Neutrosophic Sets and Systems, Vol. 46, 2021 _____________________________________________________________________________________ University of New Mexico NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Florentin Smarandache University of New Mexico Mathematics, Physical and Natural Science Division 705 Gurley Ave., Gallup, NM 87301, USA Email: [email protected] Abstract In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric space, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean 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 even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.), and the NeutroAxiom results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. Keywords: Non-Euclidean Geometries, Euclidean Geometry, Lobachevski-Bolyai-Gauss Geometry, Riemannian Geometry, NeutroManifold, AntiManifold, NeutroAlgebra, AntiAlgebra, NeutroGeometry, AntiGeometry, NeutroAxiom, AntiAxiom, Partial Function, NeutroFunction, AntiFunction, NeutroOperation, AntiOperation, NeutroAttribute, AntiAttribute, NeutroRelation, AntiRelation, NeutroStructure, AntiStructure ______________________________________________________________________________ 1. Introduction In our real world, the spaces are not homogeneous, but mixed, complex, even ambiguous. And the elements that populate them and the rules that act upon them are not perfect, uniform, or complete - but fragmentary and disparate, with unclear and conflicting information, and they do not apply in the same degree to each element. The perfect, idealistic ones exist just in the theoretical sciences. We live in a multi-space endowed with a multi-structure [35]. Neither the space’s elements nor the regulations that govern them are egalitarian, all _____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 457 _____________________________________________________________________________________ of them are characterized by degrees of diversity and variance. The indeterminate (vague, unclear, incomplete, unknown, contradictory etc.) data and procedures are surrounding us. That’s why, for example, the classical algebraic and geometric spaces and structures were extended to more realistic spaces and structures [1], called respectively NeutroAlgebra & AntiAlgebra [2019] and respectively NeutroGeometry & AntiGeometry [1969, 2021], whose elements do not necessarily behave the same, while the operations and rules onto these spaces may only be partially (not totally) true. While the Non-Euclidean 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 even of more axioms 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 [and no total negation of no axiom] from any geometric axiomatic system. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. On a given space, a classical Axiom is totally (100%) true. While a NeutroAxiom is partially true, partially indeterminate, and partially false. Also, an AntiAxiom is totally (100%) false. A classical Geometry has only totally true Axioms. While a NeutroGeometry is a geometry that has at least one NeutroAxiom and no AntiAxiom. Also, an AntiGeometry is a geometry that has at least one AntiAxiom. Below we introduce, in the first part of this article, the construction of NeutroGeometry & AntiGeometry, together with the Non-Euclidean geometries, while in the second part we recall the evolution from paradoxism to neutrosophy, and then to NeutroAlgebra & AntiAlgebra, culminating with the most general form of NeutroStructure & AntiStructure in any field of knowledge. A classical (100%) true statement on a given classical structure, may or may not be 100% true on its corresponding NeutroStructure or AntiStructure, it depends on the neutrosophication or antisophication procedures [1 – 24]. Further on, the neutrosophic triplet (Algebra, NeutroAlgebra, AntiAlgebra) was restrained or extended to all fuzzy and fuzzy extension theories (FET) triplets of the form (Algebra, NeutroFETAlgebra, AntiFETAlgebra), where FET may be: Fuzzy, Intuitionistic Fuzzy, Inconsistent Intuitionistic Fuzzy (Picture Fuzzy, Ternary Fuzzy), Pythagorean Fuzzy (Atanassov’s Intuitionistic Fuzzy of second type), q-Rung Orthopair Fuzzy, Spherical Fuzzy, n-HyperSpherical Fuzzy, Refined Neutrosophic, etc. 1.1. Concept, NeutroConcept, AntiConcept Let us consider on a given geometric space a classical geometric concept (such as: axiom, postulate, operator, transformation, function, theorem, property, theory, etc.). ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 458 _____________________________________________________________________________________ We form the following geometric neutrosophic triplet: Concept(1, 0, 0), NeutroConcept(T, I, F), AntiConcept (0, 0, 1), where (T, I, F) ∉ {(1, 0, 0), (0, 0, 1)}. { Of course, we consider only the neutrosophic triplets (Concept, NeutroConcept, AntiConcept) that make sense in our everyday life and in the real world. } Concept(1, 0, 0) means that the degree of truth of the concept is T = 1, I = 0, F = 0, or the Concept is 100% true, 0% indeterminate, and 0% false in the given geometric space. NeutroConcept (T, I, F) means that the concept is T% true, I% indeterminate, and 0% false in the given geometric space, with (T, I, F) ∈ [0, 1], and (T, I, F) ∉ {(1, 0, 0), (0, 0, 1)}. AntiConcept (0, 0, 1) means that T = 0, I = 0, and F = 1, or the Concept is 0% true, 0% indeterminate, and 100% false in the given geometric space. 1.2. Geometry, NeutroGeometry, AntiGeometry We go from the neutrosophic triplet (Algebra, NeutroAlgebra, AntiAlgebra) to a similar neutrosophic triplet (Geometry, NeutroGeometry, AntiGeometry), in the same way. Correspondingly from the algebraic structuires, with respect to the geometries, one has: In the classical (Euclidean) Geometry, on a given space, all classical geometric Concepts are 100% true (i.e. true for all elements of the space). While in a NeutroGeometry, on a given space, there is at least one NeutroConcept (and no AntiConcept). In the AntiGeometry, on a given space, there is at least one AntiConcept. 1.3. Geometric NeutroSophication and Geometric AntiSophication Similarly, as to the algebraic structures, using the process of NeutroSophication of a classical geometric structure, a NeutroGeometry is produced; while through the process of AntiSophication of a classical geometric structure produces an AntiGeometry. Let S be a classical geometric space, and <A> be a geometric concept (such as: postulate, axiom, theorem, property, function, transformation, operator, theory, etc.). The <antiA> is the opposite of <A>, while <neutA> (also called <neutroA>) is the neutral (or indeterminate) part between <A> and <antiA>. The neutrosophication tri-sections S into three subspaces: - the first subspace, denoted just by <A>, where the geometric concept is totally true [degree of truth T = 1]; we denote it by Concept(1,0,0). ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 459 _____________________________________________________________________________________ - the second subspace, denoted by <neutA>, where the geometric concept is partially true [degree of truth T], partially indeterminate [degree of indeterminacy I], and partially false [degree of falsehood F], denoted as NeutroConcept(T,I,F), where (T, I, F)  {(1,0,0), (0,0,1)}; - the third subspace, denoted by <antiA>, where the geometric concept is totally false [degree of falsehood F = 1], denoted by AntiConcept(0,0,1). The three subspaces may or may not be disjoint, depending on the application, but they are exhaustive (their union equals the whole space S). 1.4. Non-Euclidean Geometries 1.4.1. The Lobachevsky (also known as Lobachevsky-Bolyai-Gauss) Geometry, and called Hyperbolic Geometry, is an AntiGeometry, because the Fifth Euclidean Postulate (in a plane, through a point outside a line, only one parallel can be drawn to that line) is 100% invalidated in the following AntiPostulate (first version) way: in a plane through a point outside of a line, there can be drawn infinitely many parallels to that line. Or (T, I, F) = (0, 0, 1). 1.4.2. The Riemannian Geometry, which is called Elliptic Geometry, is an AntiGeometry too, since the Fifth Euclidean Postulate is 100% invalidated in the following AntiPostulate (second version) way: in a place, through a point outside of a line, no parallel can be drawn to that line. Or (T, I, F) = (0, 0, 1). 1.4.3. The Smarandache Geometries (SG) are more complex [30 – 57]. Why this type of mixed non- Euclidean geometries, and sometimes partially Non-Euclidean and partially Euclidean? Because the real geometric spaces are not pure but hybrid, and the real rules do not uniformly apply to all space’s 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 [57], 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 and a Smarandache n-manifold is a n-manifold that support a Smarandache geometry. Iseri provided a construction for Smarandache 2-manifolds by equilateral triangular disks on a plane and a more general way for Smarandache 2-manifolds on surfaces, called map geometries was presented by the author (…). However, few observations for cases of n ≥ 3 are found on the journals. As a kind of Smarandache geometries, a general way for constructing dimensional n pseudo-manifolds are presented for any integer n ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 460 _____________________________________________________________________________________ ≥ 2 in this paper. Connection and principal fiber bundles are also defined on these manifolds. Following these constructions, nearly all existent geometries, such as those of Euclid geometry, Lobachevshy-Bolyai geometry, Riemann geometry, Weyl geometry, Kahler geometry and Finsler geometry, etc. are their sub- geometries.” Iseri ([34], [39 - 40]) has constructed some Smarandache Manifolds (S-manifolds) that topologically are piecewise linear, and whose geodesics have elliptic, Euclidean, and hyperbolic behavior. An SG geometry may exhibit one or more types of negative, zero, or positive curvatures into the same given space. 1.4.3.1) If at least one axiom is validated (partially true, T > 0) and invalidated (partially false, F > 0), and no other axiom is only invalidated (AntiAxiom), then this first class of SG geometry is a NeutroGeometry. 1.4.3.2) If at least one axiom is only invalidated (or F = 1), no matter if the other axioms are classical or NeutroAxioms or AntiAxioms too, then this second class of SG geometry is an AntiGeometry. 1.4.3.3) The model of an SG geometry that is a NeutroGeometry: Bhattacharya [38] has constructed the following SG model: Fig. 1. Bhattacharya’s Model for the SG geometry as a NeutroGeometry The geometric space is a square ABCD, comprising all points inside and on its edges. ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 461 _____________________________________________________________________________________ “Point” means the classical point, for example: A, B, C, D, E, N, and M. “Line” means any segment of line connecting two points on the opposite square sides AC and BD, for example: AB, CD, CE, (u), and (v). “Parallel lines” are lines that do not intersect. Let us take a line CE and an exterior point N to it. We observe that there is an infinity of lines passing through N and parallel to CE [all lines passing through N and in between the lines (u) and (v) for example] – the hyperbolic case. Also, taking another exterior point, D, there is no parallel line passing through D and parallel to CE because all lines passing through D intersects CE – the elliptic case. Taking another exterior point M ∈ AB, then we only have one line AB parallel to CE, because only one line passes through the point M – the Euclidean case. Consequently, the Fifth Euclidean Postulate is twice invalidated, but also once validated. Being partially hyperbolic Non-Euclidean, partially elliptic Non-Euclidean, and partially Euclidean, therefore we have here a SG. This is not a Non-Euclidean Geometry (since the Euclid’s Fifth Postulate is not totally false, but only partially), but it is a NeutroGeometry. Theorem 1.4.3.3.1 If a statement (proposition, theorem, lemma, property, algorithm, etc.) is (totally) true (degree of truth T = 1, degree of indeterminacy I = 0, and degree of falsehood F = 0) in the classical geometry, the statement may get any logical values (i.e. T, I, F may be any values in [0, 1]) in a NeutroGeometry or in an AntiGeometry Proof. The logical value the statement gets in a NeutroGeometry or in an AntiGeometry depends on what classical axioms the statement is based upon in the classical geometry, and how these axioms behave in the NeutroGeometry or AntiGeometry models. Let’s consider the below classical geometric proposition P(L1, L2, L3) that is 100% true: In a 2D-Euclidean geometric space, if two lines L1 and L2 are parallel with the third line L3, then they are also parallel (i.e. L1 // L2). In Bhattacharya’s Model of an SG geometry, this statement is partially true and partially false. For example, in Fig. 1: - degree of truth: the lines AB and (u) are parallel to the line CE, then AB is parallel to (u); - degree of falsehood: the lines (u) and (v) are parallel to the line CE, but (u) and (v) are not parallel since they intersect in the point N. ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 462 _____________________________________________________________________________________ 1.4.3.4) The Model of a SG geometry that is an AntiGeometry Let us consider the following rectangular piece of land PQRS, Fig. 2. Model for an SG geometry that is an AntiGeometry whose middle (shaded) area is an indeterminate zone (a river, with swamp, canyons, and no bridge) that is impossible to cross over on the ground. Therefore, this piece of land is composed from a determinate zone and an indeterminate zone (as above). “Point” means any classical (usual) point, for example: P, Q, R, S, X, Y, Z, and W that are determinate well-known (classical) points, and I1, I2 that are indeterminate (not well-known) points [in the indeterminate zone]. “Line” is any segment of line that connects a point on the side PQ with a point on the side RS. For example, PR, QS, XY. However, these lines have an indeterminate (not well known, not clear) part that is the indeterminate zone. On the other hand, ZW is not a line since it does not connect the sides PQ and RS. ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 463 _____________________________________________________________________________________ The following geometric classical axiom: through two distinct points there always passes one single line, is totally (100%) denied in this model in the following two ways: through any two distinct points, in this given model, either no line passes (see the case of ZW), or only one partially determinate line does (see the case of XY) - therefore no fully determinate line passes. Thus, this SG geometry is an AntiGeometry. 1.5. Manifold, NeutroManifold, AntiManifold 1.5.1. Manifold The classical Manifold [29] is a topological space that, on the small scales, near each point, resembles the classical (Euclidean) Geometry Space [i.e. in this space there are only classical Axioms (totally true)]. Or each point has a neighborhood that is homeomorphic to an open unit ball of the Euclidean Space Rn (where R is the set of real numbers). Homeomorphism is a continuous and bijective function whose inverse is also continuous. “In general, any object that is near ‘flat’ on the small scale is a manifold” [29]. 1.5.2. NeutroManifold The NeutroManifold is a topological space that, on the small scales, near each point, resembles the NeutroGeometry Space [i.e. in this space there is at least a NeutroAxiom (partially true, partially indeterminate, and partially false) and no AntiAxiom]. For example, Bhattacharya’s Model for a SG geometry (Fig. 1) is a NeutroManifold, since the geometric space ABCD has a NeutroAxiom (i.e. the Fifth Euclidean Postulate, which is partially true and partially false), and no AntiAxiom. 1.5.3. AntiManifold The AntiManifold is a topological space that, on the small scales, near each point, resembles the AntiGeometry Space [i.e. in this space there is at least one AntiAxiom (totally false)]. For example, the Model for a SG geometry (Fig. 2) is an AntiManifold, since the geometric space PQRS has an AntiAxiom (i.e., through two distinct points there always passes a single line - which is totally false). *** 2. Evolution from Paradoxism to Neutrosophy then to NeutroAlgebra/AntiAlgebra and now to NeutroGeometry/AntiGeometry Below we recall and revise the previous foundations and developments that culminated with the introduction of NeutroAlgebra & AntiAlgebra as new field of research, extended then to NeutroStructure ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 464 _____________________________________________________________________________________ & AntiStructure, and now particularized to NeutroGeometry & AntiGeometry that are extensions of the Non-Euclidean Geometries. 2.1. From Paradoxism to Neutrosophy Paradoxism [58] is an international movement in science and culture, founded by Smarandache in 1980s, based on excessive use of antitheses, oxymoron, contradictions, and paradoxes. During three decades (1980-2020) hundreds of authors from tens of countries around the globe contributed papers to 15 international paradoxist anthologies. In 1995, he extended the paradoxism (based on opposites) to a new branch of philosophy called neutrosophy (based on opposites and their neutral) [59], that gave birth to many scientific branches, such as: neutrosophic logic, neutrosophic set, neutrosophic probability, neutrosophic statistics, neutrosophic algebraic structures, and so on with multiple applications in engineering, computer science, administrative work, medical research, social sciences, etc. Neutrosophy is an extension of Dialectics that have derived from the Yin-Yan Ancient Chinese Philosophy. 2.2. From Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures In 2019 Smarandache [1] generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) {whose operations and axioms are partially true, partially indeterminate, and partially false} as extensions of Partial Algebra, and to AntiAlgebraic Structures (or AntiAlgebras) {whose operations and axioms are totally false} and on 2020 he continued to develop them [2,3,4]. The NeutroAlgebras & AntiAlgebras are a new field of research, which is inspired from our real world. In classical algebraic structures, all operations are 100% well-defined, and all axioms are 100% true, but in real life, in many cases these restrictions are too harsh, since in our world we have things that only partially verify some operations or some laws. By substituting Concept with Operation, Axiom, Theorem, Relation, Attribute, Algebra, Structure etc. respectively, into the above (Concept, NeutroConcept, AntiConcept), we get the below neutrosophic triplets: 2.3. Operation, NeutroOperation, AntiOperation When we define an operation on a given set, it does not automatically mean that the operation is well- defined. There are three possibilities: 1) The operation is well-defined (also called inner-defined) for all set's elements [degree of truth T = 1] (as in classical algebraic structures; this is a classical Operation). Neutrosophically we write: Operation(1,0,0). 2) The operation if well-defined for some elements [degree of truth T], indeterminate for other elements [degree of indeterminacy I], and outer-defined 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 NeutroOperation). Neutrosophically we write: ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries Neutrosophic Sets and Systems, Vol. 46, 2021 465 _____________________________________________________________________________________ NeutroOperation(T,I,F). 3) The operation is outer-defined for all set's elements [degree of falsehood F = 1] (this is an AntiOperation). Neutrosophically we write: AntiOperation(0,0,1). An operation * on a given non-empty set S is actually a n-ary function, for integer n ≥ 1, f :Sn →S. 2.4. Function, NeutroFunction, AntiFunction Let U be a universe of discourse, A and B be two non-empty sets included in U, and f be a function: f :A→B Again, we have three possibilities: 1) The function is well-defined (also called inner-defined) for all elements of its domain A [degree of truth T = 1] (this is a classical Function), i.e. xA, f(x)B. Neutrosophically we write: Function(1,0,0). 2) The function if well-defined for some elements of its domain, i.e. xA, f(x)B [degree of truth T], indeterminate for other elements, i.e. xA, f(x)=indeterminate [degree of indeterminacy I], and outer-defined for the other elements, i.e. xA, f(x)B [degree of falsehood F], where (T,I,F) is different from (1,0,0) and from (0,0,1). This is a NeutroFunction. Neutrosophically we write: NeutroFunction(T,I,F). 3) The function is outer-defined for all elements of its domain A [degree of falsehood F = 1] (this is an AntiFunction), i.e. xA, f(x)B(all function’s values are outside of its codomain B; they may be outside of the universe of discourse too). Neutrosophically we write: AntiFunction(0,0,1). 2.5. NeutroFunction & AntiFunction vs. Partial Function We prove that the NeutroFunction & AntiFunction are extensions and alternatives of the Partial Function. Definition of Partial Function [60] A function f: A→B is sometimes called a total function, to signify that f(a) is defined for every a ∈ A. If C is any set such that C ⊇ A then f is also a partial function from C to B. Clearly if f is a function from A to B then it is a partial function from A to B, but a partial function need not be defined for every element of its domain. The set of elements of A for which f is defined is sometimes called the domain of definition. From other sites, the Partial Function means: for any a ∈ A one has: f(a) ∈ B or f(a) = undefined. Comparison ____________________________________________________________________________________ Florentin Smarandache, NeutroGeometry & AntiGeometry are alternatives and generalizations of the Non-Euclidean Geometries