**NeutroGeometry & AntiGeometry **

**as alternatives and generalizations **

**of Non-Euclidean Geometry**

In our real world there are many neutrosophic triplets of the form (<A>, <neutA>, <antiA>), where <A> is some entity, while <antiA> is its opposite, and <neutA> is the neutral (indeterminate) between them.

A classical Geometry structure has all axioms totally (100%) true. A NeutroGeometry structure has some axioms that are only partially true, and no axiom is totally (100%) false. Whereas an AntiGeometry structure has at least one axiom that is totally (100%) false.

Let's examine several neutrosophic triplets.

** 1. (<Structure>, <NeutroStructure>,
<AntiStructure>) in any
field of knowledge**

A Structure, in any field of knowledge, is composed of: a
non-empty __space__, populated by some __elements__, and both (the
space and all elements) are characterized by some __relations__ among
themselves (such as: operations, laws, axioms, properties, functions,
theorems, lemmas, consequences, algorithms, charts, hierarchies,
equations, inequalities, etc.), and by their __attributes__ (size,
weight, color, shape, location, etc.).

[Of course, when analysing a structure, it counts with respect to what space, elements, relations, and attributes we do it.]

(i) If all these categories (space,
elements, relations, attributes) are determinate and 100% true, we deal
with a **classical Structure. **

** ** (ii) If there is some indeterminacy
data with respect to any of these categories (space, elements,
relations, attributes), or some relation that is partially false, and no
relation that is totally false, we have a **NeutroStructure**.

** ** (iii)** **If at least one relation is
100% false, we have an **AntiStructure**.

** 2. (<Relation>, <NeutroRelation>, <AntiRelation>)**

(i) A classical **Relation** is a relation that is true for all elements
of the set (degree of truth T = 1). Neutrosophically we write Relation(1,0,0).

(ii) A **NeutroRelation** is a relation that is true for some of the
elements (degree of truth T), indeterminate for other elements (degree of
indeterminacy I), and false for the other elements (degree of falsehood F).
Neutrosophically we write Relation(T,I,F), where (T,I,F) is different from
(1,0,0) and (0,0,1).

(iii) An **AntiRelation** is a relation that is false for all elements
(degree of falsehood F = 1). Neutrosophically we write Relation(0,0,1).

** 3. (<Attribute>, <NeutroAttribute>, <AntiAttribute>)**

(i) A classical **Attribute** is an attribute that is true for all
elements of the set (degree of truth T = 1). Neutrosophically we write
Attribute(1,0,0).

(ii) A **NeutroAttribute** is an attribute that is true for some of the
elements (degree of truth T), indeterminate for other elements (degree of
indeterminacy I), and false for the other elements (degree of falsehood F).
Neutrosophically we write Attribute(T,I,F), where (T,I,F) is different from
(1,0,0) and (0,0,1).

(iii) An **AntiAttribute** is an attribute that is false for all
elements (degree of falsehood F = 1). Neutrosophically we write
Attribute(0,0,1).

** 4. (<Axiom>, <NeutroAxiom>, <AntiAxiom>)**

When we define an axiom on a given set, it does not
automatically mean that the axiom is true for all set elements. We have three possibilities again:

(i) The axiom is true for all set's elements (totally true) [degree of truth
T = 1] (as in classical algebraic structures; this is a classical **Axiom**).
Neutrosophically we write: Axiom(1,0,0).

(ii) The axiom if true for some elements [degree of truth T], indeterminate
for other elements [degree of indeterminacy I], and false for other elements
[degree of falsehood F], where (T,I,F) is different from (1,0,0) and from
(0,0,1) (this is **NeutroAxiom**). Neutrosophically we write
NeutroAxiom(T,I,F).

(iii) The axiom is false for all set's elements [degree of falsehood F =
1](this is **AntiAxiom**). Neutrosophically we write AntiAxiom(0,0,1).

** 5. (<Theorem>, <NeutroTheorem>, <AntiTheorem>)**

In any science, a classical Theorem,
defined on a given space, is a statement that is 100% true (i.e. true for all
elements of the space). To prove that a classical theorem is false, it is
sufficient to get a single counter-example where the statement is false.
Therefore, the classical sciences do not leave room for __ partial truth__ of
a theorem (or a statement). But, in our
real
world and in our everyday life, we have
many more examples of statements that are only partially true, than statements
that are totally true. The NeutroTheorem and AntiTheorem are generalizations and
alternatives of the classical Theorem in any science.

Let's consider a theorem, stated on a given set, endowed with some operation(s).
When we construct the theorem on a given set, it does not automatically mean
that the theorem is true for all set’s elements. We have three possibilities
again:

(i) The theorem is true for all set's elements [totally true] (as in
classical algebraic structures; this is a classical **Theorem**).
Neutrosophically we write: Theorem(1,0,0).

(ii) The theorem if true for some elements [degree of truth T], indeterminate
for other elements [degree of indeterminacy I], and false for the other elements
[degree of falsehood F], where (T,I,F) is different from (1,0,0) and from
(0,0,1) (this is a **NeutroTheorem**). Neutrosophically we write:
NeutroTheorem(T,I,F).

(iii) The theorem is false for all set's elements (this is an **AntiTheorem**).
Neutrosophically we write: AntiTheorem(0,0,1).

And similarly for (Lemma, NeutroLemma, AntiLemma), (Consequence, NeutroConsequence, AntiConsequence), (Algorithm, NeutroAlgorithm, AntiAlgorithm), (Property, NeutroProperty, AntiProperty), etc.

**6. (<Geometry>, <NeutroGeometry>, <AntiGeometry>)**

(i) A geometric structure who’s all axioms (and theorems,
propositions, etc.) are totally true, and there is no indeterminate data, is called a classical Geometric Structure (or **Geometry**).

(ii) A geometric structure that has at least one NeutroAxiom (and no AntiAxiom),
or some indeterminate data, is called a NeutroGeometric
Structure (or **NeutroGeometry**).

(iii) A geometric structure that has at least one AntiAxiom is called an
AntiGeometric Structure (or **AntiGeometry**).

Therefore, a neutrosophic triplet is formed: (<Geometry>,
<NeutroGeometry>,
<AntiGeometry>),

where <Geometry> can be any classical Euclidean, Projective, Affine,
Differential, Discrete etc. geometric structure.

**7. Real Examples of NeutroGeometry and
AntiGeometry**

**
**
**7.1. The NeutroGeometry is a generalization of the Hybrid
Geometry (SG) **

The Hybrid (or Smarandache) Geometries (SG), for the case when an axiom is partially validated and

partially invalidated, and as such it is partially Euclidean and partially Non-Euclidean in the same space).

Because the real geometric spaces are not pure, but hybrid, and the real rules do not uniformly apply to

all space elements, but they have degrees of diversity – applying to some geometrical concepts (point,

line, plane, surface, etc.) in a smaller or bigger degree.

From Prof. Dr. Linfan Mao’s arXiv.org paper Pseudo-Manifold Geometries with Applications,

Cornell University, New York City, USA, 2006, https://arxiv.org/abs/math/0610307: “A

Smarandache geometry is a geometry which has at least one Smarandachely denied axiom

(1969), i.e., an axiom behaves in at least two different ways within the same space, i.e.,

validated and invalided, or only invalided but in multiple distinct ways.”

**
7.2. The AntiGeometry is a generalization of the Non-Euclidean Geometries**

While the Non-Euclidean Geometries [hyperbolic and elliptic geometries] resulted from the total

negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total

negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem,

concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates,

Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of one or more axioms

a[and no total negation of no axiom] from any geometric axiomatic system [1].

** References**

1. Florentin Smarandache, NeutroGeometry & AntiGeometry are
alternatives and generalizations of the Non-Euclidean Geometries, Neutrosophic
Sets and Systems, vol. 46, 2021, pp. 456-477. DOI: __10.5281/zenodo.5553552,
http://fs.unm.edu/NSS/NeutroGeometryAntiGeometry31.pdf __

2. Carlos Granados, A
note on AntiGeometry and NeutroGeometry and their application to real life, Neutrosophic Sets
and Systems, Vol. 49, 2022, pp. 579-593. DOI: __10.5281/zenodo.6466520,
http://fs.unm.edu/NSS/AntiGeometryNeutroGeometry36.pdf __

__3. Florentin
Smarandache, Real Examples of NeutroGeometry & AntiGeometry, Neutrosophic Sets
and Systems, Vol. 55, 2023, pp. 568-575. DOI: 10.5281/zenodo.7879548__,
__
http://fs.unm.edu/NSS/ExamplesNeutroGeometryAntiGeometry35.pdf __

4. F. Smarandache, Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures [ http://fs.unm.edu/NA/NeutroAlgebraicStructures-chapter.pdf ], in his book Advances of Standard and Nonstandard Neutrosophic Theories, Pons Publishing House Brussels, Belgium, Chapter 6, pages 240-265, 2019; http://fs.unm.edu/AdvancesOfStandardAndNonstandard.pdf

5. Florentin Smarandache: NeutroAlgebra is a Generalization of Partial Algebra. International Journal of Neutrosophic Science (IJNS), Volume 2, 2020, pp. 8-17. DOI: http://doi.org/10.5281/zenodo.3989285 http://fs.unm.edu/NeutroAlgebra.pdf

6. Florentin Smarandache: Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited). Neutrosophic Sets and Systems, vol. 31, pp. 1-16, 2020. DOI: 10.5281/zenodo.3638232 http://fs.unm.edu/NSS/NeutroAlgebraic-AntiAlgebraic-Structures.pdf

7. Florentin Smarandache, Generalizations and Alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures, Journal of Fuzzy Extension and Applications (JFEA), J. Fuzzy. Ext. Appl. Vol. 1, No. 2 (2020) 85–87, DOI: 10.22105/jfea.2020.248816.1008 http://fs.unm.edu/NeutroAlgebra-general.pdf

8. F. Smarandache, Structure, NeutroStructure, and AntiStructure in Science, International Journal of Neutrosophic Science (IJNS), Volume 13, Issue 1, PP: 28-33, 2020; http://fs.unm.edu/NeutroStructure.pdf

9. F. Smarandache, Universal NeutroAlgebra and Universal AntiAlgebra, Chapter 1, pp. 11-15, in the collective book NeutroAlgebra Theory, Vol. 1, edited by F. Smarandache, M. Sahin, D. Bakbak, V. Ulucay, A. Kargin, Educational Publ., Grandview Heights, OH, United States, 2021, http://fs.unm.edu/NA/UniversalNeutroAlgebra-AntiAlgebra.pdf

10. L. Mao, Smarandache Geometries & Map Theories with Applications (I), Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing, P. R. China, 2006, http://fs.unm.edu/CombinatorialMaps.pdf

11. Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries (first edition - postdoctoral report to Chinese Academy of Mathematics and System Science, Beijing, China; and second editions - graduate textbooks in mathematics), 2005 and 2011, http://fs.unm.edu/Linfan.pdf, http://fs.unm.edu/Linfan2.pdf

12. L. Mao, Combinatorial Geometry with Applications to Field Theory (second edition), graduate textbook in mathematics, Chinese Academy of Mathematics and System Science, Beijing, China, 2011, http://fs.unm.edu/CombinatorialGeometry2.pdf

13. Yuhua Fu, Linfan Mao, and Mihaly Bencze, Scientific Elements - Applications to Mathematics, Physics, and Other Sciences (international book series): Vol. 1, ProQuest Information & Learning, Ann Arbor, MI, USA, 2007, http://fs.unm.edu/SE1.pdf

14. Howard Iseri, Smarandache Manifolds, ProQuest Information & Learning, Ann Arbor, MI, USA, 2002, http://fs.unm.edu/Iseri-book.pdf

15. Linfan Mao, Smarandache Multi-Space Theory (partially post-doctoral research for the Chinese Academy of Sciences), Academy of Mathematics and Systems Chinese Academy of Sciences Beijing, P. R. China, 2006, http://fs.unm.edu/S-Multi-Space.pdf

16. Yanpei Liu, Introductory Map Theory, ProQuest Information & Learning, Michigan, USA, 2010, http://fs.unm.edu/MapTheory.pdf

17. L. Kuciuk & M. Antholy, An Introduction to the Smarandache Geometries, JP Journal of Geometry & Topology, 5(1), 77-81, 2005, http://fs.unm.edu/IntrodSmGeom.pdf

18. S. Bhattacharya, A Model to A Smarandache Geometry, http://fs.unm.edu/ModelToSmarandacheGeometry.pdf

19. Howard Iseri, A Classification of s-Lines in a Closed s-Manifold, http://fs.unm.edu/Closed-s-lines.pdf

20. Howard Iseri, Partially Paradoxist Smarandache Geometries, http://fs.unm.edu/Howard-Iseri-paper.pdf

21. Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Paradoxist Geometry", Bulletin of Pure and Applied Sciences, Delhi, India, Vol. 17E, No. 1, 123-1124, 1998.

22. David E. Zitarelli, Reviews, Historia Mathematica, PA, USA, Vol. 24, No. 1, p. 114, #24.1.119, 1997.

23. Marian Popescu, "A Model for the Smarandache Paradoxist Geometry", Abstracts of Papers Presented to the American Mathematical Society Meetings, Vol. 17, No. 1, Issue 103, 1996, p. 265.

24. Popov, M. R., "The Smarandache Non-Geometry", Abstracts of Papers Presented to the American Mathematical Society Meetings, Vol. 17, No. 3, Issue 105, 1996, p. 595.

25. Brown, Jerry L., "The Smarandache Counter-Projective Geometry", Abstracts of Papers Presented to the American Mathematical Society Meetings, Vol. 17, No. 3, Issue 105, 595, 1996.

26. F. Smarandache, Degree of Negation of Euclid's Fifth Postulate, http://fs.unm.edu/DegreeOfNegation.pdf

27. M. Antholy, An Introduction to the Smarandache Geometries, New Zealand Mathematics Colloquium, Palmerston North Campus, Massey University, 3-6 December 2001. http://fs.unm.edu/IntrodSmGeom.pdf

28. L. Mao, 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, English Chinese bilingual, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing, P. R. China, http://fs.unm.edu/LetsFlyByWind-ed3.pdf

29. Ovidiu Sandru, Un model simplu de geometrie Smarandache construit exclusiv cu elemente de geometrie euclidiană, http://fs.unm.edu/OvidiuSandru-GeometrieSmarandache.pdf

30; L. Mao, A new view of combinatorial maps by Smarandache's notion, Cornell University, New York City, USA, 2005, https://arxiv.org/pdf/math/0506232

31. Linfan Mao, A generalization of Stokes theorem on combinatorial manifolds, Cornell University, New York City, USA, 2007, https://arxiv.org/abs/math/0703400

32. Linfan Mao, Combinatorial Speculations and the Combinatorial Conjecture for Mathematics, Cornell University, New York City, USA, 2006, https://arxiv.org/pdf/math/0606702

33. Linfan Mao, Geometrical Theory on Combinatorial Manifolds, Cornell University, New York City, USA, 2006, https://arxiv.org/abs/math/0612760

34. Linfan Mao, Parallel bundles in planar map geometries, Cornell University, New York City, USA, 2005, https://arxiv.org/pdf/math/0506386

35. Linfan Mao, Pseudo-Manifold Geometries with Applications, Cornell University, New York City, USA, 2006, Paper’s abstract: https://arxiv.org/abs/math/0610307, Full paper: https://arxiv.org/pdf/math/0610307

36. F. Smarandache, Indeterminacy in Neutrosophic Theories and their Applications, International Journal of Neutrosophic Science (IJNS), Vol. 15, No. 2, PP. 89-97, 2021, http://fs.unm.edu/Indeterminacy.pdf

37. Cornel Gingăraşu, Arta Paradoxistă, Sud-Est Forum, Magazine for photography, culture and visual arts, https://sud-estforum.ro/wp/2020/06/paradoxismul-curent-cultural-romanesc-sau-cand-profu-de-mate-isi-gaseste-libertatea-inpoezie

38. Erick Gonzalez-Caballero. Applications of NeutroGeometry and AntiGeometry in Real World. Journal of

International Journal of Neutrosophic Science, (2023); 21 ( 1 ): 14-32,

http://fs.unm.edu/NeutroGeometryInRealWorld.pdf

All articles, books, and other materials from this website are licensed under a
**
Creative Common
Attribute 4,0 International License**,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.