Geometry

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Recent submissions

Any replacements are listed further down

[165] viXra:1407.0198 [pdf] submitted on 2014-07-25 22:09:36

Launching the Chaotic Realm of Iso-Fractals: a Short Remark

Authors: Nathan O. Schmidt, Reza Katebi, Christian Corda
Comments: 5 pages, 1 figure, accepted in the AIP Conference Proceedings of ICNAAM 2014

In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci's numbers and (the complex moduli of) Mandelbrot's numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin's holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli's iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
Category: Geometry

[164] viXra:1407.0163 [pdf] submitted on 2014-07-21 13:47:46

Moments Defined by Subdivision Curves

Authors: Jan Hakenberg, Ulrich Reif, Scott Schaefer, Joe Warren
Comments: 19 Pages.

We derive the (d+2)-linear forms that compute the moment of degree d of the area enclosed by a subdivision curve in the plane. We circumvent the need to solve integrals involving the basis function by exploiting a recursive relation and calibration that establishes the coefficients of the form within the nullspace of a matrix. For demonstration, we apply the technique to the dual three-point scheme, the interpolatory C1 four-point scheme, and the dual C2 four-point scheme.
Category: Geometry

[163] viXra:1407.0027 [pdf] submitted on 2014-07-03 12:14:35

Area Moments Defined by Example Subdivision Curves

Authors: Jan Hakenberg
Comments: 23 Pages.

We list examples of subdivision curves together with their exact area, centroid, and inertia. We assume homogeneous mass-distribution within the space bounded by the curve, therefore the term 'area moments' is used. The subdivision curves that we consider are generated by 1) the low order B-spline schemes, 2) the generalized, interpolatory C^1 four-point scheme, as well as 3) the more recent, dual C^2 four-point scheme. The derivation of the (d+1)-linear form that computes the area moment of degree p+q=d based on the initial control points for a given subdivision scheme is deferred to a publication in the near future.
Category: Geometry

[162] viXra:1406.0165 [pdf] submitted on 2014-06-27 02:44:50

Latest Trends in Spherical Trigonometry

Authors: S.kalimuthu
Comments: 4 Pages. NA

According to Einstein and his followers space time geometry is gravity. Gravity is the manifestation of distortion of geometry of space due to presence of matter. The heart of these physical and cosmological phenomena is the line element or metric. This metric generated the field equation of Einstein general relative theory. Space time curvature, geodetic effect, frame tracking, gravitational lenses, gravitational red and blue shifts, block holes, dark matter, dark energy, big bang singularity, expansion of the universe and gravitational waves are the predictions of Einstein general relative theory. All these theoretical findings expect gravitational waves have been experimental test at to a very high degree of accuracy. In this work, the authors introduce an entirely new type of polar spherical triangle. The application of this triangle has been extended to Gabuzda- Wardle-Roberts superluminal motion equation and the consecution is noted
Category: Geometry

[161] viXra:1406.0060 [pdf] submitted on 2014-06-10 08:09:18

Volume Enclosed by Subdivision Surfaces with Sharp Creases

Authors: Jan Hakenberg, Ulrich Reif, Scott Schaefer, Joe Warren
Comments: 14 Pages.

Subdivision surfaces with sharp creases are used in surface modeling and animation. The framework that derives the volume formula for classic surface subdivision also applies to the crease rules. After a general overview, we turn to the popular Catmull-Clark, and Loop algorithms with sharp creases. We enumerate common topology types of facets adjacent to a crease. We derive the trilinear forms that determine their contribution to the global volume. The mappings grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex.
Category: Geometry

[160] viXra:1405.0324 [pdf] submitted on 2014-05-26 14:33:21

Volume Enclosed by Example Subdivision Surfaces with Sharp Creases

Authors: Jan Hakenberg
Comments: 31 Pages.

The formula for the volume enclosed by subdivision surfaces has been identified only recently. We present example meshes with cycles of edges defined as sharp creases, and state the volume enclosed by their limit surface defined by Catmull-Clark, and Loop subdivision. The article can serve as a reference for future implementations of the volume formula.
Category: Geometry

[159] viXra:1405.0260 [pdf] submitted on 2014-05-18 00:57:53

A Phenomenon in Great Circle Triangles

Authors: S.Kalimuthu
Comments: 5 Pages. NA

Great circle triangles and its related trigonometry are wider applications in astronomy, astrophysics, cosmology, engineering fields, space travel, sea voyages, electronics, architecture etc. Maxwell’s electromagnetic theory showed that light is an electromagnetic wave, Dirac’s equation revealed the existence and generation of anti particles and Einstein’s filed equations predicted bending of light rays near a massive body, gravitational time dilation, gravitational waves , gravitational lenses, black holes, dark matter, dark energy and big bang singularity. All these findings have been experimentally established except gravitational waves. In this short work, the author finds a peculiar phenomenon in great circle triangles / Euler triangles
Category: Geometry

[158] viXra:1405.0246 [pdf] submitted on 2014-05-15 04:16:44

Volume Enclosed by Example Subdivision Surfaces

Authors: Jan Hakenberg
Comments: 28 Pages.

Simple meshes such as the cube, tetrahedron, and tripod frequently appear in the literature to illustrate the concept of subdivision. The formula for the volume enclosed by subdivision surfaces has only recently been identified. We specify simple meshes and state the volume enclosed by the corresponding limit surfaces. We consider the subdivision schemes Doo-Sabin, Midedge, Catmull-Clark, and Loop.
Category: Geometry

[157] viXra:1405.0215 [pdf] submitted on 2014-05-11 19:02:40

Un Model de Geometrie Smarandache Care Unește Geometria Euclidiană cu Geometria Neeuclidiană Eliptică și Geometria Neeuclidiană Hiperbolică

Authors: Ion Patrascu
Comments: 2 Pages.

O geometrie Smarandache este o geometrie în care cel puțin o axiomă este fie validată și invalidată, sau numai invalidată dar în multiple feluri (în cadrul aceluiași spațiu geometric). Vom construi un model de geometrie Smarandache în care axioma paralelelor este validată pentru unele drepte și puncte, și invalidată pentru alte drepte și puncte.
Category: Geometry

[156] viXra:1405.0012 [pdf] submitted on 2014-05-02 10:39:37

Volume Enclosed by Subdivision Surfaces

Authors: Jan Hakenberg, Ulrich Reif, Scott Schaefer, Joe Warren
Comments: 15 Pages.

We present a framework to derive the coefficients of trilinear forms that compute the exact volume enclosed by subdivision surfaces. The coefficients depend only on the local mesh topology, such as the valence of a vertex, and the subdivision rules. The input to the trilinear form are the initial control points of the mesh.
Our framework allows us to explicitly state volume formulas for surfaces generated by the popular subdivision algorithms Doo-Sabin, Catmull-Clark, and Loop. The trilinear forms grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex.
The approach extends to higher order momentums such as the center of gravity, and the inertia of the volume enclosed by subdivision surfaces.
Category: Geometry

[155] viXra:1404.0409 [pdf] submitted on 2014-04-18 01:04:07

On the System Analysis of the Foundations of Trigonometry

Authors: Temur Z. Kalanov
Comments: 22 Pages.

Analysis of@@ the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Category: Geometry

[154] viXra:1404.0205 [pdf] submitted on 2014-04-16 06:20:33

Mixt-Linear Circles Adjointly Ex-Inscribed Associated to a Triangle

Authors: Ion Patrascu, Florentin Smarandache
Comments: 6 Pages.

In [1] we introduced the mixt-linear circles adjointly inscribed associated to a triangle, with emphasizes on some of their properties. Also, we’ve mentioned about mixt-linear circles adjointly ex-inscribed associated to a triangle. In this article we’ll show several basic properties of the mixt-linear circles adjointly exinscribed associate to a triangle.
Category: Geometry

[153] viXra:1404.0125 [pdf] submitted on 2014-04-15 04:26:52

The Analysis Techniques for Convexity: Cat-Spaces (3)

Authors: Cheng Tianren
Comments: 4 Pages.

we consider the global differential geometry of polar closed convex cueves in the spherical. we study the regularity of geodesic Ptolemy spaces and apply our findings to metric fixed point theory. we provide examples of non-locally compact geodesic Ptolemy metric space which are not uniquely geodesic.
Category: Geometry

[152] viXra:1404.0080 [pdf] submitted on 2014-04-11 00:15:03

The Generalized Helices of Consecutively Order is Mannheim Pairs

Authors: Abel Cavaşi
Comments: 2 Pages.

A relationship between generalized helices and Mannheim pairs.
Category: Geometry

[151] viXra:1404.0018 [pdf] submitted on 2014-04-02 21:22:03

Equidistant Curve Coordinate System(inversions(9))

Authors: Morio Kikuchi
Comments: 12 Pages.

We generalize inversion mathematically(4).
Category: Geometry

[150] viXra:1402.0024 [pdf] submitted on 2014-02-04 22:32:15

The Ptolemy Theorem in Conics (2)

Authors: Cheng Tianren
Comments: 16 Pages.

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. we study kiepert triangles and their iterations ,the kiepert triangles relative to kiepert triangle .for arbitrary and ,we show that .we also introduce the parasix configuration ,which consists of two congruent triangles. at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category: Geometry

[149] viXra:1402.0013 [pdf] submitted on 2014-02-02 19:55:25

Equidistant Curve Coordinate System(inversions(8))

Authors: Morio Kikuchi
Comments: 8 Pages.

We generalize inversion mathematically(3).
Category: Geometry

[148] viXra:1402.0001 [pdf] submitted on 2014-02-01 04:48:44

The Ptolemy Theorem in Conics (1)

Authors: Cheng Tianren
Comments: 20 Pages.

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. this paper use the barycentric coordinate of a circle to study the lester circle,and we give some applications of these coordinates.we also prove two conditions for a tangential quadrilateral to be cyclic.at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category: Geometry

[147] viXra:1401.0219 [pdf] submitted on 2014-01-29 16:43:10

A New Slant on Lebesgue’s Universal Covering Problem

Authors: Philip E Gibbs
Comments: 24 Pages.

Lebesgue’s universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypothesis based on the conjectures. A new upper bound of 0.844112 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.
Category: Geometry

[146] viXra:1401.0210 [pdf] submitted on 2014-01-29 02:10:07

Application of the Ptolemy Theorem (2)

Authors: Cheng Tianren
Comments: 23 Pages.

We study the figure of a triangle with a rectangle attached to each side.and we prove some interesting results on inscribed triangles which are isotomic,where we apply the proof of the harcourt’s theorem.we also strengthen floor van lamoen’s theorem that the 6 circumcenters of the cevasix configuration of the centroid of a triangle are concyclic by giving a proof shows that the converse is also true.at last,we apply the three results we get to the malfatti circles and the lucas circles.
Category: Geometry

[145] viXra:1401.0206 [pdf] submitted on 2014-01-28 07:28:58

Reading Report On Differential Forms

Authors: Ren Shiquan
Comments: 16 Pages. this is a reading report which may include mistakes. Thanks

In this report, we study differential forms on a manifold M. We first give the definition of differential forms. Then the exterior derivative, Lie derivative, and integrations of differential forms are discussed. Finally we will look at a special family of differential forms, called harmonic forms. This report is a preparation for de Rham cohomology and Hodge theorem that will be studied in the second report on topology of manifolds.
Category: Geometry

[144] viXra:1401.0186 [pdf] submitted on 2014-01-27 02:24:27

Application of the Ptolemy Theorem (1)

Authors: Cheng Tianren
Comments: 24 Pages.

We give a simple construction of the Apollonius circle without invoking the excircles. and we give a construction of the circular hull of the excircles of a triangle as a tucker circle.and we also give an example to describe the generalized Ptolemy theorem,and its application to the Lester circle.
Category: Geometry

[143] viXra:1401.0131 [pdf] submitted on 2014-01-17 19:37:45

A Proof of the Kepler’s Conjecture

Authors: Zhang Tianshu
Comments: 16 Pages.

Heap together equivalent spheres into a cube up to most possible, then variant general volumes of equivalent spheres inside the cube depend on variant arrangements of equivalent spheres fundamentally. This π/√18 which the Kepler’s conjecture mentions is the ratio of the general volume of equivalent spheres under the maximum to the volume of the cube. We will do a closer arrangement of equivalent spheres inside a cube. Further let a general volume of equivalent spheres to getting greater and greater, up to tend upwards the super-limit, in pace with which each of equivalent spheres is getting smaller and smaller, and their amount is getting more and more. We will prove the Kepler’s conjecture by such a way in this article.
Category: Geometry

[142] viXra:1401.0011 [pdf] submitted on 2014-01-02 00:21:23

Equidistant Curve Coordinate System(inversions(7))

Authors: Morio Kikuchi
Comments: 12 Pages.

We generalize inversion mathematically(2).
Category: Geometry

[141] viXra:1312.0172 [pdf] submitted on 2013-12-21 22:02:52

Логические основания многомерных пространств (Logical Basis Multidimensional Spaces)

Authors: Putenikhin P.V.
Comments: 18 Pages. rus (русский)

The understanding of Reality as a multidimensional space education has some logical difficulties. If Reality have four or more spatial coordinates, then would have to be observed phenomenon, that contrary to the known physical laws, logic and common sense.
Путенихин П.В. Представления о Реальности, как многомерном пространственном образовании, имеют некоторые логические сложности. Если бы Реальность имела четыре или более пространственных координат, то в ней должны были бы наблюдаться явления, противоречащие известным физическим законам, логике и обыденному здравому смыслу.
Category: Geometry

[140] viXra:1312.0153 [pdf] submitted on 2013-12-20 07:45:13

Тороподобные поверхности (The Surface, Resembling Torus)

Authors: Putenikhin P.V.
Comments: 4 Pages. rus (русский)

There are a large number of surfaces, on which are performed hyperbolic Lobachevsky's geometry. Here are investigated the surfaces, resembling torus with locally constant negative curvature.
Путенихин Петр Васильевич. Существует большое число поверхностей, на которых осуществляется гиперболическая геометрия Лобачевского. Рассмотрены тороподобные поверхности с локально постоянной отрицательной кривизной.
Category: Geometry

[139] viXra:1312.0146 [pdf] submitted on 2013-12-19 23:09:35

Пространственное линзирование (The Spatial Lensing)

Authors: Putenikhin P.V.
Comments: 9 Pages. rus (русский)

The empty curves space possesses the properties of lenses. It is possible snap the flat Euclidean space by the procedures of gluing, with the preservation of local Euclidean metric, without giving them the properties of lenses. Gluing of the extra dimensions in string theory can eliminate from it a Calabi-Yau manifold and to reduce landscape theory.
Путенихин Петр Васильевич. Кривые пустые пространства обладают свойствами линзы. Плоское пространство Евклида можно замкнуть процедурой отождествления с сохранением локальной евклидовой метрики, не наделяя его свойствами линзы. Отождествление дополнительных измерений в теории струн может устранить из неё многообразия Калаби-Яу и сократить ландшафт теории.
Category: Geometry

[138] viXra:1312.0109 [pdf] submitted on 2013-12-15 19:44:05

Equidistant Curve Coordinate System(inversions(6))

Authors: Morio Kikuchi
Comments: 11 Pages.

We generalize inversion mathematically
Category: Geometry

[137] viXra:1312.0105 [pdf] submitted on 2013-12-16 03:48:13

Тайна третьего постулата Евклида (The Mystery of Euclid’s Third Postulate)

Authors: Putenikhin P.V.
Comments: 10 Pages. rus (русский)

The geometry of Euclid is the original, primary geometry of smooth недеформированного space. Only there is indeed a direct and really plane. The geometry of Euclid is possible to deform and get the geometry of Lobachevsky and the Riemann - the geometry on the twisted, deformed Euclidean planes. The third postulate is a necessary and sufficient condition for the justice of the fifth postulate. If there is a third postulate, only then the fifth postulate has the force strictly in the formulation of Euclid, is its consequence.
Путенихин Петр Васильевич. Геометрия Евклида – это исходная, первичная геометрия гладкого недеформированного пространства. Только в ней существует действительно прямая и действительно плоскость. Геометрию Евклида можно деформировать и получить геометрии Лобачевского и Римана – геометрии на искривлённых, деформированных евклидовых плоскостях. Третий постулат является необходимым и достаточным условием справедливости пятого постулата. Если существует третий постулат, то и пятый имеет силу строго в формулировке Евклида, то есть является его следствием.
Category: Geometry

[136] viXra:1312.0075 [pdf] submitted on 2013-12-10 18:37:49

Notes on Noncommutative Geometry

Authors: Igor Nikolaev
Comments: Pages.

The text is a trailer of an approximately 300 pages book comprising a foreword and the table of contents; part III is not written yet but remarks are welcome!
Category: Geometry

[135] viXra:1311.0192 [pdf] submitted on 2013-11-28 13:21:22

Effective Iso-Radius of Dynamic Iso-Sphere Holographic Rings

Authors: Nathan O. Schmidt
Comments: 17 pages, 5 figures, submitted to the Gulf Journal of Mathematics

In this work, we introduce the "effective iso-radius" for dynamic iso-sphere Inopin holographic rings (IHR) as the iso-radius varies, which facilitates a heightened characterization of these emerging, cutting-edge iso-spheres as they vary in size and undergo "iso-transitions" between "iso-states". The initial results of this exploration fuel the construction of a new "effective iso-state" platform with a potential for future scientific application, but this emerging dynamic iso-architecture warrants further development, scrutiny, collaboration, and hard work in order to advance it as such.
Category: Geometry

[134] viXra:1311.0141 [pdf] submitted on 2013-11-19 18:27:21

Equidistant Curve Coordinate System(inversions(5))

Authors: Morio Kikuchi
Comments: 11 Pages.

We generalize inversion.
Category: Geometry

[133] viXra:1311.0038 [pdf] submitted on 2013-11-06 00:56:16

Variance of Topics of Plane Geometry

Authors: Ion Patrascu, Florentin Smarandache
Comments: 112 Pages.

This book contains 21 papers of plane geometry. It deals with various topics, such as: quasi-isogonal cevians, nedians, polar of a point with respect to a circle, anti-bisector, aalsonti-symmedian, anti-height and their isogonal. A nedian is a line segment that has its origin in a triangle’s vertex and divides the opposite side in n equal segments. The papers also study distances between remarkable points in the 2D-geometry, the circumscribed octagon and the inscribable octagon, the circles adjointly ex-inscribed associated to a triangle, and several classical results such as: Carnot circles, Euler’s line, Desargues theorem, Sondat’s theorem, Dergiades theorem, Stevanovic’s theorem, Pantazi’s theorem, and Newton’s theorem. Special attention is given in this book to orthological triangles, biorthological triangles, ortho-homological triangles, and trihomological triangles. Each paper is independent of the others. Yet, papers on the same or similar topics are listed together one after the other. The book is intended for College and University students and instructors that prepare for mathematical competitions such as National and International Mathematical Olympiads, or for the AMATYC (American Mathematical Association for Two Year Colleges) student competition, Putnam competition, Gheorghe Ţiţeica Romanian competition, and so on. The book is also useful for geometrical researchers.
Category: Geometry

[132] viXra:1310.0049 [pdf] submitted on 2013-10-07 10:11:42

Nedians and Triangles with the Same Coefficient of Deformation

Authors: Ion Patrascu, Florentin Smarandache
Comments: 11 Pages.

In [1] Dr. Florentin Smarandache generalized several properties of the nedians. Here, we will continue the series of these results and will establish certain connections with the triangles which have the same coefficient of deformation.
Category: Geometry

[131] viXra:1310.0033 [pdf] submitted on 2013-10-05 20:37:38

The Polar of a Point with Respect to a Circle

Authors: Ion Patrascu, Florentin Smarandache
Comments: 4 Pages.

In this article we establish a connection between the notion of the symmedian of a triangle and the notion of polar of a point in rapport to a circle
Category: Geometry

[130] viXra:1309.0155 [pdf] submitted on 2013-09-23 08:23:22

An Algebraic Journey in to Geometric Forest

Authors: S.Kalimuthu
Comments: 3 Pages. No Comments

Once the famous French mathematician Lagrange remarked that as long as algebra and geometry are not inter linked,one can not expect good results. Keeping this in mind, the author has attempted to establish an interesting classical Euclidean theorem by applying the algebra of matrices.
Category: Geometry

[129] viXra:1308.0126 [pdf] submitted on 2013-08-23 07:00:35

A Circle Without Pie

Authors: O. V. Vijimon
Comments: 40 Pages. 25 figures

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "Pie" is simply unthinkable but it’s going to be a reality very soon. "Pie" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "Tau". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category: Geometry

[128] viXra:1308.0076 [pdf] submitted on 2013-08-15 05:36:22

On the Fifth Euclidean Postulate

Authors: S.Kalimuthu
Comments: 3 Pages. NA

Matrices and determinants are widely used to solve problems in electronics, statics , robotics , linear programming , optimization , intersections of planes , genetics, physics , cosmology and all other areas of science and engineering. In this work, we attempt to deduce E5 from E1 to E4 by applying determinants.
Category: Geometry

[127] viXra:1307.0109 [pdf] submitted on 2013-07-23 02:28:17

Path-Dependent Functions

Authors: Khrapko R
Comments: 9 Pages. Theoretical and Mathematical Physics Volume 65, Issue 3, December 1985 p. 1196

Various path-dependent functions are described in a uniform manner by means of a series expansion of Taylor type. For this, "path integrals" and "path tensors" are introduced. They are systems of multicomponent quantities whose values are defined for an arbitrary path in a coordinated region of space in such a way that they carry sufficient information about the shape of the path. These constructions are regarded as elementary path-dependent functions and are used instead Of the power monomials of an ordinary Taylor series. The coefficients of such expansions are interpreted as partial derivatives, which depend on the order of differentiation, or as nonstandard covariant derivatives, called two-point derivatives. Examples of path-dependent functions are given. We consider the curvature tensor of a space whose geometrical properties are specified by a translator of parallel transport of general type (nontransitive). A covariant operation leading to "extension" of tensor fields is described
Category: Geometry

[126] viXra:1307.0066 [pdf] submitted on 2013-07-15 11:06:49

Supermatematica Profesorului Şelariu

Authors: Florentin Smarandache
Comments: 7 Pages.

Acest articol este o scurtă trecere în revistă a cărţii “SuperMatematica. Fundamente”, Vol. 1, 2012, care constituie un domeniu nou de cercetare şi cu multe aplicaţii, iniţiat de profesorul universitar Mircea Eugen Şelariu. Lucrarea sa este unică în literatura mondială, deoarece combină matematica centrică cu matematica excentrică.
Category: Geometry

[125] viXra:1306.0233 [pdf] submitted on 2013-06-29 12:21:12

The Projective Line as a Meridian

Authors: Kelly McKennon
Comments: 102 Pages.

We investigate that mathematical idea which in algebra is known as a cross ratio, in one-dimensional geometry as a projective line, in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus. We view each of these in its natural habitat, and show how each is an avatar of one Platonic object, which object we term a meridian.
Category: Geometry

[124] viXra:1306.0190 [pdf] submitted on 2013-06-21 12:44:04

Dual Structures in Cube Nets Disclosed

Authors: Klaus Lange
Comments: 6 Pages. 8 figures

It will be shown how the well known eleven nets for three dimensional cubes, separated in 10 + 1 forms, are hiding a special dual 3-6-1-structure. Implications for space - time models in theoretical physics will be questioned.
Category: Geometry

[123] viXra:1306.0155 [pdf] submitted on 2013-06-19 01:52:08

Visualization of Fundamental Symmetries in Nature

Authors: Eckhard Hitzer, Christian Perwass
Comments: 6 Pages. 7 figures, 4 tables. Proceedings of Fuzzy System Symposium (FSS 2009), Tsukuba, Japan, 14-16 Jul. 2009.

Most matter in nature and technology is composed of crystals and crystal grains. A full understanding of the inherent symmetry is vital. A new interactive software tool is demonstrated, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) is a script for the open source visual CLUCalc, which fully supports geometric algebra computation. In our presentation we will first give some insights into the geometric algebra description of space groups. The symmetry generation data are stored in an XML file, which is read by a special CLUScript in order to generate the visualization. Then we will use the Space Group Visualizer to demonstrate space group selection and give a short interactive computer graphics presentation on how reflections combine to generate all 230 three-dimensional space groups.
Category: Geometry

[122] viXra:1306.0134 [pdf] submitted on 2013-06-17 05:10:48

Tutorial on Reflections in Geometric Algebra

Authors: Eckhard Hitzer
Comments: 22 Pages. 16 figures, 6 tables. In K. Tachibana (ed.) Tutorial on Reflections in Geometric Algebra, Lecture notes of the international Workshop for “Computational Science with Geometric Algebra” (FCSGA2007), Nagoya Univ., Japan, 14-21 Feb. 2007, pp. 34-44 (2007).

This tutorial focuses on describing the implementation and use of reflections in the geometric algebras of three-dimensional (3D) Euclidean space and in the five-dimensional (5D) conformal model of Euclidean space. In the latter reflections at parallel planes serve to implement translations as well. Combinations of reflections allow to implement all isometric transformations. As a concrete example we treat the symmetries of (2D and 3D) space lattice crystal cells. All 32 point groups of three dimensional crystal cells (10 point groups in 2D) are exclusively described by vectors (two for each cell in 2D, three for one particular cell in 3D) taken from the physical cell. Geometric multiplication of these vectors completely generates all symmetries, including reflections, rotations, inversions, rotary reflections and rotary-inversions. The inclusion of translations with the help of the 5D conformal model of 3D Euclidean space allows the full formulation of the 230 crystallographic space groups in geometric algebra. The sets of vectors necessary are illustrated in drawings and all symmetry group elements are listed explicitly as geometric vector products. Finally a new free interactive software tool is introduced, that visualizes all symmetry transformations in the way described in the main geometrical part of this tutorial.
Category: Geometry

[121] viXra:1306.0119 [pdf] submitted on 2013-06-17 03:27:06

Learning about Conic Sections with Geometric Algebra and Cinderella

Authors: Eckhard Hitzer
Comments: 16 Pages. 8 figures, 1 table. Proc. of the Symposium Innovative Teaching of Mathematics with Geometric Algebra 2003, Nov. 20-22, 2003, RIMS, University of Kyoto, Japan, pp. 89-104 (2003).

Over time an astonishing and sometimes confusing variety of descriptions of conic sections has been developed. This article will give a brief overview over some interesting descriptions, showing formulations in the three geometric algebras of Euclidean three space, projective geometry and the conformal model of Euclidean space. Some illustrations with Cinderella created Java applets will be given. I think a combined geometric algebra & illustration approach can motivate students to explorative learning.
Category: Geometry

[120] viXra:1306.0118 [pdf] submitted on 2013-06-17 03:33:05

Conic Sections Through Five Points Classical, Projective, Conformal

Authors: Eckhard Hitzer
Comments: 6 Pages. 2 figures, 1 table. Proc. of the International Symposium 2003 of Advanced Mechanical Engineering, Pukyong National Univ., Busan, Korea, 22-25 Nov. 2003, pp. 109-114 (2003).

In the so-called conformal model of Euclidean space of geometric algebra, circles receive a very elegant description by the outer product of three general points of that circle, forming what is called a tri-vector. Because circles are a special kind of conic section, the question arises, whether in general some kind of third order outer product of five points on a conic section (or certain linear combinations) may be able to describe other types of conic sections as well. The main idea pursued in this paper is to follow up a formula of Grassmann for conic sections through five points and implement it in the conformal model. Grassmann obviously based his formula on Pascal’s theorem. At the end we consider a simple linear combination of circle tri-vectors.
Category: Geometry

[119] viXra:1306.0115 [pdf] submitted on 2013-06-17 04:05:54

Play with Geometry - Animated and Interactive, Free, Instant Access, Online Geometric Algebra Java Applets

Authors: Eckhard Hitzer, Luca Redaelli
Comments: 6 Pages. 18 figures. Proceedings of Fukui University International Congress, International Symposium on Advanced Mechanical Engineering, 11-13 Sep. 2002, pp. 7-12 (2002).

Conventional illustrations of elementary relations and physical applications of geometric algebra are helpful, but restricted in communicating full generality and time dependence. The main restrictions are one special perspective in each graph and the static character of such illustrations. Several attempts have been made to overcome such restrictions. But up till now very little animated and interactive, free, instant access, online material is available. This talk presents therefore a set of well over 60 newly developed (freely online accessible[1]) JAVA applets. These applets range from the elementary concepts of vector, bivector, outer product and rotations to triangle relationships, oscillations and polarized waves. A special group of 21 applets illustrates three geometrically different approaches to the representation of conics; and even more ways to describe ellipses. Finally Clifford's circle chain theorem is illustrated for two to eight primary circles. The interactive geometry software Cinderella[2] was used for creating these applets. Some construction principles will be explained and a number of applets will be demonstrated. The interactive features of many of the applets invite the user to freely explore by a few mouse clicks as many different special cases and perspectives as he likes. This is of great help in "visualizing" the geometry encoded in the concepts and formulas of Geometric Algebra.
Category: Geometry

[118] viXra:1306.0052 [pdf] submitted on 2013-06-08 09:44:49

Version of Proof of Morley's Trisector Theorem

Authors: Michael Pogorsky
Comments: 2 Pages.

The properties of trisected triangle are utilized in this proof in the way different from other known proofs.
Category: Geometry

[117] viXra:1306.0037 [pdf] submitted on 2013-06-06 10:14:00

Smarandache Half-Groups

Authors: Arun S. Muktibodh
Comments: 5 Pages.

In this paper we introduce the concept of half-groups. This is a totally new concept and demands considerable attention. R.H.Bruck [1] has defined a half groupoid. We have imposed a group structure on a half groupoid wherein we have an identity element and each element has a unique inverse. Further, we have defined a new structure called Smarandache half-group. We have derived some important properties of Smarandache half- groups. Some suitable examples are also given.
Category: Geometry

[116] viXra:1305.0022 [pdf] submitted on 2013-05-03 23:36:36

The Solution of the Problem of Relation Between Geometry and Natural Sciences

Authors: Temur Z. Kalanov
Comments: 11 Pages.

@@The work is devoted to solution of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the method of attack is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the parallel axiom (Euclid’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid’s, Lobachevski’s, and Riemann’s geometries are proposed.
Category: Geometry

[115] viXra:1305.0013 [pdf] submitted on 2013-05-03 01:15:25

The Critical Analysis of the Pythagorean Theorem and of the Problem of Irrational Numbers

Authors: Temur Z. Kalanov
Comments: 10 Pages.

@@The critical analysis of the Pythagorean theorem and of the problem of irrational numbers is proposed. Methodological basis for the analysis is the unity of formal logic and of rational dialectics. It is shown that: 1) the Pythagorean theorem represents a conventional (conditional) theoretical proposition because, in some cases, the theorem contradicts the formal-logical laws and leads to the appearance of irrational numbers; 2) the standard theoretical proposition on the existence of incommensurable segments is a mathematical fiction, a consequence of violation of the two formal-logical laws: the law of identity of geometrical forms and the law of lack of contradiction of geometrical forms; 3) the concept of irrational numbers is the result of violation of the dialectical unity of the qualitative aspect (i.e. form) and quantitative aspect (i.e. content: length, area) of geometric objects. Irrational numbers represent a calculation process and, therefore, do not exist on the number scale. There are only rational numbers.
Category: Geometry

[114] viXra:1304.0016 [pdf] submitted on 2013-04-04 04:06:02

Localization Formulas About Two Killing Vector Fields

Authors: Xu Chen
Comments: 9 Pages.

In this article, we will discuss the smooth $(X_{M}+\sqrt{-1}Y_{M})$-invariant forms on M and to establish a localization formulas. As an application, we get a localization formulas for characteristic numbers.
Category: Geometry

[113] viXra:1303.0146 [pdf] submitted on 2013-03-19 23:14:58

Equidistant Curve Coordinate System(inversions(3))

Authors: Morio Kikuchi
Comments: 14 Pages.

The types of inversions are made clear.
Category: Geometry

[112] viXra:1303.0130 [pdf] submitted on 2013-03-17 17:44:41

The Problem of Points on a Parabola

Authors: Edigles Guedes
Comments: 4 pages

By means of geometrical problem of how many points can you find on the (half) parabola, such that the distance between any pair of them is rational, we construct some parametric equations.
Category: Geometry

[111] viXra:1303.0104 [pdf] submitted on 2013-03-14 11:07:48

Parabole si Paraboloizi

Authors: Mircea Eugen Selariu
Comments: 21 Pages.

EXIS TĂ O LEGATURĂ ÎNTRE PARABOLA CA POVESTIRE Ş I PARABOLA DIN MATEMATICĂ ? “ Exis tă ! Există şi între parabolele centrice şi parabolele excentrice sau excentricele parabolice !
Category: Geometry

[110] viXra:1303.0015 [pdf] submitted on 2013-03-03 10:51:13

Eccentricity, Space Bending, Dimmension

Authors: Marian Nitu, Florentin Smarandache, Mircea Eugen Selariu
Comments: 23 Pages.

This work’s central idea is to present new transformations, previously non - existent in Ordinary mathematics, named centric mathematics ( CM) but that became possible due to new born eccentric mathematics, and, implicit, to supermathematics. As shown in this work, the new geometric transformations, named conversion or transfiguration, wipes the boundaries between discrete and continuous geometric forms, showing that the first ones are also continuous, being just apparently discontinuous.
Category: Geometry

[109] viXra:1301.0143 [pdf] submitted on 2013-01-23 10:28:29

Presentation on Perspective Drawing & Design

Authors: Andrew Nassif
Comments: 10 Pages.

Linear Perspective allows you the ability to work by representing light passing through a scene in a rectangular base, this method is often used in some paintings or modern day sketches.
Category: Geometry

[108] viXra:1211.0134 [pdf] submitted on 2012-11-22 21:32:20

Law of Sums of the Squares of Areas, Volumes and Hyper Volumes of Regular Polytopes from Clifford Polyvectors

Authors: Carlos Perelman, Fang Fang, Garret Sadler, Klee Irwin
Comments: 9 Pages.

Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Category: Geometry

[107] viXra:1211.0099 [pdf] submitted on 2012-11-18 14:50:51

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 12 Pages.

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case. Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative. A standard method, for applying the above defined product of distributions to polyhedra vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[106] viXra:1211.0024 [pdf] submitted on 2012-11-05 13:29:38

Several Metrical Relations Regarding the Anti-Bisectrix, the Anti-Symmedian, the Anti-Height and their Isogonal

Authors: Ion Patrascu, Florentin Smarandache
Comments: 4 Pages.

We’ll prove now that there is a similar relation for the isometric cevians as Steiner's relation for the isogonal cevians.
Category: Geometry

[105] viXra:1211.0023 [pdf] submitted on 2012-11-05 13:31:16

The Duality and the Euler’s Line

Authors: Ion Patrascu, Florentin Smarandache
Comments: 3 Pages.

In this article we’ll discuss about a theorem which results from a duality transformation of a theorem and the configuration in relation to the Euler’s line.
Category: Geometry

[104] viXra:1210.0006 [pdf] submitted on 2012-10-01 22:56:01

A First Study On Theory Of Connections Of Fibre Bundles

Authors: Ren Shiquan
Comments: 11 Pages. this is a draft of review.

We give a review on the connection theory of fibre bundles, according to our study procedure.
Category: Geometry

[103] viXra:1210.0005 [pdf] submitted on 2012-10-01 23:08:15

Holonomy And De Rham Decomposition Of Manifolds

Authors: Ren Shiquan
Comments: 8 Pages.

This is a review of our study on the holonomy group and De Rham Decomposition of manifolds.
Category: Geometry

[102] viXra:1209.0108 [pdf] submitted on 2012-09-29 11:14:31

An Important Application of the Computation of the Distances Between Remarkable Points in the Triangle Geometry

Authors: Ion Patrascu, Florentin Smarandache
Comments: 5 Pages.

In this article we’ll prove through computation the Feuerbach’s theorem relative to the tangent to the nine points circle, the inscribed circle, and the ex-inscribed circles of a given triangle.
Category: Geometry

[101] viXra:1208.0070 [pdf] submitted on 2012-08-16 17:24:24

From a Problem of Geometrical Construction to the Carnot Circles

Authors: Ion Patrascu, Florentin Smarandache
Comments: 4 Pages.

In this article we’ll give solution to a problem of geometrical construction and we’ll show the connection between this problem and the theorem relative to Carnot’s circles.
Category: Geometry

[100] viXra:1205.0092 [pdf] submitted on 2012-05-23 20:05:38

Definirea FSM-ce Hipoelementere DE VARIABILĂ EXCENTRICĂ \theta ŞI CENTRICĂ \alpha

Authors: Mircea Eugen Şelariu
Comments: 23 Pages.

Prezentarea ar trebui să începă cu funcţiile beta excentrice, deoarece ele vor fi utilizate în continuare şi la definirea şi prezentarea următoarelor FSM-CE, care sunt funcţiile amplitudine excentrică, funcţii asemănătoare din multe puncte de vedere cu funcţiile eliptice Jacobi amplitudine sau amplitudinus am(u,k). Dar va începe cu fucţia “rege” radial excentric rexθ şi Rexα.
Category: Geometry

[99] viXra:1205.0060 [pdf] submitted on 2012-05-13 16:00:45

An Alternative Cosine's Law Deduction

Authors: Hilário Fernandes de Araújo Júnior
Comments: 3 Pages.

The cosine's law shows that, if we have a triangle with sides a, b and c, and an angle α between the sides b and c, this relationship is right: a²=b²+c²−2bc[cos α].Will be shown here this law deduction through the trigonometry's fundamental relation.
Category: Geometry

[98] viXra:1205.0055 [pdf] submitted on 2012-05-11 20:15:25

π's Representation as an Infinite Sum

Authors: Hilário Fernandes de Araújo Júnior
Comments: 4 Pages.

In this article, is developed a π representation as an infinite sum, through a definite integral.
Category: Geometry

[97] viXra:1205.0051 [pdf] submitted on 2012-05-09 07:44:01

Right Triangle in Wich the Sum of the Legs is Close to pi .

Authors: Alberto Coe
Comments: 3 Pages.

Using elementary geometry we have performed an approach to Pi .this agrees to the fifth decimal place .
Category: Geometry

[96] viXra:1205.0003 [pdf] submitted on 2012-05-02 23:20:08

Proof of Euclid's Fifth Postulate

Authors: Jay Yoon
Comments: 5 Pages.

I will present a proof of Euclid’s fifth postulate (I.Post.5) that proves, as an intermediate step, a proposition equivalent to it (I.32); namely, that in any triangle, the sum of the three interior angles of the triangle equals two right angles. The proof that I.32 implies I.Post.5 and vice versa is well-established and will be omitted for the sake of brevity. The proof technique is somewhat unorthodox in that it proves I.33, which states that straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel, before establishing I.32, contrary to the order in which the propositions are demonstrated in Euclid’s Elements. Two triangle congruence theorems, namely the side-angle-side (I.4) and side-side-side congruence theorems (I.8) are employed in order to prove I.33 without recourse to I.Post.5 or any of its equivalent formulations. In addition, a parallelogram is constructed by an unorthodox method; namely, by defining the diagonals upon which the parallelogram’s sides will be determined prior to the sides themselves. The proof assumes the five common notions stated in Book I of The Elements without explicitly making a reference to them when they are used. Furthermore, a figure is presented with color-coded angles and sides, with angles of the same color being equal in measure and sides of both the same color and the same number of tick marks being equal in length. The sides GH and EJ enclosed by brackets are indicated to be equal in length, the reason for the different notation being that the tick marks were used in reference to the halves of GH, namely OG and OH. The tick marks then refer to the parts of GH, and the bracket refers to the whole of GH; the latter is then equated to EJ by I.33, which is proven before its use.
Category: Geometry

[95] viXra:1204.0097 [pdf] submitted on 2012-04-27 09:27:47

Quasi-Isogonal Cevians

Authors: Ion Patrascu, Florentin Smarandache
Comments: 3 Pages.

In this article we will introduce the quasi-isogonal Cevians and we’ll emphasize on triangles in which the height and the median are quasi-isogonal Cevians.
Category: Geometry

[94] viXra:1204.0096 [pdf] submitted on 2012-04-27 09:29:37

Another Proof of the I. Pătraşcu’s Theorem

Authors: Florentin Smarandache
Comments: 2 Pages.

In [1] professor Ion Pătraşcu proves the following theorem: The Brocard’s point of an isosceles triangle is the intersection of the medians and the perpendicular bisectors constructed from the vertexes of the triangle’s base, and reciprocal. We’ll provide below a different proof of this theorem than the proof given in [1] and [2].
Category: Geometry

[93] viXra:1203.0006 [pdf] submitted on 2012-03-02 10:47:14

Energy Laws , Follow Moulds of Euclidean Geometry and the Six , Triple Concurrency Points , Line

Authors: Marcos Georgallides
Comments: 21 Pages.

Article < The Six , Triple Concurrency Points , Line > is an extension of two Fundamental branches of geometry that of Perspectivity ( Desargues`s theorem , where 3 concurrency Points in a center of Perspectivity and 3 concurrency points on a line of Perspectivity , per two sides ) and that of Projective geometry ( Pascal`s theorem , with the 3 concurrency Points on a line , per two sides ) . Analyzing Extremum Principle ( Extrema ) on lines and Points , it was found that in any triangle ( three points only , which form a Plane ) and on the circumcircle exist one Inscribed and one Circumscribed , Extrema Triangle , such that on the six Extrema lines ( with a common concurrency point ) , both Perspectivity and Projective geometry concurrence on Common points on Extrema Lines . i.e. 18 lines concurrence in Six Points , per three , on a line , six triple concurrency points line. This Compact logic of Extrema exists on Points and in lines of Euclidean geometry. Article < Energy Laws follow Properties of Euclidean geometry > , is the deeper concept of Pythagoras theorem , where Conservation laws , referred to Physics and Mechanics , follow Euclidean moulds because these Principles belong to geometry as Points and Spaces ( geometry ) create Quantities and Qualities . Analyzing Euclid Spaces , it was found that on any two Equal and perpendicular , One dimensional Units , exists a Plane Formation ( A changeable and constant Tensor ) of constructing Squares , such that the Sum of Areas of the two Changeable Squares ( the Sum of the Squares of sides) is constant and equal to that of the circumscribed Square .The same also exists in Space Formation , where then , The Total Resultant Volume (cube of Resultant Sphere ) is the Sum of Changeable Volumes ( the Sum of the Cubes of Spheres of sides ). In Space Formation Changeable Volumes are Perpendicular each other , meaning that Conservation in Space ( Solid geometry ) occurs on Perpendiculars since first dimensional Units are Vectors . This geometrical mould of Conservation , is followed by Energy in Mechanics and Physics . i.e. The referred Energy Conservation laws in Mechanics and Physics , follow the Principle ( mould ) of Conserved Areas for Pythagoras` theorem on the moving machine of the two changeable Squares , and Conserved Perpendicular Volumes for Spaces on the Three Changeable Spheres .
Category: Geometry

[92] viXra:1203.0001 [pdf] submitted on 2012-03-01 05:51:13

Calculation of the Area of a Sampled Boundary Using Fourier Components

Authors: David Proffitt
Comments: 3 Pages.

A reformulation of the area of a planar two-dimensional object in the frequency domain allows for the computation of the true area of a band-limited boundary to be calculated.
Category: Geometry

[91] viXra:1202.0032 [pdf] submitted on 2012-02-11 17:14:46

Cardinal Functions and Integral Functions

Authors: Mircea Selariu, Florentin Smarandache, Marian Nitu
Comments: 14 Pages.

This paper presents the correspondences of the eccentric mathematics of cardinal and integral functions and centric mathematics, or ordinary mathematics. Centric functions will also be presented in the introductory section, because they are, although widely used in undulatory physics, little known.
Category: Geometry

[90] viXra:1201.0061 [pdf] submitted on 2012-01-15 22:14:03

The Geometry of Homological Triangles

Authors: Florentin Smarandache, Ion Patrascu
Comments: 244 Pages.

This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of triangles as a “filter” through which remarkable notions and theorems from the geometry of the triangle are unitarily passed. Our research is structured in seven chapters, the first four are dedicated to the homology of the triangles while the last ones to their applications.
Category: Geometry

[89] viXra:1201.0047 [pdf] submitted on 2012-01-10 09:29:35

The Decreasing Tunnel by Professor Florentin Smarandache

Authors: Markos Georgallides
Comments: 12 Pages.

In this work is given a new approach to the Open Question of professor Florentine Smarandache concerning the decreasing Tunnel for Orthocenter H on any triangle ABC . Circumcenter O , Centroid K and Ortocenter H lie on Euler line OH . The midpoint N of segment OH is the center of the nine - points circle which is passing from the three midpoints of each side and from the three feet of the altitudes , so this point N is orthic`s triangle circum center . This property of point N ( as it is the first link of a chain ) connects segment ( bar ) OH with an infinite set of segments OnHn of the orthic triangles where On coincides with point Nn-1 , that of each time midpoint of segments . This chain is the locus of point N and that of the repetitive ( rotating ) segment OnHn . On any triangle ABC and on the vertices of the triangle , is constructed an orthogonal hyperbola which passes from orthocenter and provides two fix points ( the foci ) in plane . As a result is the Axial Symmetry to the two axis , the orthogonal x,y and that of asymptotes . Since orthocenter H changes position , then AH is altering magnitude and direction , therefore AH is a repetitive damped Vector Quantity which assumes its extreme in the opposite direction relative to the first or prior positions . The above property results to a Central Symmetry to one of the vertices A , B , C with the two hyperbolas and after following the greatest of sides a , b , c . Damped Vector AHn can then convergent to Hn which is the Orthocenter of AnBnCn and it is the extreme in opposite direction . i.e. Orthocenter H… Hn limits to a point on a chain ( straight line or curved ) through A .
Category: Geometry

[88] viXra:1111.0092 [pdf] submitted on 24 Nov 2011

Smooth Infinitesimal Analysis Based Model of Multidimensional Geometry

Authors: Alexander Egoyan
Comments: 6 pages.

In this work a new approach to multidimensional geometry based on smooth infinitesimal analysis (SIA) is proposed. An embedded surface in this multidimensional geometry will look different for the external and internal observers: from the outside it will look like a composition of infinitesimal segments, while from the inside like a set of points equipped by a metric. The geometry is elastic. Embedded surfaces possess dual metric: internal and external. They can change their form in the bulk without changing the internal metric.
Category: Geometry

[87] viXra:1110.0072 [pdf] submitted on 28 Oct 2011

The Hyperbolic Smarandache Theorem in the Poincaré Upper Half-Plane Model of Hyperbolic Geometry

Authors: Nilgün Sönmez, Catalin Barbu
Comments: 4 pages.

In this study, we give a hyperbolic version of the Smarandache's theorem in the Poincaré upper half-plane model.
Category: Geometry

[86] viXra:1110.0066 [pdf] submitted on 25 Oct 2011

Smaransache Multi-Space Theory

Authors: Linfan Mao
Comments: 377 pages

Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means theWORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds.
Category: Geometry

[85] viXra:1108.0008 [pdf] submitted on 4 Aug 2011

A New Approach on Smarandache tn1 Curves in terms of Spacelike Biharmonic Curves with a Timelike Binormal in the Lorentzian Heisenberg Group Heis

Authors: T Körpinar, E Turhan
Comments: 8 pages

In this paper, we study spacelike biharmonic curve with a timelike binormal in the Lorentzian Heisenberg group Heis. We define a special case of such curves and call it Smarandache tn1 curves in the Lorentzian Heisenberg group Heis. We construct parametric equations of Smarandache tn1 curves in terms of spacelike biharmonic curves with a timelike binormal in the Lorentzian Heisenberg group Heis
Category: Geometry

[84] viXra:1107.0008 [pdf] submitted on 4 Jul 2011

Generalized Fermat's Last Theorem (6)

Authors: Chun-Xuan Jiang
Comments: 3 pages

In this paper we prove ...(see paper) part 6
Category: Geometry

[83] viXra:1107.0007 [pdf] submitted on 4 Jul 2011

Generalized Fermat's Last Theorem (5)

Authors: Chun-Xuan Jiang
Comments: 3 pages

In this paper we prove ...(see paper) part 5
Category: Geometry

[82] viXra:1107.0006 [pdf] submitted on 4 Jul 2011

Generalized Fermat's Last Theorem (4)

Authors: Chun-Xuan Jiang
Comments: 3 pages

In this paper we prove ...(see paper) part 4
Category: Geometry

[81] viXra:1107.0005 [pdf] submitted on 3 Jul 2011

Lobe Exterioare si Cvazilobe Interioare Cercului Unitate

Authors: Mircea Eugen Selariu
Comments: 18 pages, In Romanian

LOBE EXTERIOARE SI CVAZILOBE INTERIOARE CERCULUI UNITATE
Category: Geometry

[80] viXra:1106.0058 [pdf] submitted on 27 Jun 2011

Teorema S a Bisectoarelor Unui Patrulater Inscriptibil si Teoremele S Ale Triunghiului

Authors: Mircea Selariu
Comments: 20 pages.

TEOREMA S A BISECTOARELOR UNUI PATRULATER INSCRIPTIBIL SI TEOREMELE S ALE TRIUNGHIULUI
Category: Geometry

[79] viXra:1106.0057 [pdf] submitted on 27 Jun 2011

Noi Linii Concurente si un Nou Punct DE Intersectie Intr-un Triunghi

Authors: Mircea Selariu
Comments: 13 pages.

NOI LINII CONCURENTE SI UN NOU PUNCT DE INTERSECTIE INTR-UN TRIUNGHI
Category: Geometry

[78] viXra:1104.0079 [pdf] submitted on 19 Apr 2011

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

Authors: Linfan Mao
Comments: 16 pages

A tendering is a negotiating process for a contract through by a tenderer issuing an invitation, bidders submitting bidding documents and the tenderer accepting a bidding by sending out a notification of award. As a useful way of purchasing, there are many norms and rulers for it in the purchasing guides of the World Bank, the Asian Development Bank,..., also in contract conditions of various consultant associations. In China, there is a law and regulation system for tendering and bidding. However, few works on the mathematical model of a tendering and its evaluation can be found in publication. The main purpose of this paper is to construct a Smarandache multi-space model for a tendering, establish an evaluation system for bidding based on those ideas in the references [7] and [8] and analyze its solution by applying the decision approach for multiple objectives and value engineering. Open problems for pseudo-multi-spaces are also presented in the final section.
Category: Geometry

[77] viXra:1104.0078 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(IV)

Authors: Linfan Mao
Comments: 26 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part IV)
Category: Geometry

[76] viXra:1104.0077 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(III)

Authors: Linfan Mao
Comments: 74 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multi-rings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers (part III)
Category: Geometry

[75] viXra:1104.0076 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(II)

Authors: Linfan Mao
Comments: 78 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n &t; 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.
Category: Geometry

[74] viXra:1104.0075 [pdf] submitted on 19 Apr 2011

Smarandache Multi-Space Theory(I)

Authors: Linfan Mao
Comments: 47 pages

A Smarandache multi-space is a union of n different spaces equipped with some different structures for an integer n ≥ 2, which can be both used for discrete or connected spaces, particularly for geometries and spacetimes in theoretical physics. This monograph concentrates on characterizing various multi-spaces including three parts altogether. The first part is on algebraic multi-spaces with structures, such as those of multi-groups, multirings, multi-vector spaces, multi-metric spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold,..., etc.. The second discusses Smarandache geometries, including those of map geometries, planar map geometries and pseudo-plane geometries, in which the Finsler geometry, particularly the Riemann geometry appears as a special case of these Smarandache geometries. The third part of this book considers the applications of multi-spaces to theoretical physics, including the relativity theory, the M-theory and the cosmology. Multi-space models for p-branes and cosmos are constructed and some questions in cosmology are clarified by multi-spaces. The first two parts are relative independence for reading and in each part open problems are included for further research of interested readers.
Category: Geometry

[73] viXra:1104.0074 [pdf] submitted on 19 Apr 2011

On Multi-Metric Spaces

Authors: Linfan Mao
Comments: 9 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with classical metric spaces, the conception of multi-metric space is introduced. Some characteristics of a multi-metric space are obtained and Banach's fixed-point theorem is generalized in this paper.
Category: Geometry

[72] viXra:1104.0073 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Vector Spaces

Authors: Linfan Mao
Comments: 7 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with linear vector spaces in classical linear algebra, the conception of multi-vector spaces is introduced. Some characteristics of a multi-vector space are obtained in this paper.
Category: Geometry

[71] viXra:1104.0072 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Ring Spaces

Authors: Linfan Mao
Comments: 8 pages

A Smarandache multi-space is a union of n spaces A1,A2,...,An with some additional conditions holding. Combining Smarandache multispaces with rings in classical ring theory, the conception of multi-ring spaces is introduced. Some characteristics of a multi-ring space are obtained in this paper
Category: Geometry

[70] viXra:1104.0071 [pdf] submitted on 19 Apr 2011

On Algebraic Multi-Group Spaces

Authors: Linfan Mao
Comments: 8 pages

A Smarandache multi-space is a union of n spaces A1,A2, ... ,An with some additional conditions holding. Combining classical of a group with Smarandache multi-spaces, the conception of a multi-group space is introduced in this paper, which is a generalization of the classical algebraic structures, such as the group, filed, body,..., etc.. Similar to groups, some characteristics of a multi-group space are obtained in this paper.
Category: Geometry

[69] viXra:1104.0070 [pdf] submitted on 19 Apr 2011

A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

Authors: Linfan Mao
Comments: 16 pages

As we known, the Seifert-Van Kampen theorem handles fundamental groups of those topological spaces (see paper)
Category: Geometry

[68] viXra:1104.0069 [pdf] submitted on 19 Apr 2011

A Generalization of Stokes Theorem on Combinatorial Manifolds

Authors: Linfan Mao
Comments: 16 pages

For an integer m > 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for(...) there is a local chart (see paper) where Bnij is an nij -ball for integers 1 < j < s(p) < m. Integral theory on these smoothly combinatorial manifolds are introduced. Some classical results, such as those of Stokes' theorem and Gauss' theorem are generalized to smoothly combinatorial manifolds in this paper.
Category: Geometry

[67] viXra:1104.0068 [pdf] submitted on 19 Apr 2011

Geometrical Theory on Combinatorial Manifolds

Authors: Linfan Mao
Comments: 37 pages

For an integer m ≥ 1, a combinatorial manifold fM is defined to be a geometrical object fM such that for (...), there is a local chart (see paper) where Bnij is an nij -ball for integers 1 ≤ j ≤ s(p) ≤ m. Topological and differential structures such as those of d-pathwise connected, homotopy classes, fundamental d-groups in topology and tangent vector fields, tensor fields, connections, Minkowski norms in differential geometry on these finitely combinatorial manifolds are introduced. Some classical results are generalized to finitely combinatorial manifolds. Euler-Poincare characteristic is discussed and geometrical inclusions in Smarandache geometries for various geometries are also presented by the geometrical theory on finitely combinatorial manifolds in this paper.
Category: Geometry

[66] viXra:1104.0062 [pdf] submitted on 20 Apr 2011

Pseudo-Manifold Geometries with Applications

Authors: Linfan Mao
Comments: 15 pages.

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 in [9]-[10] and [12]. 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 ≥ 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, Kähler geometry and Finsler geometry, ...,etc., are their sub-geometries.
Category: Geometry

[65] viXra:1104.0061 [pdf] submitted on 20 Apr 2011

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

Authors: Linfan Mao
Comments: 19 pages.

Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an more important work for mathematician is to apply combinatorics to other mathematics and other sciences not merely to find combinatorial behavior for objectives. Recently, such research works appeared on journals for mathematics and theoretical physics on cosmos. The main purpose of this paper is to survey these thinking and ideas for mathematics and cosmological physics, such as those of multi-spaces, map geometries and combinatorial cosmoses, also the combinatorial conjecture for mathematics proposed by myself in 2005. Some open problems are included for the 21th mathematics by a combinatorial speculation.
Category: Geometry

[64] viXra:1104.0060 [pdf] submitted on 20 Apr 2011

Parallel Bundles in Planar Map Geometries

Authors: Linfan Mao
Comments: 16 pages.

Parallel lines are very important objects in Euclid plane geometry and its behaviors can be gotten by one's intuition. But in a planar map geometry, a kind of the Smarandache geometries, the situation is complex since it may contains elliptic or hyperbolic points. This paper concentrates on the behaviors of parallel bundles in planar map geometries, a generalization of parallel lines in plane geometry and obtains characteristics for parallel bundles.
Category: Geometry

[63] viXra:1104.0059 [pdf] submitted on 20 Apr 2011

A New View of Combinatorial Maps by Smarandache's Notion

Authors: Linfan Mao
Comments: 19 pages.

On a geometrical view, the conception of map geometries is introduced, which is a nice model of the Smarandache geometries, also new kind of and more general intrinsic geometry of surfaces. Some open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are presented.
Category: Geometry

[62] viXra:1104.0054 [pdf] submitted on 18 Apr 2011

Microscopes and Telescopes for Theoretical Physics : How Rich Locally and Large Globally is the Geometric Straight Line ?

Authors: Elemér E Rosinger
Comments: 31 pages.

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but one of the possible mathematical models chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers, among others. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analysis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. Consequently, one can define an extension of the usual Differential Calculus. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it and general can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called "infinities in physics" can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one : theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL. One of the consequences of considering the essential difference between the GSL and its various mathematical models is that what appears to be the definitive answer is given to the intriguing question raised by Penrose : "Why is it that physics never uses spaces with a cardinal larger than that of the continuum ?".
Category: Geometry

[61] viXra:1104.0053 [pdf] submitted on 17 Apr 2011

A New Proof of Menelaus's Theorem of Hyperbolic Quadrilaterals in the Poincaré Model of Hyperbolic Geometry

Authors: Catalin Barbu, Florentin Smarandache
Comments: 6 pages.

In this study, we present a proof of the Menelaus theorem for quadrilaterals in hyperbolic geometry, and a proof for the transversal theorem for triangles.
Category: Geometry

[60] viXra:1103.0119 [pdf] submitted on 31 Mar 2011

The Trisection of Any Angle

Authors: Markos Georgallides
Comments: 7 pages.

Universe is following Euclid Spaces. In Euclidean geometry points do not exist , but their position and correlation is doing geometry and physics . The universe cannot be created , because becomes and never is . According to Euclidean geometry , and since the position of points ( empty Space ) creates geometry and Spaces , the trisection of any angle exists in these Spaces and in this way. Infinite points exist always between points.
Category: Geometry

[59] viXra:1103.0076 [pdf] submitted on 19 Mar 2011

Geometry: Problems of Dividing Objects. Thales`s Theorem and an Idea Which Can Arose When You Applied the Theorem, for Solving an Interesting and Simple Problem in Geometry.

Authors: Martiros Khurshudyan
Comments: 3 pages.

Geometry it is not a word, moreover it is not just mathematical research area. It is art, it is the base of our Nature, it is language of Nature. The aim of this article is to present how Thales`s theorem is working for simple cases, when we need to divide a geometrical object into equal parts: mainly, we considered the problem of dividing a straight segment of length N into n equal parts. On the base of this simple case, we proposed a generalizations of the problem. We presented they as questions. Purpose of this article is to ask to find solutions for the questions. It seems, that for the positive answer, here must be developed geometrical techniques.
Category: Geometry

[58] viXra:1103.0043 [pdf] submitted on 13 Mar 2011

The Euclidean Philosophy of Universe ( Nature )

Authors: Markos Georgallides
Comments: 6 pages

It is not Accidental the fact that the Perception and Order of Elements of the Euclidean Geometry are with so much conceptual importance . This will appear clearly with the analysis which follows
Category: Geometry

[57] viXra:1103.0042 [pdf] submitted on 13 Mar 2011

The Parallel Postulate is Depended on the Other Axioms

Authors: Markos Georgallides
Comments: 20 pages

This article was sent to some specialists in Euclidean Geometry for criticism . The geometrical solution of this problem is based on the four Postulates for Constructions in Euclid geometry
Category: Geometry

[56] viXra:1103.0035 [pdf] submitted on 11 Mar 2011

A Property of the Circumscribed Octogon

Authors: Ion Patrascu, Florentin Smarandache
Comments: 3 pages

In this article we'll obtain through the duality method a property in relation to the contact cords of the opposite sides of a circumscribable octagon.
Category: Geometry

[55] viXra:1103.0034 [pdf] submitted on 11 Mar 2011

Triplets of Tri-Homological Triangles

Authors: Ion Patrascu, Florentin Smarandache
Comments: 9 pages

In this article will prove some theorems in relation to the triplets of homological triangles two by two. These theorems will be used later to build triplets of triangles two by two trihomological.
Category: Geometry

[54] viXra:1102.0015 [pdf] submitted on 10 Feb 2011

Un Model Pentru O Geometrie Smarandache

Authors: S. Bhattacharya
Comments: 2 pages. Romanian language.

Prezentam aici un model simplu al geometriei Smarandache si invitam cititorul, ca o distractie matematica, sa compuna alte modele.
Category: Geometry

[53] viXra:1102.0014 [pdf] submitted on 10 Feb 2011

O Introducere În Geometriile Smarandache

Authors: L. Kuciuk, M. Antholy
Comments: 4 pages. Romanian language.

O Geometrie Smarandache este o geometrie care are cel putin o axioma negata in mod smarandachean (1969). Spunem ca o axioma este negata smarandachean daca axioma se comporta cel putin in doua moduri diferite in acelasi spatiu (i.e. validata si negata, sau numai negata dar in mai multe moduri diferite).
Category: Geometry

[52] viXra:1102.0006 [pdf] submitted on 5 Feb 2011

Un Model Simplu de Geometrie Smarandache Construit Exclusiv cu Elemente de Geometrie Euclidiana

Authors: Ovidiu Sandru
Comments: 3 pages.

A model formed by two parallel plans is constructed which behaves the Smarandache geometries.
Category: Geometry

[51] viXra:1101.0093 [pdf] submitted on 28 Jan 2011

Fast Approximation of π Using Regular Polyon

Authors: Jongsoo Park
Comments: 17 pages, In Korean

Fast Approximation of π Using Regular Polyon
Category: Geometry

[50] viXra:1101.0068 [pdf] submitted on 22 Jan 2011

Construction of 50o Using Ruler and Pencil Only

Authors: Don Jojan
Comments: 4 pages

Here I am presenting the construction of an angle of 50o without using a compass or a protractor.
Category: Geometry

[49] viXra:1101.0067 [pdf] submitted on 22 Jan 2011

Construction of 120o Using Ruler and Pencil Only

Authors: Don Jojan
Comments: 4 pages

Here I am presenting the construction of an angle of 120o without using a compass or a protractor.
Category: Geometry

[48] viXra:1101.0046 [pdf] submitted on 14 Jan 2011

Construction of 60o Using Ruler and Pencil Only

Authors: Don Jojan
Comments: 4 pages

Here I am presenting the construction of an angle of 60o without using a compass or a protractor.
Category: Geometry

[47] viXra:1011.0028 [pdf] submitted on 20 Mar 2010

Generalizations of Degargues Theorem

Authors: Florentin Smarandache
Comments: 1 pages

Let's consider the points...
Category: Geometry

[46] viXra:1010.0060 [pdf] submitted on 28 Oct 2010

Let's Flying by Wing

Authors: Linfan Mao
Comments: 83 pages, in Chinese

Mathematical Combinatorics & Smarandache Multi-Spaces
Category: Geometry

[45] viXra:1010.0055 [pdf] submitted on 20 Mar 2010

Generalization of the Theorem of Menelaus Using a Self-Recurrent Method

Authors: Florentin Smarandache
Comments: 3 pages

This generalization of the Theorem of Menelaus from a triangle to a polygon with n sides is proven by a self-recurrent method which uses the induction procedure and the Theorem of Menelaus itself.
Category: Geometry

[44] viXra:1010.0050 [pdf] submitted on 20 Mar 2010

Limits of Recursive Triangle and Polygon Tunnels

Authors: Florentin Smarandache
Comments: 5 pages

In this paper we present unsolved problems that involve infinite tunnels of recursive triangles or recursive polygons, either in a decreasing or in an increasing way. The "nedians or order i in a triangle" are generalized to "nedians of ratio r" and "nedians of angle α" or "nedians at angle β", and afterwards one considers their corresponding "nedian triangles" and "nedian polygons". This tunneling idea came from physics.
Category: Geometry

[43] viXra:1010.0038 [pdf] submitted on 25 Oct 2010

Two Applications of Desargues' Theorem

Authors: Florentin Smarandache, Ion Pătraşcu
Comments: 6 pages

In this article we will use the Desargues' theorem and its reciprocal to solve two problems.
Category: Geometry

[42] viXra:1010.0008 [pdf] submitted on 4 Oct 2010

Two Triangles with the Same Orthocenter and a Vectorial Proof of Stevanovic's Theorem

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 4 pages

In this article we'll emphasize on two triangles and provide a vectorial proof of the fact that these triangles have the same orthocenter. This proof will, further allow us to develop a vectorial proof of the Stevanovic's theorem relative to the orthocenter of the Fuhrmann's triangle.
Category: Geometry

[41] viXra:1009.0046 [pdf] submitted on 12 Sep 2010

Pantazi's Theorem Regarding the Bi-orthological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 5 pages

In this article we'll present an elementary proof of a theorem of Alexandru Pantazi (1896-1948), Romanian mathematician, regarding the bi-orthological triangles.
Category: Geometry

[40] viXra:1009.0015 [pdf] submitted on 13 Mar 2010

Smarandache's Concurrent Lines Theorem

Authors: M. Khoshnevisan
Comments: 2 pages.

In this paper we present the Smarandache's Concurrent Lines Theorem in the geometry of the triangle.
Category: Geometry

[39] viXra:1009.0013 [pdf] submitted on 13 Mar 2010

Smarandache's Cevians Theorem (II)

Authors: M. Khoshnevisan
Comments: 2 pages.

In this paper we present the Smarandache's Cevians Theorem (II) in the geometry of the triangle.
Category: Geometry

[38] viXra:1009.0012 [pdf] submitted on 13 Mar 2010

Smarandache's Cevians Theorem (I)

Authors: M. Khoshnevisan
Comments: 2 pages.

We present the Smarandache's Cevians Theorem in the geometry of the triangle.
Category: Geometry

[37] viXra:1009.0011 [pdf] submitted on 13 Mar 2010

Smarandache's Ratio Theorem

Authors: M. Khoshnevisan
Comments: 2 pages.

In this paper we present the Smarandache's Ratio Theorem in the geometry of the triangle.
Category: Geometry

[36] viXra:1009.0010 [pdf] submitted on 13 Mar 2010

The Smarandache-Pătraşcu Theorem of Orthohomological Triangles

Authors: Mihai Dicu
Comments: 1 page.

The Smarandache-Pătraşcu Theorem of Orthohomological Triangles is the folllowing:
Category: Geometry

[35] viXra:1009.0009 [pdf] submitted on 13 Mar 2010

Smarandache's Orthic Theorem

Authors: Ion Pătraşcu
Comments: 3 pages.

We present the Smarandache's Orthic Theorem in the geometry of the triangle.
Category: Geometry

[34] viXra:1009.0006 [pdf] submitted on 2 Sep 2010

Two Remarkable Ortho-Homological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 10 pages

In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to orthological triangles, we emphasize on four important collinear points in the geometry of the triangle.
Category: Geometry

[33] viXra:1008.0081 [pdf] submitted on 28 Aug 2010

Smarandache's Minimum Theorem in the Einstein Relativistic Velocity Model of Hyperbolic Geometry

Authors: Catalin Barbu
Comments: 3 pages

In this note, we present a proof to the Smarandache's Minimum Theorem in the Einstein Relativistic Velocity Model of Hyperbolic Geometry.
Category: Geometry

[32] viXra:1008.0043 [pdf] submitted on 16 Aug 2010

Differentiable Structures on Real Grassmannians

Authors: Jeidsan A. C. Pereira
Comments: 10 Pages.

Given a vector space V of dimension n and a natural number k < n, the grassmannian Gk(V) is defined as the set of all subspaces W ⊂ V such that dim(W) = k. In the case of V = Rn, Gk(V) is the set of k-fl ats in Rn and is called real grassmannian [1]. Recently the study of these manifolds has found applicability in several areas of mathematics, especially in Modern Differential Geometry and Algebraic Geometry. This work will build two differential structures on the real grassmannian, one of which is obtained as a quotient space of a Lie group [1], [3], [2], [7]
Category: Geometry

[31] viXra:1008.0037 [pdf] submitted on 12 Aug 2010

A Direct Proof of the Yff's Conjecture

Authors: Marian Dincă
Comments: 2 Pages.

In this paper it is given proof Yff's conjecture using convexity arguments.
Category: Geometry

[30] viXra:1008.0031 [pdf] submitted on 11 Aug 2010

Another Proof of a Theorem Relative to the Orthological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 3 pages

In [1] we proved, using barycentric coordinates, the following theorem
Category: Geometry

[29] viXra:1008.0030 [pdf] submitted on 11 Aug 2010

Proof Wolstenholme-Lenhard Ciclic Inequality for Real Numbers and L.fejes Tóth Conjecture

Authors: Marian Dincă
Comments: 4 Pages.

In this paper an elementary proof of the Wolstenholme-Lenhard ciclic inequality for real numbers and L.Fejes T&oactute;th conjecture( equivalent by Erdis-Mordell inequality for polygon) is given, using a remarcable identity We give the following:
Category: Geometry

[28] viXra:1007.0035 [pdf] submitted on 23 Jul 2010

A New Proof of an Inequality of Oppenheim

Authors: Marian Dincă, J. L. Díaz-Barrero
Comments: 4 pages.

In this short note a new proof of a classical inequality involving the areas of a pair of triangles is presented.
Category: Geometry

[27] viXra:1007.0011 [pdf] submitted on 8 Jul 2010

A New Proof Daniel Pedoe and Oene Bottema Inequalities

Authors: Marian Dincă, Şcoala Generală
Comments: 1 page.

In the paper given a new proof the two inequalities using unitary method.
Category: Geometry

[26] viXra:1006.0069 [pdf] submitted on 30 Jun 2010

An Application of Sondat's Theorem Regarding the Orthohomological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 4 pages.

In this article we prove the Sodat's theorem regarding the orthohomological triangle and then we use this theorem and Smarandache-Patrascu's theorem in order to obtain another theorem regarding the orthohomological triangles.
Category: Geometry

[25] viXra:1006.0059 [pdf] submitted on 13 Mar 2010

Properties of a Hexagon Circumscribed to a Circle

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 3 pages.

In this paper we analyze and prove two properties of a hexagon circumscribed to a circle
Category: Geometry

[24] viXra:1006.0058 [pdf] submitted on 13 Mar 2010

A Multiple Theorem with Isogonal and Concyclic Points

Authors: Florentin Smarandache, Ion Pătraşcu
Comments: 3 pages.

A Multiple Theorem with Isogonal and Concyclic Points
Category: Geometry

[23] viXra:1006.0024 [pdf] submitted on 13 Mar 2010

A Theorem about Simultaneous Orthological and Homological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 13 pages.

In this paper we prove that if P1,P2 are isogonal points in the triangle ABC , and if A1B1C1 and A2B2C2 are their ponder triangle such that the triangles ABC and A1B1C1 are homological (the lines AA1 , BB1 , CC1 are concurrent), then the triangles ABC and A2B2C2 are also homological.
Category: Geometry

[22] viXra:1006.0015 [pdf] submitted on 11 Mar 2010

An Economics Model for the Smarandache Anti-Geometry

Authors: Roberto Torretti
Comments: 3 pages

The Smarandache anti-geometry is a non-euclidean geometry that denies all Hilbert's twenty axioms, each axiom being denied in many ways in the same space. In this paper one finds an economics model to this geometry by making the following correlations: (i) A point is the balance in a particular checking account, expressed in U.S. currency. (Points are denoted by capital letters). (ii) A line is a person, who can be a human being. (Lines are denoted by lower case italics). (iii) A plane is a U.S. bank, affiliated to the FDIC. (Planes are denoted by lower case boldface letters).
Category: Geometry

[21] viXra:1006.0004 [pdf] submitted on 3 Jun 2010

A Class of Orthohomological Triangles

Authors: Claudiu Coandă, Florentin Smarandache, Ion Pătraşcu
Comments: 5 pages

In this article we propose to determine the triangles' class... (see paper for full abstract)
Category: Geometry

[20] viXra:1006.0003 [pdf] submitted on 3 Jun 2010

The Hyperbolic Menelaus Theorem in The Poincaré Disc Model of Hyperbolic Geometry

Authors: Florentin Smarandache, Catalin Barbu
Comments: 4 pages

In this note, we present the hyperbolic Menelaus theorem in the Poincaré disc of hyperbolic geometry.
Category: Geometry

[19] viXra:1005.0053 [pdf] submitted on 11 Mar 2010

Solved Problems of Geometry and Trigonometry for College Students.

Authors: Florentin Smarandache
Comments: 171 pages

Solved problems of geometry and trigonometry for college students.
Category: Geometry

[18] viXra:1005.0016 [pdf] submitted on 5 May 2010

Generalization of a Remarkable Theorem

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 3 pages

In [1] Professor Claudiu Coandă proves the following theorem using the barycentric coordinates.
Category: Geometry

[17] viXra:1004.0137 [pdf] submitted on 10 Mar 2010

An Introduction to the Smarandache Geometries

Authors: L. Kuciuk, M. Antholy
Comments: 23 pages.

In this paper we make a presentation of these exciting geometries and present a model for a particular one.
Category: Geometry

[16] viXra:1004.0050 [pdf] submitted on 8 Apr 2010

A Proof of Smarandache-Pătrașcu's Theorem using Barycentric Coordinates

Authors: Claudiu Coandă
Comments: 4 pages

In this article we prove the Smarandache-Pătrașcu's Theorem in relation to the inscribed orthohomological triangles using the barycentric coordinates.
Category: Geometry

[15] viXra:1004.0025 [pdf] submitted on 3 Apr 2010

An Application of a Theorem of Orthohomological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 3 pages

In this note we prove a problem given at a Romanian student mathematical competition, and we obtain an interesting result by using a Theorem of Orthohomological Triangles.
Category: Geometry

[14] viXra:1004.0003 [pdf] submitted on 8 Mar 2010

Super-Mathematics Functions

Authors: Mircea Eugen Șelariu
Comments: 14 pages, translated from Romanian by Marian Nitu and Florentin Smarandache

In this paper we talk about the so-called Super-Mathematics Functions (SMF), which often constitute the base for generating technical, neo-geometrical, therefore less artistic objects. These functions are the results of 38 years of research, which began at University of Stuttgart in 1969. Since then, 42 related works have been published, written by over 19 authors, as shown in the References.
Category: Geometry

[13] viXra:1003.0272 [pdf] submitted on 8 Mar 2010

Eight Solved and Eight Open Problems in Elementary Geometry

Authors: Florentin Smarandache
Comments: 9 pages

In this paper we review eight previous proposed and solved problems of elementary 2D geometry [1], and we extend them either from triangle to polygons or from 2D to 3D-space and make some comments about them.
Category: Geometry

[12] viXra:1003.0256 [pdf] submitted on 8 Mar 2010

Degree of Negation of an Axiom

Authors: Florentin Smarandache
Comments: 4 pages

In this article we present the two classical negations of Euclid's Fifth Postulate (done by Lobachevski-Bolyai-Gauss, and respectively by Riemann), and in addition of these we propose a partial negation (or a degree of negation) of an axiom in geometry. The most important contribution of this article is the introduction of the degree of negation (or partial negation) of an axiom and, more general, of a scientific or humanistic proposition (theorem, lemma, etc.) in any field - which works somehow like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood) or like the negation in neutrosophic logic [with a degree of truth, a degree of falsehood, and a degree of neutrality (i.e. neither truth nor falsehood, but unknown, ambiguous, indeterminate)].
Category: Geometry

[11] viXra:1003.0254 [pdf] submitted on 26 Mar 2010

Smarandache's Cevian Triangle Theorem in The Einstein Reletivistic Velocity Model of Hyperbolic Geometry

Authors: Cătălin Barbu
Comments: 4 pages

In this note, we present a proof of Smarandache's cevian triangle hyperbolic theorem in the Einstein relativistic velocity model of hyperbolic geometry.
Category: Geometry

[10] viXra:1003.0245 [pdf] submitted on 25 Mar 2010

Smarandache's Pedal Polygon Theorem in The Poincaré Disc Model of Hyperbolic Geometry

Authors: Cătălin Barbu
Comments: 4 pages

In this note, we present a proof of the hyperbolic a Smarandache's pedal polygon theorem in the Poincaré disc model of hyperbolic geometry.
Category: Geometry

[9] viXra:1003.0227 [pdf] submitted on 7 Mar 2010

Automorphism Groups of Maps, Surfaces and Smarandache Geometries

Authors: Linfan Mao
Comments: 124 pages

A combinatorial map is a connected topological graph cellularly embedded in a surface. As a linking of combinatorial configuration with the classical mathematics, it fascinates more and more mathematician's interesting. Its function and role in mathematics are widely accepted by mathematicians today.
Category: Geometry

[8] viXra:1003.0221 [pdf] submitted on 7 Mar 2010

Combinatorial Geometry with Applications to Field Theory

Authors: Linfan Mao
Comments: 499 pages

Anyone maybe once heard the proverb of the six blind men with an elephant, in which these blind men were asked to determine what an elephant looks like by touch different parts of the elephant's body. The man touched its leg, tail, trunk, ear, belly or tusk claims that the elephant is like a pillar, a rope, a tree branch, a hand fan, a wall or a solid pipe, respectively. Each of them insisted his view right. They entered into an endless argument. All of you are right! A wise man explains to them: why are you telling it differently is because each one of you touched the different part of the elephant. So, actually the elephant has all those features what you all said.
Category: Geometry

[7] viXra:1003.0187 [pdf] submitted on 6 Mar 2010

A Generalization of a Leibniz Geometrical Theorem

Authors: Mihály Bencze, Florin Popovici, Florentin Smarandache
Comments: 5 pages

In this article we present a generalization of a Leibniz's theorem in geometry and an application of this.
Category: Geometry

[6] viXra:1003.0164 [pdf] submitted on 6 Mar 2010

Generalizations of Ceva's Theorem and Applications

Authors: Florentin Smarandache
Comments: 7 pages

In these paragraphs one presents three generalizations of the famous theorem of Ceva
Category: Geometry

[5] viXra:1003.0162 [pdf] submitted on 6 Mar 2010

A Generalization in Space of Jung's Theorem

Authors: Florentin Smarandache
Comments: 2 pages

In this short note we will prove a generalization of Joung's theorem in space.
Category: Geometry

[4] viXra:1003.0116 [pdf] submitted on 6 Mar 2010

Mixed Noneuclidean Geometries

Authors: Florentin Smarandache
Comments: 23 pages

The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the apparently unscientific ideas into scientific ones, and to develop their study (The Theory of Imperfections). And finally, to interconnect opposite (and not only) human fields of knowledge into as-heterogeneous-as-possible another fields.
Category: Geometry

[3] viXra:1003.0058 [pdf] submitted on 6 Mar 2010

The Dual of the Orthopole Theorem

Authors: Ion Pătraşcu
Comments: 5 pages, Translated by Prof. Florentin Smarandache

In this article we prove the theorems of the orthopole and we obtain, through duality, its dual, and then some interesting specific examples of the dual of the theorem of the orthopole.
Category: Geometry

[2] viXra:1003.0057 [pdf] submitted on 6 Mar 2010

DE Longchamps' Point, Line and Circle

Authors: Ion Pătraşcu
Comments: 7 pages, Translated by Prof. Florentin Smarandache

The purpose of this article is to familiarize the reader with these notions, emphasizing on connections between them.
Category: Geometry

[1] viXra:1003.0056 [pdf] submitted on 6 Mar 2010

The Dual Theorem Relative to the Simson's Line

Authors: Ion Pătraşcu
Comments: 5 pages, Translated by Prof. Florentin Smarandache

In this article we elementarily prove some theorems on the poles and polars theory, we present the transformation using duality and we apply this transformation to obtain the dual theorem relative to the Samson's line.
Category: Geometry

Replacements of recent Submissions

[48] viXra:1407.0027 [pdf] replaced on 2014-07-23 12:52:51

Moments Defined by Example Subdivision Curves

Authors: Jan Hakenberg
Comments: 25 Pages.

We list examples of 2-dimensional domains bounded by subdivision curves together with their exact area, centroid, and inertia. We assume homogeneous mass-distribution within the space bounded by the curve. The subdivision curves that we consider are generated by 1) the low order B-spline schemes, 2) the generalized, interpolatory C^1 four-point scheme, as well as 3) the more recent dual C^2 four-point scheme. The derivation of the (d + 2)-linear form that computes the area moment of degree p + q = d based on the initial control points for a given subdivision scheme is deferred to a publication in the near future.
Category: Geometry

[47] viXra:1404.0409 [pdf] replaced on 2014-06-04 23:42:12

On the System Analysis of the Foundations of Trigonometry

Authors: Temur Z. Kalanov
Comments: 22 Pages.

Analysis of the foundations of standard trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of trigonometry: standard trigonometry is a false theory.
Category: Geometry

[46] viXra:1404.0018 [pdf] replaced on 2014-04-03 21:57:25

Equidistant Curve Coordinate System(inversions(9))

Authors: Morio Kikuchi
Comments: 12 Pages.

We generalize inversion mathematically(4).
Category: Geometry

[45] viXra:1402.0024 [pdf] replaced on 2014-02-25 05:42:37

The Ptolemy Theorem in Conics (2)

Authors: Cheng Tianren
Comments: 16 Pages.

We define and study a transformation in the triangle plane called the orthocorrespondence.this transformation leads to the consideration of a family of circular circumcubics containing the neuberg cubic. we study kiepert triangles and their iterations ,the kiepert triangles relative to kiepert triangle .for arbitrary and ,we show that .we also introduce the parasix configuration ,which consists of two congruent triangles. at last,we apply the property of the aberrancy of a plane curve,and also use the problem known as the “twisted cylinder” and the “sweeping tangent” to parameterize the conics we get above.
Category: Geometry

[44] viXra:1402.0013 [pdf] replaced on 2014-03-26 21:50:04

Equidistant Curve Coordinate System(inversions(8))

Authors: Morio Kikuchi
Comments: 8 Pages.

We generalize inversion mathematically(3).
Category: Geometry

[43] viXra:1401.0219 [pdf] replaced on 2014-02-01 05:34:17

A New Slant on Lebesgue’s Universal Covering Problem

Authors: Philip E Gibbs
Comments: 24 Pages.

Lebesgue’s universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypothesis based on the conjectures. A new upper bound of 0.844112 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous best results by removing corners from a regular hexagon at a small slant angle to the edges of the dodecahedron used before. Simulations indicate that the minimum area for a convex universal cover is likely to be around 0.84408.
Category: Geometry

[42] viXra:1401.0206 [pdf] replaced on 2014-02-07 21:18:57

Differential Forms, de Rham Cohomology of Manifolds and Applications (Reading Report)

Authors: Ren Shiquan
Comments: 35 Pages. This is the reading report I and II on differential forms and de Rham cohomology of manifolds respectively. These reports are results of our group discussion and may include mistakes. Thanks.

In report I, we study differential forms on a manifold. We first give the definition of differential forms. Then the exterior derivative, Lie derivative, and integrations of differential forms are discussed. Finally we will look at a special family of differential forms, called harmonic forms. In report II, we study topological structures of manifolds using differential forms. We first state the de Rham cohomology Theorem and introduce Cech cohomology as a tool. Then we discuss about Hodge Theorem. Finally, we study some applications of these theorems.
Category: Geometry

[41] viXra:1401.0011 [pdf] replaced on 2014-04-02 21:18:40

Equidistant Curve Coordinate System(inversions(7))

Authors: Morio Kikuchi
Comments: 12 Pages.

We generalize inversion mathematically(2).
Category: Geometry

[40] viXra:1312.0109 [pdf] replaced on 2014-01-13 19:55:46

Equidistant Curve Coordinate System(inversions(6))

Authors: Morio Kikuchi
Comments: 12 Pages.

We generalize inversion mathematically.
Category: Geometry

[39] viXra:1312.0075 [pdf] replaced on 2014-02-23 13:54:55

Notes on Noncommutative Geometry

Authors: Igor Nikolaev
Comments: 181 Pages. Chapters 4 and 6 are posted

The text consists of an introduction, table of contents and Chapters 1, 4, 5 and 6 of a 300 pages book
Category: Geometry

[38] viXra:1311.0141 [pdf] replaced on 2014-01-13 19:53:33

Equidistant Curve Coordinate System(inversions(5))

Authors: Morio Kikuchi
Comments: 11 Pages.

We generalize inversion.
Category: Geometry

[37] viXra:1311.0115 [pdf] replaced on 2013-11-28 13:07:36

The Iso-Dual Tesseract

Authors: Nathan O. Schmidt
Comments: 16 pages, 4 figures, accepted in Algebras, Groups and Geometries

In this work, we deploy Santilli's iso-dual iso-topic lifting and Inopin's holographic ring (IHR) topology as a platform to introduce and assemble a tesseract from two inter-locking, iso-morphic, iso-dual cubes in Euclidean triplex space. For this, we prove that such an "iso-dual tesseract" can be constructed by following a procedure of simple, flexible, topologically-preserving instructions. Moreover, these novel results are significant because the tesseract's state and structure are directly inferred from the one initial cube (rather than two distinct cubes), which identifies a new iso-geometrical inter-connection between Santilli's exterior and interior dynamical systems.
Category: Geometry

[36] viXra:1309.0158 [pdf] replaced on 2013-09-23 14:12:48

Kissing Number Cells and Integral Conjecture

Authors: Antony Ryan
Comments: 7 Pages.

Kissing Numbers (1) appear to be the product of dimension number and the dimension’s simplex vertex number for 0-3 Euclidean spatial dimensions, but depart from the linear product of dimension and dimension+1 relationship at 4-dimensions and above increasing away from this exponentially. For 0-8 dimensions there is a Coxeter Number root system type relationship. The author proposes a very simple relationship which satisfies both aforementioned patterns, but extends from dimension 0 infinitely upwards. The conjecture is seen to satisfy the non-root system 24-dimensions and leads to prediction. The simplex nature of this work may be utilised in Quantum Gravity theories similar to Causal Dynamical Triangulation.
Category: Geometry

[35] viXra:1308.0126 [pdf] replaced on 2014-05-18 23:13:07

A Circle Without Pi

Authors: O. V. Vijimo
Comments: 40 Pages. Undergoing Review in a International Peer Reviewed Journal

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "$\pi$" is simply unthinkable but it’s going to be a reality very soon. "$\pi$" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "$\tau$". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category: Geometry

[34] viXra:1308.0126 [pdf] replaced on 2013-09-25 19:30:39

A Circle Without Pi

Authors: O. V. Vijimo
Comments: 40 Pages; 25 Figures; Under Review in an Int. Peer Reviewed Jrnl.

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "$\pi$" is simply unthinkable but it’s going to be a reality very soon. "$\pi$" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "$\tau$". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity
Category: Geometry

[33] viXra:1308.0126 [pdf] replaced on 2013-08-31 19:20:30

A Circle Without Pi

Authors: O. V. Vijimon
Comments: 40 Pages; 25 Figures; Under Review in an Int. Peer Reviewed Jrnl.

This paper provides the proof of invalidity of the most fundamental constant known to mankind. Imagining a circle without "{Pi}" is simply unthinkable but it’s going to be a reality very soon. "{Pi}" is not a true circle constant. This paper explores this idea and proposes a new constant in the process which gives the correct measure of a circle. It is given by "{Tau}". As a result, it redefines the area of the circle. The circle area currently accounted is wrong and therefore needs correction. This has serious implications for science. I have also discovered the fundamental geometrical ratio b/w a circle and a square in which it’s inscribed and have also discovered a new circle formula. This paper makes this strong case with less ambiguity.
Category: Geometry

[32] viXra:1307.0066 [pdf] replaced on 2013-07-31 10:30:23

Supermatematica Profesorului Şelariu

Authors: Florentin Smarandache
Comments: 10 Pages.

Acest articol este o scurtă trecere în revistă a cărţii “SuperMatematica. Fundamente”, Vol. 1 şi Vol. 2, ediţia a II-a, 2012, care constituie un domeniu nou de cercetare şi cu multe aplicaţii, iniţiat de profesorul universitar Mircea Eugen Şelariu. Lucrarea sa este unică în literatura mondială, deoarece combină matematica centrică cu matematica excentrică.
Category: Geometry

[31] viXra:1306.0233 [pdf] replaced on 2013-12-31 19:09:35

The Projective Line as a Meridian

Authors: Kelly McKennon
Comments: 114 pages and 24 figures.

Descriptions of 1-dimensional projective space in terms of the cross ratio, in one-dimensional geometry as a projective line, in two-dimensional geometry as a circle, and in three-dimensional geometry as a regulus. A characterization of projective 3-space is given in terms of polarity. This paper differs from the original version by addition of a section showing that the circle is distinguished from other meridians by its compactness and the existence of exponential functions.
Category: Geometry

[30] viXra:1304.0074 [pdf] replaced on 2013-04-19 20:49:23

Equidistant Curve Coordinate System(inversions(4))

Authors: Morio Kikuchi
Comments: 8 Pages.

There are two directions in inversion.
Category: Geometry

[29] viXra:1303.0146 [pdf] replaced on 2013-04-03 19:52:20

Equidistant Curve Coordinate System(inversions(3))

Authors: Morio Kikuchi
Comments: 14 Pages.

The types of inversions are made clear.
Category: Geometry

[28] viXra:1303.0015 [pdf] replaced on 2013-03-03 12:08:50

Eccentricity, Space Bending, Dimmension

Authors: Marian Nitu, Florentin Smarandache, Mircea Eugen Selariu
Comments: 23 Pages.

This work central idea is to present new transformations, previously non - existent in Ordinary mathematics, named centric mathematics ( CM) but that became possible due to new born eccentric mathematics, and, implicit, to supermathematics. As shown in this work, the new geometric transformations, named conversion or transfiguration, wipes the boundaries between discrete and continuous geometric forms, showing that the first ones are also continuous, being just apparently discontinuous.
Category: Geometry

[27] viXra:1211.0134 [pdf] replaced on 2012-11-26 06:15:14

Law of Sums of the Squares of Areas, Volumes and Hyper Volumes of Regular Polytopes from Clifford Polyvectors

Authors: Carlos Perelman, Fang Fang, Garret Sadler, Klee Irwin
Comments: 9 Pages.

Inspired by the recent sums of the squares law obtained by Kovacs-Fang-Sadler-Irwin we derive the law of the sums of the squares of the areas, volumes and hyper-volumes associated with the faces, cells and hyper-cells of regular polytopes in diverse dimensions after using Clifford algebraic methods.
Category: Geometry

[26] viXra:1211.0099 [pdf] replaced on 2013-07-14 16:04:32

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 16 Pages.

A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method for applying the above defined product of distributions to polyhedron vertices is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[25] viXra:1211.0099 [pdf] replaced on 2013-04-23 12:53:29

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 11 Pages.

A method for dealing with the product of step discontinuous and delta functions is proposed. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[24] viXra:1211.0099 [pdf] replaced on 2013-02-15 11:34:01

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 20 Pages.

A method for dealing with the product of step discontinuous and delta functions is proposed. A new space of generalised functions extending the space D', together with a well defined product, is constructed. The new space of generalized functions is used to prove interesting equalities involving products among elements of D'. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the well known defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[23] viXra:1211.0099 [pdf] replaced on 2013-01-03 09:05:50

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 19 Pages.

A method for dealing with the product of step discontinuous and delta function is proposed. A new space of generalised function, extending the space D', is constructed. The new space of generalised functions is used to show why it is not possible to define the most general product, among steps, deltas and delta derivatives. The new space of generalized function is used also to prove interesting equalities involving products among elements of D'. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[22] viXra:1211.0099 [pdf] replaced on 2012-12-18 17:45:07

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 14 Pages.

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective. The method is extended to the product of more general step discontinuous distributions and to the product of distributions in a multidimensional case. A space extension of generalised functions, in which product of step and delta functions is commutative and associative, is constructed. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[21] viXra:1211.0099 [pdf] replaced on 2012-12-17 17:51:53

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 14 Pages.

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective. The method is extended to the product of more general step discontinuous distributions and to the product of distributions in a multidimensional case. A space extension of generalised functions, in which product of step and delta functions is commutative and associative, is constructed. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus.
Category: Geometry

[20] viXra:1211.0099 [pdf] replaced on 2012-11-30 14:28:45

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 12 Pages.

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to the Colombeau theory but different in the formalism and the perspective, which make it particularly suitable for applications in differential geometry. The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case. Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus. Key Words: distribution theory, product of distributions, discrete differential geometry.
Category: Geometry

[19] viXra:1211.0099 [pdf] replaced on 2012-11-21 16:25:23

Product of Distributions Applied to Discrete Differential Geometry

Authors: Vincenzo Nardozza
Comments: 14 Pages. reason for new issue: fix of minor typo. examples added in the appendix

A method for dealing with the product of step discontinuities and Dirac delta functions, related each other by a continuous function, is proposed. The proposed method is similar, for many aspects, to Colombeau theory but different in the formalism and the perspective, which make it particularly suitable for applications in differential geometry. The method is extended to the product of more general distributions and to the product of distributions in a multidimensional case. Further points on product of distributions are discussed showing, among other thing, that the proposed product is associative and commutative. A standard method, for applying the above defined product of distributions to polyhedron vertices, is analysed and the method is applied to a special case where the famous defect angle formula, for the discrete curvature of polyhedra, is derived using the tools of tensor calculus. An elementary application to the theory of differential equations is discussed in the appendix.
Category: Geometry

[18] viXra:1205.0088 [pdf] replaced on 2012-07-24 23:40:57

Equidistant Curve Coordinate System(klein)

Authors: Morio Kikuchi
Comments: 8 Pages.

A point in the disk is represented by an intersection of two semiellipses in two directions.
Category: Geometry

[17] viXra:1204.0016 [pdf] replaced on 2013-02-24 19:54:30

Equidistant Curve Coordinate System(3 Dimensions(3), Inversions(2), Pseudo-Sphere)

Authors: Morio Kikuchi
Comments: 8 Pages.

A constant of length in an orthogonal sphere agrees with a constant of length in a plane which passes through origin.
Category: Geometry

[16] viXra:1203.0049 [pdf] replaced on 2012-07-24 23:33:01

Equidistant Curve Coordinate System(length, Area)

Authors: Morio Kikuchi
Comments: 15 Pages.

In equidistant curve coordinate system, the two expressions of the length between two points in disk and upper half-plane are the same.
Category: Geometry

[15] viXra:1201.0090 [pdf] replaced on 2013-02-10 20:57:47

Equidistant Curve Coordinate System(inversions)

Authors: Morio Kikuchi
Comments: 10 Pages.

In the inversion between two coordinate spheres, a ratio of length and constant of length is invariable.
Category: Geometry

[14] viXra:1112.0073 [pdf] replaced on 2013-01-03 19:57:22

Equidistant Curve Coordinate System(3 Dimensions(2), Hyperboloid)

Authors: Morio Kikuchi
Comments: 14 Pages.

In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry

[13] viXra:1111.0113 [pdf] replaced on 2012-07-24 23:23:58

Equidistant Curve Coordinate System(differential Forms)

Authors: Morio Kikuchi
Comments: 7 Pages.

The differential forms of two kinds in equidistant curve coordinate system can be changed into the differential forms in spherical orthogonal coordinate system by making a radius of the infinity sphere smaller limitlessly and making a constant of length larger limitlessly.
Category: Geometry

[12] viXra:1111.0005 [pdf] replaced on 2012-05-23 01:12:23

Equidistant Curve Coordinate System (3 Dimensions)

Authors: Morio Kikuchi
Comments: 15 Pages.

In three-dimensional equidistant curve coordinate system, a constant of length on a sphere depends upon its radius. Equidistant curve and round line have the relation of the inversion between a plane and a coordinate sphere, generally between a coordinate sphere and another coordinate sphere.
Category: Geometry

[11] viXra:1109.0061 [pdf] replaced on 2012-05-23 01:05:10

Equidistant Curve Coordinate System (Exterior Disk)

Authors: Morio Kikuchi
Comments: 7 Pages.

Two points on disk and exterior disk which have the same equidistant curve coordinates have the relation of the inversion on a circle which divides both regions. An isometry is realized between exterior disk and lower half-plane.
Category: Geometry

[10] viXra:1108.0023 [pdf] replaced on 2012-05-23 00:58:20

Equidistant Curve Coordinate System

Authors: Morio Kikuchi
Comments: 13 Pages.

An isometry is realized between disk of which radius is not limited to 1 and upper half-plane. Metrics are the same in both regions when equidistant curve coordinate system is used.
Category: Geometry

[9] viXra:1106.0062 [pdf] replaced on 2013-02-09 21:07:35

Spherical Orthogonal Coordinate System(3 Dimensions)

Authors: Morio Kikuchi
Comments: 10 Pages.

Product of metric coefficient and radius of round line is constant in spherical orthogonal coordinate system. Coordinates and so forth are constant in the coordinate transformation from orthogonal coordinates into spherical orthogonal coordinates if the value is special.
Category: Geometry

[8] viXra:1103.0077 [pdf] replaced on 2012-04-04 03:24:03

Spherical Orthogonal Coordinate System

Authors: Morio Kikuchi
Comments: 4 Pages.

Spherical orthogonal coordinate system agrees with plane orthogonal coordinate system in coordinates, length, and angle of an intersection. Using spherical orthogonal coordinate system, we can realize complex sphere to which complex number is indicated with no stereographic projection. By the coordinate transformation of the inversion which is characterized by swap of origin and point at infinity, three-dimensional orthogonal coordinates are transformed into new coordinates, namely three-dimensional spherical orthogonal coordinates, however coordinates and so forth are constant.
Category: Geometry

[7] viXra:1103.0043 [pdf] replaced on 23 May 2011

The Euclidean Philosophy of Universe ( Nature )

Authors: Markos Georgallides
Comments: 12 pages.

This article explains what is a Point, a Positive Space and a negative Anti-Space for their equilibrium, how points exist and their correlation also in Spaces . Any two points A,B on Spaces consist the first dimensional Unit AB, which has infinite bounded Spaces, Anti-Spaces and Sub-Spaces on unit AB . It is proved that when points A, B exist in a constant distance ds = AB, which is then a Restrained System of this Unit, then equilibrium under equal and opposite Impulses Pa, Pb on points A, B . This means that any distance AB of the Space is a DIPOLE or [ FMD = AB - Pa, Pb ], which is the first material unit . The unique case where at the points of Space and Anti-Space exist null Impulses, then is the Primary Neutral Space and it is obvious that the infinite Dipole ds = 0 → AB → ∞ move in this P.N.S . The position of points on Space /Space, Anti-Space/ Anti-Space Space / Anti-Space, Anti-Space / Space, creates (+) matter (-) antimatter (±) Neutral matter which moves in this Space with finite velocity and in case of the bounded Neutral Space AB, which may have zero Inertia, moves with infinite velocity . Since Neutral Space is the interval between Impulse ( which Impulse is the Principle of movement ) and Spaces ( which Spaces are the medium of movement ), therefore, Motion can alternatively occur itself as that of a Dipole = matter ( which is particle ) and as that of Impulses Pa, Pb ( which is a wave ) in the Neutral matter and Neutral Anti-matter . [ The one thing, say the light, is then as Particle and as Wave Structure ] Following the principle < Cause on → Communicator → the Obvious > is then explained that, Monads, can reproduce themselves through their bounded Communicator ( we may refer this as the DNA of the Monad ) . Following Euclidean logic for Spaces, and since one may use them as the first dimensional Unit ds = 0 → AB → ∞ in Geometry, Algebra, etc either as Dipole ds = AB, [ FMD = AB - Pa, Pb ] and since also Primary Neutral Space is proved to be Homogenous and Isotropic, then also in Mechanics and Physics and in all laws of Universe .
Category: Geometry

[6] viXra:1103.0035 [pdf] replaced on 13 Mar 2011

A Property of the Circumscribed Octagon

Authors: Ion Patrascu, Florentin Smarandache
Comments: 3 pages

In this article we'll obtain through the duality method a property in relation to the contact cords of the opposite sides of a circumscribable octagon.
Category: Geometry

[5] viXra:1101.0093 [pdf] replaced on 31 Jan 2011

Fast Approximation of π Using Regular Polyon

Authors: Jongsoo Park
Comments: 18 pages, In Korean

Fast Approximation of π Using Regular Polyon
Category: Geometry

[4] viXra:1009.0006 [pdf] replaced on 5 Sep 2010

Two Remarkable Ortho-Homological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 11 pages

In a previous paper [5] we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to orthological triangles, we emphasize on four important collinear points in the geometry of the triangle. Orthological / homological / orthohomological triangles in the 2D-space are generalized to orthological / homological / orthohomological polygons in 2D-space, and even more to orthological / homological / orthohomological triangles, polygons, and polyhedrons in 3D-space.
Category: Geometry

[3] viXra:1009.0006 [pdf] replaced on 4 Sep 2010

Two Remarkable Ortho-Homological Triangles

Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 10 pages

In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given triangle ABC and its first Brocard's triangle), and using the Sondat's theorem relative to orthological triangles, we emphasize on four important collinear points in the geometry of the triangle.
Category: Geometry

[2] viXra:1005.0016 [pdf] replaced on 2012-04-18 09:35:00

Generalization of a Remarkable Theorem

Authors: Ion Patrascu, Florentin Smarandache
Comments: 3 Pages.

Professor Claudiu Coandă proved, using the barycentric coordinates, a remarkable theorem. We generalize this theorem using some results from projective geometry relative to the pole and polar notions.
Category: Geometry

[1] viXra:1003.0272 [pdf] replaced on 3 Apr 2010

Eight Solved and Eight Open Problems in Elementary Geometry

Authors: Florentin Smarandache
Comments: 12 pages

In this paper we review nine previous proposed and solved problems of elementary 2D geometry [4] and [6], and we extend them either from triangles to polygons or polyhedrons,or from circles to spheres (from 2D-space to 3D-space), and make some comments about them.
Category: Geometry