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[5] viXra:2212.0188 [pdf] replaced on 2023-01-13 12:44:44
Authors: K. Idicula Koshy
Comments: 6 Pages. The article is expected to encourage further research on Ellipse Perimeter Approximation.
Abstract In this article, the author presents a new formula for Ellipse Perimeter Approximation. This formula, with two parameters, is unique in form among all published formulae on Ellipse Perimeter Approximation. Of the two parameters, one is a constant and the other is a polynomial of the aspect ratio, which is dependent on the chosen constant. We were able to reduce the Absolute Relative Error to less than 1.83 parts per million (ppm) for any ellipse, by suitable choice of the parameters.
Category: Geometry
[4] viXra:2212.0065 [pdf] submitted on 2022-12-08 02:24:10
Authors: Florentin Smarandache
Comments: 14 Pages. In Spanish
In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry results from the partial negation of one or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented.
Category: Geometry
[3] viXra:2212.0064 [pdf] submitted on 2022-12-07 07:04:15
Authors: Florentin Smarandache
Comments: 22 Pages.
In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. 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.
Category: Geometry
[2] viXra:2212.0048 [pdf] submitted on 2022-12-06 02:05:54
Authors: Antoine Warnery
Comments: 23 Pages. In French
The purpose of this paper is to present an idea of geometry described by set theory. This method can describe the axioms of the different geometric representations of space. The axioms of Euclid will be described through straight, segment or sphere subsets, for example the axiom of parallels will be described through a straight set and the definition of the acute angle. The axioms of algebra will be described in the same way using subsets of space with original properties. This description by set theory makes it possible to make a theoretical link between geometry and algebra, and to make a practical link between formulas from different mathematical universes such as trigonometry, algebra and geometry. Apart from the description of axioms, this paper makes it possible to reformulate and explain the meaning of theorems (trigonometric formula, Euler formula, etc.) in an original way, in order to find coherent and efficient methods of describing space.
Category: Geometry
[1] viXra:2212.0017 [pdf] submitted on 2022-12-03 01:41:30
Authors: Janko Kokosar
Comments: 9 Pages.
In the article, I show a visual representation of the formula $pi^3=31.00627..$, respectively, visualization how $pi^3$ is close to 31. I show this using the area of a circle with radius $pi$ that is compared with the area that is quite simply composed of squares and triangles. In the same way, the Ramanujan formula $pi^4=97.5-1/11+1.2491..x10^{-7}$ is visualized. At the end, I mention once again the challenge to explain the Ramanujan formula for $pi^4$.
Category: Geometry
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