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 (or Hybrid 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. 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.

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, scientific or humanistic proposition) 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 in 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 inR .

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-points, s-lines, s-planes, s-spaces, s-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 R3 for 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.

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.

[Dr. Linfan Mao, Chinese Academy of Sciences, Beijing, P. R. China, 2005-2011]

Smarandache Geometries (paradoxist, non-geometry, counter-projective, anti-geometry)

Books

Articles

Articles on Smarandache Curves (or Hybrid Curves)

[ A curve in a Smarandache Geometry is called a Smarandache Curve.

These are hybrid curves, because they verify some axioms (or theorems, properties etc.) from a type of curves,

and other axioms (or theorems, properties etc.) from other types of curves. ]

Emad Solouma: Equiform Spacelike Smarandache Curves of Anti-Eqiform Salkowski Curve According to Equiform FrameInternational Journal of Mathematical Analysis, Vol. 15, 2021, no. 1, 43-59, 17 p.; DOI: 10.12988/ijma.2021.912141

Soukaina Ouarab: NC-Smarandache Ruled Surface and NW-Smarandache Ruled Surface according to Alternative Moving Frame in E3. Hindawi: Journal of Mathematics, Volume 2021, Article ID 9951434, 6 p.; DOI: 10.1155/2021/9951434

Zuhal Kucukarslan Yuzbasi: On Characterizations of Curves in The Galilean PlanePalestine Journal of Mathematics, Vol. 10(1) (2021), 308–311, 4 p.

Shankar Lal: Parallel Transport Frame of Smarandache Curves in Euclidean Space. J. Mountain Res., Vol. 16(1), (2021), 225-233, 9 p.; DOI:

10.51220/jmr.v16i1.23

Mustafa Altin, Ahmet Kazan, H.Bayram Karadag: Ruled and Rotational Surfaces Generated by Non-Null Curves with Zero Weighted CurvatureInternational Electronic Journal of Geometry, Volume 13, no. 2 11–29 (2020), 19 p.; DOI: 10.36890/IEJG.599817

Tanju Kahraman, Hasan Huseyin Ugurlu: Smarandache Curves of a Spacelike Curve lying on Unit dual Lorentzian SphereCBU J. of Sci., Volume 11, Issue 2, p 93-105, 13 p.

Kahraman Esen Ozen, Murat Tosun: Trajectories Generated by Special Smarandache Curves According to Positional Adapted Frame. 12 p.;

www.researchgate.net/publication/348759696

F. Almaz, & M.A. Kulahci. (2019). A Note on Special Smarandache Curves in The Null Cone Q3. Acta Universitatis Apulensis, No. 56/2018, pp. 111-124, http://doi.org/10.5281/zenodo.2987357

M. Khalifa Saad and R. A. Abdel-Baky, On Ruled Surfaces According to Quasi-Frame in Euclidean 3-Space [on Smarandache curves], Aust. J. Math. Anal. Appl. Vol. 17 (2020), No. 1, Art. 11, 16 pp.

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 (2017) 382–390. http://doi.org/10.5281/zenodo.2987485

Tanju Kahraman, Mehmet Onder, & H. Huseyin Ugurlu. (2019). Dual Smarandache Curves and Smarandache Ruled Surfaceshttp://doi.org/10.5281/zenodo.2987568

Süleyman Senyurt, Yasin Altun, Ceyda Cevahir, & Huseyin Kocayigit. (2019). On The Sabban Frame Belonging To Involute-Evolute Curveshttp://doi.org/10.5281/zenodo.2989788

Mervat Elzawy. (2017). Smarandache curves in Euclidean 4- space E4. Journal of the Egyptian Mathematical Society 25 / 2017, 268–271. http://doi.org/10.5281/zenodo.2989884

Suleyman Senyurt, & Abdussamet Calskan. (2015). Smarandache Curves in Terms of Sabban Frame of Spherical Indicatrix Curves. Gen. Math. Notes, Vol. 31, No. 2, December 2015, pp.1-15, http://doi.org/10.5281/zenodo.2990072

M. Elzawy, & S. Mosa. (2017). Smarandache curves in the Galilean 4-space G4. Journal of the Egyptian Mathematical Society 25 / 2017, 53–56 http://doi.org/10.5281/zenodo.2990158

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir, & Huseyin Kocayigit. (2019). Some Special Curves Belonging to Mannheim Curves Pairhttp://doi.org/10.5281/zenodo.2990510

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

E.M. Solouma. (2019). Special equiform Smarandache curves in Minkowski space-time. Journal of the Egyptian Mathematical Society 25 (2017) 319–325, http://doi.org/10.5281/zenodo.2990660

Elham Mehdi-Nezhad, & Amir M. Rahimi. (2010). The Smarandache Vertices of The Comaximal Graph of A Commutative Ringhttp://doi.org/10.5281/zenodo.2990970

Elham Mehdi-Nezhad, Amir M. Rahimi: The Smarandache Vertices of the Comaximal Graph of a Commutative Ring. 12 p. DOI: https://doi.org/10.5281/zenodo.1419756

Mihriban Kulahci, Fatma Almaz:  . 12 p. http://doi.org/10.5281/zenodo.1412498

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Elarslan, Emilija Nesovic: On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space. In International Journal of Mathematics and Mathematical Sciences, 2013, 8 pages.  http://doi.org/10.5281/zenodo.1413399

H. S. Abdel-Aziz, & M. Khalifa Saad:  On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC, vol.54, no.3, pp.229-249, 2018. http://doi.org/10.5281/zenodo.1413401

Tanju Kahraman:  Smarandache Curves of Null Quaternionic Curves in Minkowski 3-space. In MANAS Journal of Engineering (MJEN),  Volume 6, Issue 1, 2018, pp. 1-6. http://doi.org/10.5281/zenodo.1413905

H. S. Abdel-Aziz, & M. Khalifa Saad:  On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space. CMC, vol.54, no.3, pp.229-249, 2018. http://doi.org/10.5281/zenodo.1413401

Tevk Sahin, Merve Okur:  Special Smarandache Curves with Respect to Darboux Frame in Galilean 3-Space. 2018, 15 pages. http://doi.org/10.5281/zenodo.1413956

Roberto Torretti:  A model for the Smarandache anti-geometry. In Int. Journal of Social Economics, vol. 29, nr. 11, 2002, 886-896. http://doi.org/10.5281/zenodo.1412417

Gulnur Saffak Atalay:  Surfaces family with a common Mannheim geodesic curve. In Journal of Applied Mathematics and Computation (JAMC), 2018, 2(4), pp. 155-165. http://doi.org/10.5281/zenodo.1413970

V. Ramachandran: , International J. Math. Combin. Vol. 2 (2018), pp. 114-121. DOI:

R. Ponraj and M. Maria Adaickalam: , International J. Math. Combin. Vol. 2 (2018), pp. 122-128. DOI:

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

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

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

M. Subramanian, T. Subramanian: , International J. Math. Combin. Vol. 3 (2017), pp. 116-118.

U M Prajapati, R M Gajjar: , International J. Math. Combin. Vol. 3 (2017), pp. 90-115.

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

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

K. Muthugurupackiam, S. Ramya: , International J. Math. Combin. Vol. 1 (2018), pp. 75-82. DOI:

Linfan Mao: , International J. Math. Combin. Vol. 4 (2017), pp. 19-45. DOI:

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

P.S.K. Reddy, K.N. Prakasha, Gavirangaiah K.: , International J. Math. Combin. Vol. 3 (2017), pp. 22-31. DOI:

Rajendra P., R. Rangarajan: , International J. Math. Combin. Vol. 1 (2018), pp. 97-108. DOI:

P. S. K. Reddy, K. N. Prakasha, Gavirangaiah K.: , International J. Math. Combin. Vol. 3 (2017), pp. 81-89.

T. Deepa, M. Venkatachalam: , International J. Math. Combin. Vol. 2 (2018), pp. 97-113. DOI:

Akram Alqesmah, Anwar Alwardi, R. Rangarajan: , International J. Math. Combin. Vol. 4 (2017), pp. 110-120. DOI:

A. Nellai Murugan, P. Iyadurai Selvaraj: , International J. Math. Combin. Vol. 3 (2017), pp. 119-124. DOI:

Ujwala Deshmukh, Smita A. Bhatavadeka: , International J. Math. Combin. Vol. 4 (2017), pp. 151-156. DOI:

V. Lokesha, P. S. Hemavathi, S. Vijay: , International J. Math. Combin. Vol. 2 (2018), pp. 80-86. DOI:

Tanju Kahraman, Hasan Huseyin Ugurlu: …, International J. Math. Combin. Vol. 3 (2017), pp. 1-9.

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

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

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

Ahmed M. Naji and Soner Nandappa D.: , International J. Math. Combin. Vol. 4 (2017), pp. 91-102. DOI:

Suleyman Senyurt, Selin Sivas: An Application of Smarandache Curve. In Ordu Univ. Bil. Tek. Derg., Cilt: 3, Sayi:1, 2013,46-60/Ordu Univ. J. Sci. Tech., Vol. 3, No. 1 (2013), 15 pages.

H.S. Abdel-Aziz, M. Khalifa Saad: Computation of Smarandache curves according to Darboux frame in Minkowski 3-space. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 382-390, 9 pages.

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

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

M. Khalifa Saad: Spacelike and timelike admissible Smarandache curves in pseudo-Galilean space. In Journal of the Egyptian Mathematical Society, 24 (2016), pp. 416-423, 8 pages.

E.M. Solouma: Special equiform Smarandache curves in Minkowski space-time. In Journal of the Egyptian Mathematical Society, 25 (2017), pp. 319-325, 7 pages.

E.M. Solouma: Special timelike Smarandache curves in Minkowski 3-space. Al Imam Mohammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, KSA, Riyadh, 16 pages.

b−Smarandache m1m2 Curves of Biharmontic New Type b−Slant Helices According to Bishop Frame in the Sol Space Sol3, by Talat Korpinar, Essin Turhan. In International J.Math. Combin. Vol. 4 (2012), pp. 33-39. DOI:10.5281/zenodo.825679, https://doi.org/10.5281/zenodo.825678

Isotropic Smarandache Curves in Complex Space C3, by Suha Yilmaz. In International J.Math. Combin. Vol. 4 (2016), 7 pages. DOI:10.5281/zenodo.826790, https://doi.org/10.5281/zenodo.826789

Dual Smarandache Curves and Smarandache Ruled Surfaces, by Tanju Kahraman, Mehmet Onder, H. Huseyin Ugurlu, In Mathematical Sciences and Applications E-Notes, Volume 2 No. 1, pp. 83/98 (2014), 16 pages. DOI:10.5281/zenodo.835438, https://doi.org/10.5281/zenodo.835437

Mannheim Partner Curve a Different View, Suleyman Senyurt, Yasin Altun, Ceyda Cevahir. In International J.Math. Combin. Vol. 2 (2017), pp. 84-91. DOI:10.5281/zenodo.831975, https://doi.org/10.5281/zenodo.831974

Ruled surfaces generated by some special curves in Euclidean 3-Space, by Ahmad T. Ali, Hossam S. Abdel Aziz, Adel H. Sorour. In Journal of the Egyptian Mathematical Society (2013) 21, pp. 285–294. DOI:10.5281/zenodo.835445, https://doi.org/10.5281/zenodo.835444

N*C*-Smarandache Curve of Bertrand Curves Pair According to Frenet Frame, by Suleyman Senyurt, Abdussamet Caliskan, Unzile Celik. In International J.Math. Combin. Vol. 1 (2016), 7 pages. DOI:10.5281/zenodo.815716, https://doi.org/10.5281/zenodo.815715

N*C*-Smarandache Curves of Mannheim Curve Couple According to Frenet Frame, by Suleyman Senyurt, Abdussamet Caliskan. In International J.Math. Combin. Vol. 1 (2015), 13 pages. DOI:10.5281/zenodo.815112, https://doi.org/10.5281/zenodo.815111

New type surfaces in terms of B-Smarandache Curves in Sol3, by Talat Korpinar. In Acta Scientiarum. Technology, Maringá, v. 37, n. 3, July-Sept. 2015, pp. 389-393. DOI:10.5281/zenodo.835440, https://doi.org/10.5281/zenodo.835439

On Pseudospherical Smarandache Curves in Minkowski 3-Space, by Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic. In Journal of Applied Mathematics, Volume 2014, Article ID 404521, 14 pages. DOI:10.5281/zenodo.835443, https://doi.org/10.5281/zenodo.835442

Smarandache Curves According to Bishop Frame in Euclidean 3-Space, by Muhammed Cetin, Yilmaz Tuncer, Murat Kemal Karacan. arXiv:1106.3202v1 [math.GM] 16 Jun 2011, 19 pages. DOI:10.5281/zenodo.835447, https://doi.org/10.5281/zenodo.835446

Smarandache Curves According to Sabban Frame on S2, by Kemal Taskopru, Murat Tosun. arXiv:1206.6229v3 [math.DG] 20 Jul 2012, 8 pages. DOI:10.5281/zenodo.835452, https://doi.org/10.5281/zenodo.835451

Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space, by Suha Yilmaz, Umit Ziya Savcı. In International J.Math. Combin. Vol. 2 (2016), 15 pages. DOI:10.5281/zenodo.822231, https://doi.org/10.5281/zenodo.822230

Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space, by H.S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v1 [math.DG] 21 Jan 2015, 15 pages. DOI:10.5281/zenodo.835456, https://doi.org/10.5281/zenodo.835455

Smarandache Curves in Euclidean 4- space E4, by Mervat Elzawy. In Journal of the Egyptian Mathematical Society (2017), 4 pages. DOI:10.5281/zenodo.835458, https://doi.org/10.5281/zenodo.835457

Smarandache Curves in Minkowski Space-time, by Melih Turgut, Suha Yilmaz. In International J.Math. Combin. Vol.3 (2008), pp. 51-55. DOI:10.5281/zenodo.823501, https://doi.org/10.5281/zenodo.823500

Smarandache Curves in Terms of Sabban Frame of Fixed Pole Curve, by Suleyman Senyurt, Abdussamet Caliskan. In Bol. Soc. Paran. Mat. (3s.) v. 34 2 (2016), pp. 53–62. DOI:10.5281/zenodo.835460, https://doi.org/10.5281/zenodo.835459

Smarandache curves in the Galilean 4-space G4, by M. Elzawy, S. Mosa. In Journal of the Egyptian Mathematical Society (2016), 4 pages. DOI:10.5281/zenodo.835462, https://doi.org/10.5281/zenodo.835459

Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2, by Yasin Unluturk, Suha Yilmaz. In International J.Math. Combin. Vol. 4 (2016), pp. 29-43. DOI:10.5281/zenodo.826804, https://doi.org/10.5281/zenodo.826803

Smarandache curves of some special curves in the Galilean 3-space, by H. S. Abdel-Aziz, M. Khalifa Saad. arXiv:1501.05245v2 [math.DG] 19 Feb 2015, 11 pages. DOI: 10.5281/zenodo.835464, https://doi.org/10.5281/zenodo.835463

Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space, by Mahmut Mak, Hasan Altinbas. In International J.Math. Combin. Vol. 3 (2016), 16 pages. DOI:10.5281/zenodo.825056, https://doi.org/10.5281/zenodo.825055

Special Smarandache Curves in R31, by Nurten (Bayrak) Gurses, Ozcan Bektas, Salim Yuce. In Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Volume 65, Number 2, 2016, pp. 143-160. DOI:10.5281/zenodo.835466, https://doi.org/10.5281/zenodo.835465

Special Smarandache Curves According to Bishop Frame in Euclidean Spacetime, by E. M. Solouma, M. M. Wageeda. In International J.Math. Combin. Vol. 1 (2017), 9 pages. DOI:10.5281/zenodo.815770, https://doi.org/10.5281/zenodo.815769

Special Smarandache Curves in the Euclidean Space, by Ahmad T. Ali. In International J.Math. Combin. Vol. 2 (2010), pp. 30-36. DOI:10.5281/zenodo.821048, https://doi.org/10.5281/zenodo.821047

Spherical Images of Special Smarandache Curves in E3, by Vahide Bulut, Ali Caliskan. In International J.Math. Combin. Vol. 3 (2015), pp. 43-54. DOI:10.5281/zenodo.825027, https://doi.org/10.5281/zenodo.825026

The Smarandache Curves on H02,, by Murat Savas, Atakan Tugkan Yakut, Tugba Tamirci. In Gazi University Journal of Science GU J Sci. 29(1) (2016), pp. 69-77. DOI:10.5281/zenodo.835469, https://doi.org/10.5281/zenodo.835468

The Smarandache Curves on S21 and Its Duality on H20, by Atakan Tulkan Yakut, Murat Savas, Tugba Tamirci. In Journal of Applied Mathematics, Volume 2014, Article ID 193586, 12 pages. DOI:10.5281/zenodo.835471, https://doi.org/10.5281/zenodo.835470

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