoxist Smarandache Geometries. Mansfield University, 8 p.

Roberto Torretti (2002). An Economics Model for the Smarandache Anti-Geometry. Universidad de Chile, 12 p.

Clifford Singer (2001). Engineering A Visual Field. New York, presentation.

Smarandache Curves and Surfaces

Articles

M. Khalifa Saad, R. A. Abdel-Baky (2020). On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space. Aust. J. Math. Anal. Appl. 17(1), Art. 11, 16 p.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, Huseyin Kocayigit (2019). On The Sabban Frame Belonging To Involute-Evolute Curves. Ordu University, 11 p. DOI:10.5281/zenodo.2989788.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, Huseyin Kocayigit (2019). Some Special Curves Belonging to Mannheim Curves Pair. Ordu University, 10 p. DOI: 10.5281/zenodo.2990510.

F. Almaz, M.A. Kulahci (2018). A Note on Special Smarandache Curves in The Null Cone Q3. Acta Universitatis Apulensis 56, 111-124. DOI: 10.5281/zenodo.2987357.

A. Lourdusamy, Sherry George (2018). . International J. Math. Combin. (IJMC) 1, 109-126. DOI:

H. S. Abdel-Aziz, M. Khalifa Saad (2018). On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC 54(3), 229-249.

Tanju Kahraman (2018). Smarandache Curves of Null Quaternionic Curves in Minkowski 3-space. MANAS Journal of Engineering (MJEN) 6(1), 6 p. DOI: 10.5281/zenodo.1413905.

Tevk Sahin, Merve Okur (2018). . Int. J. Adv. Appl. Math. and Mech. 5(3), 15-26.

Gulnur Saffak Atalay (2018). Surfaces family with a common Mannheim geodesic curve. J. Appl. Math. Comp. (JAMC) 2(4), 155-165.

V. Ramachandran (2018). . International J. Math. Combin. (IJMC) 2, 114-121.

R. Ponraj, M. Maria Adaickalam (2018). . International J. Math. Combin. (IJMC) 2, 122-128.

R. Ponraj, K. Annathurai, R. Kala (2018). . International J. Math. Combin. (IJMC) 1, 138-145. DOI:

B. Basavanagoud, Sujata Timmanaikar (2018). . International J. Math. Combin. (IJMC) 2, 87-96.

Rajesh Kumar T.J., Mathew Varkey T.K. (2018). . International J. Math. Combin. (IJMC) 1, 90-96. DOI:

T. Chalapathi, R.V M S S Kiran Kumar (2018). . International J. Math. Combin. (IJMC) 1, 127-137.

K. Praveena, M. Venkatachalam (2018). . International J. Math. Combin. (IJMC) 2, 24-32. DOI:

K. Muthugurupackiam, S. Ramya (2018). . International J. Math. Combin. (IJMC) 1, 75-82.

V. Lokesha, P. S. Hemavathi, S. Vijay (2018). . International J. Math. Combin. (IJMC) 2, 80-86.

Rajendra P., R. Rangarajan (2018). . International J. Math. Combin. (IJMC) 1, 97-108.

T. Deepa, M. Venkatachalam (2018). . International J. Math. Combin. (IJMC) 2, 97-113.

Tanju Kahraman, Hasan Huseyin Ugurlu (2017). . International J. Math. Combin. (IJMC) 3, 1-9.

R. Ponraj, Rajpal Singh, R. Kala (2017). . International J. Math. Combin. (IJMC) 3, 125-135.

M. H. Akhbari, F. Movahedi, S. V. R. Kulli (2017). . International J. Math. Combin. (IJMC) 4, 138-150.

Samir K. Vaidya, Raksha N. Mehta (2017). . International J. Math. Combin. (IJMC) 3, 72-80.

Ahmed M. Naji, Soner Nandappa D. (2017). . International J. Math. Combin. (IJMC) 4, 91-102.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir (2017). International J.Math. Combin. (IJMC) 2, 84-91.

H.S. Abdel-Aziz, M. Khalifa Saad (2017). Computation of Smarandache curves according to Darboux frame in Minkowski 3-space. Journal of the Egyptian Mathematical Society 25, 382-390. DOI: 10.5281/zenodo.2987485.

Mervat Elzawy (2017). Smarandache curves in Euclidean 4-space E4. Journal of the Egyptian Mathematical Society 25, 268-271. , DOI:

M. Elzawy, S. Mosa (2017). Smarandache curves in the Galilean 4-space G4. Journal of the Egyptian Mathematical Society 25, 53-56, DOI:

E.M. Solouma (2017). Special equiform Smarandache curves in Minkowski space-time. Journal of the Egyptian Mathematical Society 25, 319-325

Akram Alqesmah, Anwar Alwardi, R. Rangarajan. International J. Math. Combin. (IJMC) 4, 110-120. DOI:

A. Nellai Murugan, P. Iyadurai Selvaraj. International J. Math. Combin. (IJMC) 3, 119-124.

Ujwala Deshmukh, Smita A. BhatavadekaInternational J. Math. Combin. (IJMC) 4, 151-156.

E. M. Solouma, M. M. Wageeda . International J.Math. Combin. (IJMC) 1, 1-9.

M. Subramanian, T. Subramanian . International J. Math. Combin. (IJMC) 3, 116-118.

U. M. Prajapati, R. M. Gajjar. International J. Math. Combin. (IJMC) 3, 90-115.

Linfan Mao. International J. Math. Combin. (IJMC) 4, 19-45.

P.S.K. Reddy, K.N. Prakasha, Gavirangaiah K.. International J. Math. Combin. (IJMC) 3, 22-31.

P. S. K. Reddy, K. N. Prakasha, Gavirangaiah K. (2017). Minimum Equitable Dominating Randic Energy of a Graph. International J. Math. Combin. (IJMC) 3, 81-89.

M. Khalifa Saad (2016). Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space. Journal of the Egyptian Mathematical Society 24, 416-423. DOI: 10.5281/zenodo.2990574.

Suleyman Senyurt, Abdussamet Caliskan . Bol. Soc. Paran. Mat. 34(2), 53–62. DOI: .

Yasin Unluturk, Suha Yilmaz . International J.Math. Combin. (IJMC) 4, 29-43.

Murat Savas, Atakan Tugkan Yakut, Tugba Tamirci . Gazi University Journal of Science 29(1), 69-77.

Nurten (Bayrak) Gurses, Ozcan Bektas, Salim Yuce . Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 65(2), 143-160. DOI:10.5281/zenodo.835466.

Suleyman Senyurt, Abdussamet Caliskan, Unzile Celik . International J.Math. Combin. (IJMC) 1, 1-7.

Suha Yilmaz, Umit Ziya Savci . International J.Math. Combin. (IJMC) 2, 1-15.

Mahmut Mak, Hasan Altinbas . International J.Math. Combin. (IJMC) 3, 1-16.

Suha Yilmaz . International J.Math. Combin. (IJMC) 4, 1-7.

Suleyman Senyurt, Abdussamet Calskan (2015). Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves. Gen. Math. Notes 31(2), 1-15. DOI: 10.5281/zenodo.2990072.

H. S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v2 [math.DG] 19 Feb 2015, 11 p. DOI: .

Vahide Bulut, Ali Caliskan. International J.Math. Combin. (IJMC) 3, 43-54.

Suleyman Senyurt, Abdussamet Caliskan. International J.Math. Combin. (IJMC) 1, 1-13.

Talat Korpinar. Acta Scientiarum. Technology 37(3), 389-393. DOI: .

H.S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v1 [math.DG] 21 Jan 2015, 15 p.

Tanju Kahraman, Mehmet Onder, H. Huseyin Ugurlu (2014). Dual Smarandache Curves and Smarandache Ruled Surfaces. Mathematical Sciences and Applications E-Notes 2(1), 83-98. DOI:

Atakan Tulkan Yakut, Murat Savas, Tugba Tamirci. Journal of Applied Mathematics, Article ID 193586, 12 p.

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic. Journal of Applied Mathematics, Article ID 404521, 14 p. DOI:

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic (2013). On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space. International Journal of Mathematics and Mathematical Sciences, 8 p. DOI: 10.5281/zenodo.1413399.

Suleyman Senyurt, Selin Sivas. Ordu Univ. J. Sci. Tech. 3(1), 46-60.

Ahmad T. Ali, Hossam S. Abdel Aziz, Adel H. Sorour. Journal of the Egyptian Mathematical Society 21, 285–294. DOI:

Kemal Taskopru, Murat TosunarXiv:1206.6229v3 [math.DG] 20 Jul 2012, 8 p.

Talat Korpinar, Essin Turhan. International J.Math. Combin. (IJMC) 4, 33-39.

Muhammed Cetin, Yilmaz Tuncer, Murat Kemal KaracanarXiv:1106.3202v1 [math.GM] 16 Jun 2011, 19 p.

Elham Mehdi-Nezhad, Amir M. Rahimi (2010). The Smarandache Vertices of The Comaximal Graph of A Commutative Ring. Stellenbosch University, 12 p. DOI: 10.5281/zenodo.2990970.

Ahmad T. Ali. International J.Math. Combin. (IJMC) 2, 30-36.

Melih Turgut, Suha Yilmaz. International J.Math. Combin. (IJMC) 3, 51-55.

Roberto Torretti (2002). A model for the Smarandache anti-geometry. Int. Journal of Social Economics 29(11), 886-896. DOI: 10.5281/zenodo.1412417.

E.M. Solouma (2002). . Al Imam Mohammad Ibn Saud Islamic University, 16 p.

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