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. 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 ∈ 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-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.

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

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]

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Smarandache Geometries
(paradoxist, non-geometry, counter-projective, anti-geometry)
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Books:
Linfan Mao (2017). Let’s Flying by Wing - Mathematical Combinatorics & Smarandache Multi-Spaces (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). Combinatorial Geometry with Applications to Field Theory. 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). Smarandache Geometries & Map Theory with Applications (I). Chinese Branch Xiquan House, 200 p.

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

Linfan Mao (2005). Automorphism Groups of Maps, Surfaces and Smarandache Geometries. American Research Press, 114 p.

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

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

Linfan Mao (2007). A generalization of Stokes theorem on combinatorial manifolds. 16 p.; http://lanl.arxiv.org/abs/math/0703400v1; http://xxx.lanl.gov/pdf/math/0703400v1.

Linfan Mao (2006). Combinatorial Speculations and the Combinatorial Conjecture for Mathematics. 19 p.; http://lanl.arxiv.org/abs/math/0606702v2; http://xxx.lanl.gov/pdf/math/0606702v2.

Linfan Mao (2006). Pseudo-Manifold Geometries with Applications. 15 p.; http://lanl.arxiv.org/abs/math/0610307v1; http://xxx.lanl.gov/pdf/math/0610307v1.

Linfan Mao (2006). Geometrical Theory on Combinatorial Manifolds. 37 p.; http://lanl.arxiv.org/abs/math/0612760v1; http://xxx.lanl.gov/pdf/math/0612760v1.

Linfan Mao (2005). A new view of combinatorial maps by Smarandache's notion. 19 p.; http://lanl.arxiv.org/abs/math/0506232v1; http://xxx.lanl.gov/pdf/math/0506232v1

Linfan Mao (2005). Parallel bundles in planar map geometries. 16 p.; http://lanl.arxiv.org/abs/math/0506386v1; http://xxx.lanl.gov/pdf/math/0506386v1.

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.



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Smarandache Curves
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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). Linear Cyclic Snakes as Super Vertex Mean Graphs. International J. Math. Combin. (IJMC) 1, 109-126. DOI: 10.5281/zenodo.1418960.

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). Special Smarandache Curves with Respect to Darboux Frame in Galilean 3-Space. 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). (1,N)-Arithmetic Labelling of Ladder and Subdivision of Ladder. International J. Math. Combin. (IJMC) 2, 114-121.

R. Ponraj, M. Maria Adaickalam (2018). 3-Difference Cordial Labeling of Corona Related Graphs. International J. Math. Combin. (IJMC) 2, 122-128.

R. Ponraj, K. Annathurai, R. Kala (2018). 4-Remainder Cordial Labeling of Some Graphs. International J. Math. Combin. (IJMC) 1, 138-145. DOI: 10.5281/zenodo.1418786.

B. Basavanagoud, Sujata Timmanaikar (2018). Accurate Independent Domination in Graphs. International J. Math. Combin. (IJMC) 2, 87-96.

Rajesh Kumar T.J., Mathew Varkey T.K. (2018). Adjacency Matrices of Some Directional Paths and Stars. International J. Math. Combin. (IJMC) 1, 90-96. DOI: 10.5281/zenodo.1418808.

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

K. Praveena, M. Venkatachalam (2018). Equitable Coloring on Triple Star Graph Families. International J. Math. Combin. (IJMC) 2, 24-32. DOI: 10.5281/zenodo.1418902.

K. Muthugurupackiam, S. Ramya (2018). Even Modular Edge Irregularity Strength of Graphs. International J. Math. Combin. (IJMC) 1, 75-82.

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

Rajendra P., R. Rangarajan (2018). Minimum Equitable Dominating Randic Energy of a Graph. International J. Math. Combin. (IJMC) 1, 97-108.

T. Deepa, M. Venkatachalam (2018). On r-Dynamic Coloring of the Triple Star Graph Families. International J. Math. Combin. (IJMC) 2, 97-113.

Tanju Kahraman, Hasan Huseyin Ugurlu (2017). Smarandache Curves of Curves lying on Lightlike Cone. International J. Math. Combin. (IJMC) 3, 1-9.

R. Ponraj, Rajpal Singh, R. Kala (2017). Some New Families of 4-Prime Cordial Graphs. International J. Math. Combin. (IJMC) 3, 125-135.

M. H. Akhbari, F. Movahedi, S. V. R. Kulli (2017). Some Parameters of Domination on the Neighborhood Graph. International J. Math. Combin. (IJMC) 4, 138-150.

Samir K. Vaidya, Raksha N. Mehta (2017). Strong Domination Number of Some Cycle Related Graphs. International J. Math. Combin. (IJMC) 3, 72-80.

Ahmed M. Naji, Soner Nandappa D. (2017). The k-Distance Degree Index of Corona, Neighborhood Corona Products and Join of Graphs. International J. Math. Combin. (IJMC) 4, 91-102.

Suleyman Senyurt, Yasin Altun, Ceyda Cevahir (2017). Mannheim Partner Curve a Different View. 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: 10.5281/zenodo.2989884.

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

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

Akram Alqesmah, Anwar Alwardi, R. Rangarajan (2017). On the Distance Eccentricity Zagreb Indeices of Graphs. International J. Math. Combin. (IJMC) 4, 110-120. DOI: 10.5281/zenodo.1418934.

A. Nellai Murugan, P. Iyadurai Selvaraj (2017). Path Related n-Cap Cordial Graphs. International J. Math. Combin. (IJMC) 3, 119-124.

Ujwala Deshmukh, Smita A. Bhatavadeka (2017). Primeness of Supersubdivision of Some Graphs. International J. Math. Combin. (IJMC) 4, 151-156.

E. M. Solouma, M. M. Wageeda (2017). Special Smarandache Curves According to Bishop Frame in Euclidean Spacetime. International J.Math. Combin. (IJMC) 1, 1-9.

M. Subramanian, T. Subramanian (2017). A Study on Equitable Triple Connected Domination Number of a Graph. International J. Math. Combin. (IJMC) 3, 116-118.

U. M. Prajapati, R. M. Gajjar (2017). Cordiality in the Context of Duplication in Web and Armed Helm. International J. Math. Combin. (IJMC) 3, 90-115.

Linfan Mao (2017). Hilbert Flow Spaces with Operators over Topological Graphs. International J. Math. Combin. (IJMC) 4, 19-45.

P.S.K. Reddy, K.N. Prakasha, Gavirangaiah K. (2017). Minimum Dominating Color Energy of a Graph. 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 (2016). Smarandache Curves in Terms of Sabban Frame of Fixed Pole Curve. Bol. Soc. Paran. Mat. 34(2), 53–62. DOI: 10.5281/zenodo.835460.

Yasin Unluturk, Suha Yilmaz (2016). Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2. International J.Math. Combin. (IJMC) 4, 29-43.

Murat Savas, Atakan Tugkan Yakut, Tugba Tamirci (2016). The Smarandache Curves on H20. Gazi University Journal of Science 29(1), 69-77.

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

Suleyman Senyurt, Abdussamet Caliskan, Unzile Celik (2016). N*C*-Smarandache Curve of Bertrand Curves Pair According to Frenet Frame. International J.Math. Combin. (IJMC) 1, 1-7.

Suha Yilmaz, Umit Ziya Savci (2016). Smarandache Curves and Applications According to Type-2 Bishop Frame in Euclidean 3-Space. International J.Math. Combin. (IJMC) 2, 1-15. DOI: 10.5281/zenodo.822230.

Mahmut Mak, Hasan Altinbas (2016). Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space. International J.Math. Combin. (IJMC) 3, 1-16. DOI: 10.5281/zenodo.825056.

Suha Yilmaz (2016). Isotropic Smarandache Curves in Complex Space C3. 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 (2015). Smarandache curves of some special curves in the Galilean 3-space. arXiv:1501.05245v2 [math.DG] 19 Feb 2015, 11 p. DOI: 10.5281/zenodo.835464.

Vahide Bulut, Ali Caliskan (2015). Spherical Images of Special Smarandache Curves in E3. International J.Math. Combin. (IJMC) 3, 43-54.

Suleyman Senyurt, Abdussamet Caliskan (2015). N*C*-Smarandache Curves of Mannheim Curve Couple According to Frenet Frame. International J.Math. Combin. (IJMC) 1, 1-13.

Talat Korpinar (2015). New type surfaces in terms of B-Smarandache Curves in Sol3. Acta Scientiarum. Technology 37(3), 389-393. DOI: 10.5281/zenodo.835440.

H.S. Abdel-Aziz, M. Khalifa Saad (2015). Smarandache Curves and Spherical Indicatrices in the Galilean 3-Space. 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: 10.5281/zenodo.2987568.

Atakan Tulkan Yakut, Murat Savas, Tugba Tamirci (2014). The Smarandache Curves on S21 and Its Duality on H20. Journal of Applied Mathematics, Article ID 193586, 12 p.

Esra Betul Koc Ozturk, Ufuk Ozturk, Kazim Ilarslan, Emilija Nesovic (2014). On Pseudospherical Smarandache Curves in Minkowski 3-Space. Journal of Applied Mathematics, Article ID 404521, 14 p. DOI: 10.5281/zenodo.835443.

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 (2013). An Application of Smarandache Curve. Ordu Univ. J. Sci. Tech. 3(1), 46-60.

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

Kemal Taskopru, Murat Tosun (2012). Smarandache Curves According to Sabban Frame on S2. arXiv:1206.6229v3 [math.DG] 20 Jul 2012, 8 p.

Talat Korpinar, Essin Turhan (2012). b-Smarandache m1m2 Curves of Biharmontic New Type b-Slant Helices According to Bishop Frame in the Sol Space Sol3. International J.Math. Combin. (IJMC) 4, 33-39.

Muhammed Cetin, Yilmaz Tuncer, Murat Kemal Karacan (2011). Smarandache Curves According to Bishop Frame in Euclidean 3-Space. arXiv: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 (2010). Special Smarandache Curves in the Euclidean Space. International J.Math. Combin. (IJMC) 2, 30-36.

Melih Turgut, Suha Yilmaz (2008). Smarandache Curves in Minkowski Space-time. International J.Math. Combin. (IJMC) 3, 51-55. DOI: 10.5281/zenodo.823501.

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). Special timelike Smarandache curves in Minkowski 3-space. Al Imam Mohammad Ibn Saud Islamic University, 16 p.