Previous months:
2010 - 1003(11) - 1004(4) - 1005(2) - 1006(7) - 1007(2) - 1008(5) - 1009(8) - 1010(5) - 1011(1)
2011 - 1101(4) - 1102(3) - 1103(7) - 1104(18) - 1106(3) - 1107(4) - 1108(2) - 1109(1) - 1110(2) - 1111(3) - 1112(1)
2012 - 1201(4) - 1202(1) - 1203(3) - 1204(3) - 1205(6) - 1208(1) - 1209(1) - 1210(2) - 1211(4)
2013 - 1301(1) - 1303(4) - 1304(2) - 1305(2)
Any replacements are listed further down
[129] viXra:1305.0022 [pdf] submitted on 2013-05-03 23:36:36
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
[128] viXra:1305.0013 [pdf] submitted on 2013-05-03 01:15:25
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
[127] viXra:1304.0074 [pdf] submitted on 2013-04-16 20:10:05
Authors: Morio Kikuchi
Comments: 8 Pages.
There are two directions in an inversion.
Category: Geometry
[126] viXra:1304.0016 [pdf] submitted on 2013-04-04 04:06:02
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
[125] viXra:1303.0146 [pdf] submitted on 2013-03-19 23:14:58
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[124] viXra:1303.0130 [pdf] submitted on 2013-03-17 17:44:41
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
[123] viXra:1303.0104 [pdf] submitted on 2013-03-14 11:07:48
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
[122] viXra:1303.0015 [pdf] submitted on 2013-03-03 10:51:13
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
[121] viXra:1301.0143 [pdf] submitted on 2013-01-23 10:28:29
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
[120] viXra:1211.0134 [pdf] submitted on 2012-11-22 21:32:20
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
[119] viXra:1211.0099 [pdf] submitted on 2012-11-18 14:50:51
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
[118] viXra:1211.0024 [pdf] submitted on 2012-11-05 13:29:38
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
[117] viXra:1211.0023 [pdf] submitted on 2012-11-05 13:31:16
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
[116] viXra:1210.0006 [pdf] submitted on 2012-10-01 22:56:01
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
[115] viXra:1210.0005 [pdf] submitted on 2012-10-01 23:08:15
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
[114] viXra:1209.0108 [pdf] submitted on 2012-09-29 11:14:31
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
[113] viXra:1208.0070 [pdf] submitted on 2012-08-16 17:24:24
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
[112] viXra:1205.0092 [pdf] submitted on 2012-05-23 20:05:38
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
[111] viXra:1205.0088 [pdf] submitted on 2012-05-23 00:46:33
Authors: Morio Kikuchi
Comments: 15 Pages.
A point in the disk is represented by an intersection of two semiellipses in two directions.
Category: Geometry
[110] viXra:1205.0060 [pdf] submitted on 2012-05-13 16:00:45
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
[109] viXra:1205.0055 [pdf] submitted on 2012-05-11 20:15:25
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
[108] viXra:1205.0051 [pdf] submitted on 2012-05-09 07:44:01
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
[107] viXra:1205.0003 [pdf] submitted on 2012-05-02 23:20:08
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
[106] viXra:1204.0097 [pdf] submitted on 2012-04-27 09:27:47
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
[105] viXra:1204.0096 [pdf] submitted on 2012-04-27 09:29:37
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
[104] viXra:1204.0016 [pdf] submitted on 2012-04-04 03:08:17
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
[103] viXra:1203.0049 [pdf] submitted on 2012-03-14 23:17:33
Authors: Morio Kikuchi
Comments: 15 Pages.
In equidistant curve coordinate system, the two equations of a length between two points in disk and upper half-plane are the same.
Category: Geometry
[102] viXra:1203.0006 [pdf] submitted on 2012-03-02 10:47:14
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
[101] viXra:1203.0001 [pdf] submitted on 2012-03-01 05:51:13
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
[100] viXra:1202.0032 [pdf] submitted on 2012-02-11 17:14:46
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
[99] viXra:1201.0090 [pdf] submitted on 2012-01-24 22:36:38
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
[98] viXra:1201.0071 [pdf] submitted on 2012-01-18 07:30:21
Authors: Garreth H. Tembo
Comments: 3 Pages.
The inspiration for this theorem was entirely the result of observations of the catenary
curves of power cables and telephone lines near my high School and home
that afforded me the imaginary problem of trying to solve for their centers thus leading to
this theorem.
(Garreth h. Gothaven)
TEMBO’S THEOREM:
Category: Geometry
[97] viXra:1201.0061 [pdf] submitted on 2012-01-15 22:14:03
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
[96] viXra:1201.0047 [pdf] submitted on 2012-01-10 09:29:35
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
[95] viXra:1112.0073 [pdf] submitted on 2011-12-26 02:16:22
Authors: Morio Kikuchi
Comments: 13 Pages.
In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry
[94] viXra:1111.0113 [pdf] submitted on 29 Nov 2011
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 infinity sphere smaller limitlessly and making a constant of length larger
limitlessly.
Category: Geometry
[93] viXra:1111.0092 [pdf] submitted on 24 Nov 2011
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
[92] viXra:1111.0005 [pdf] submitted on 2 Nov 2011
Authors: Morio Kikuchi
Comments: 16 Pages.
PIn 3-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
[91] viXra:1110.0072 [pdf] submitted on 28 Oct 2011
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
[90] viXra:1110.0066 [pdf] submitted on 25 Oct 2011
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
[89] viXra:1109.0061 [pdf] submitted on 28 Sep 2011
Authors: Morio Kikuchi
Comments: 7 pages
Two points on Poincaré 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
[88] viXra:1108.0023 [pdf] submitted on 18 Aug 2011
Authors: Morio Kikuchi
Comments: 14 pages
A isometry is realized between Poincare disk of which radius is not limited to 1 and
upper half-plane. Poincare metrics are the same in both regions when equidistant curve coordinate system is used.
Category: Geometry
[87] viXra:1108.0008 [pdf] submitted on 4 Aug 2011
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
[86] viXra:1107.0008 [pdf] submitted on 4 Jul 2011
Authors: Chun-Xuan Jiang
Comments: 3 pages
In this paper we prove ...(see paper) part 6
Category: Geometry
[85] viXra:1107.0007 [pdf] submitted on 4 Jul 2011
Authors: Chun-Xuan Jiang
Comments: 3 pages
In this paper we prove ...(see paper) part 5
Category: Geometry
[84] viXra:1107.0006 [pdf] submitted on 4 Jul 2011
Authors: Chun-Xuan Jiang
Comments: 3 pages
In this paper we prove ...(see paper) part 4
Category: Geometry
[83] viXra:1107.0005 [pdf] submitted on 3 Jul 2011
Authors: Mircea Eugen Selariu
Comments: 18 pages, In Romanian
LOBE EXTERIOARE SI CVAZILOBE INTERIOARE
CERCULUI UNITATE
Category: Geometry
[82] viXra:1106.0062 [pdf] submitted on 28 Jun 2011
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
[81] viXra:1106.0058 [pdf] submitted on 27 Jun 2011
Authors: Mircea Selariu
Comments: 20 pages.
TEOREMA S A BISECTOARELOR UNUI PATRULATER INSCRIPTIBIL SI TEOREMELE S ALE TRIUNGHIULUI
Category: Geometry
[80] viXra:1106.0057 [pdf] submitted on 27 Jun 2011
Authors: Mircea Selariu
Comments: 13 pages.
NOI LINII CONCURENTE SI UN NOU PUNCT DE INTERSECTIE
INTR-UN TRIUNGHI
Category: Geometry
[79] viXra:1104.0079 [pdf] submitted on 19 Apr 2011
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
[78] viXra:1104.0078 [pdf] submitted on 19 Apr 2011
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
[77] viXra:1104.0077 [pdf] submitted on 19 Apr 2011
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
[76] viXra:1104.0076 [pdf] submitted on 19 Apr 2011
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
[75] viXra:1104.0075 [pdf] submitted on 19 Apr 2011
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
[74] viXra:1104.0074 [pdf] submitted on 19 Apr 2011
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
[73] viXra:1104.0073 [pdf] submitted on 19 Apr 2011
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
[72] viXra:1104.0072 [pdf] submitted on 19 Apr 2011
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
[71] viXra:1104.0071 [pdf] submitted on 19 Apr 2011
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
[70] viXra:1104.0070 [pdf] submitted on 19 Apr 2011
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
[69] viXra:1104.0069 [pdf] submitted on 19 Apr 2011
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
[68] viXra:1104.0068 [pdf] submitted on 19 Apr 2011
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
[67] viXra:1104.0062 [pdf] submitted on 20 Apr 2011
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
[66] viXra:1104.0061 [pdf] submitted on 20 Apr 2011
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
[65] viXra:1104.0060 [pdf] submitted on 20 Apr 2011
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
[64] viXra:1104.0059 [pdf] submitted on 20 Apr 2011
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
[63] viXra:1104.0054 [pdf] submitted on 18 Apr 2011
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
[62] viXra:1104.0053 [pdf] submitted on 17 Apr 2011
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
[61] viXra:1103.0119 [pdf] submitted on 31 Mar 2011
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
[60] viXra:1103.0077 [pdf] submitted on 19 Mar 2011
Authors: Morio Kikuchi
Comments: 3 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, 3-dimensional
orthogonal coordinates are transformed into new coordinates, namely 3-dimensional
spherical orthogonal coordinstes, however coordinates and so forth are constant.
Category: Geometry
[59] viXra:1103.0076 [pdf] submitted on 19 Mar 2011
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
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
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
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
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
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
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
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
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
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
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
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
Authors: Florentin Smarandache
Comments: 1 pages
Let's consider the points...
Category: Geometry
[46] viXra:1010.0060 [pdf] submitted on 28 Oct 2010
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Authors: Ion Pătraşcu, Florentin Smarandache
Comments: 3 pages
NIn 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
[66] viXra:1304.0074 [pdf] replaced on 2013-04-19 20:49:23
Authors: Morio Kikuchi
Comments: 8 Pages.
There are two directions in inversion.
Category: Geometry
[65] viXra:1303.0146 [pdf] replaced on 2013-04-03 19:52:20
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[64] viXra:1303.0146 [pdf] replaced on 2013-04-01 20:19:43
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[63] viXra:1303.0146 [pdf] replaced on 2013-03-31 20:22:39
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[62] viXra:1303.0146 [pdf] replaced on 2013-03-27 20:39:47
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[61] viXra:1303.0146 [pdf] replaced on 2013-03-24 23:20:31
Authors: Morio Kikuchi
Comments: 14 Pages.
The types of inversions are made clear.
Category: Geometry
[60] viXra:1303.0015 [pdf] replaced on 2013-03-03 12:08:50
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
[59] viXra:1211.0134 [pdf] replaced on 2012-11-26 06:15:14
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
[58] viXra:1211.0099 [pdf] replaced on 2013-04-23 12:53:29
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
[57] viXra:1211.0099 [pdf] replaced on 2013-02-15 11:34:01
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
[56] viXra:1211.0099 [pdf] replaced on 2013-01-03 09:05:50
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
[55] viXra:1211.0099 [pdf] replaced on 2012-12-18 17:45:07
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
[54] viXra:1211.0099 [pdf] replaced on 2012-12-17 17:51:53
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
[53] viXra:1211.0099 [pdf] replaced on 2012-11-30 14:28:45
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
[52] viXra:1211.0099 [pdf] replaced on 2012-11-21 16:25:23
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
[51] viXra:1205.0088 [pdf] replaced on 2012-07-24 23:40:57
Authors: Morio Kikuchi
Comments: 8 Pages.
A point in the disk is represented by an intersection of two semiellipses in two directions.
Category: Geometry
[50] viXra:1204.0016 [pdf] replaced on 2013-02-24 19:54:30
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
[49] viXra:1204.0016 [pdf] replaced on 2013-01-03 20:01:43
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
[48] viXra:1204.0016 [pdf] replaced on 2012-07-24 23:38:02
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
[47] viXra:1204.0016 [pdf] replaced on 2012-05-23 01:25:26
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
[46] viXra:1203.0049 [pdf] replaced on 2012-07-24 23:33:01
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
[45] viXra:1203.0049 [pdf] replaced on 2012-05-23 01:18:55
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
[44] viXra:1203.0049 [pdf] replaced on 2012-04-04 04:15:21
Authors: Morio Kikuchi
Comments: 14 Pages.
In equidistant curve coordinate system, the two equations of a length between two points in disk and upper half-plane are the same.
Category: Geometry
[43] viXra:1201.0090 [pdf] replaced on 2013-02-10 20:57:47
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
[42] viXra:1201.0090 [pdf] replaced on 2012-04-04 04:10:54
Authors: Morio Kikuchi
Comments: 9 Pages.
In the inversion between two coordinate spheres, a ratio of length and constant of length is invariable.
Category: Geometry
[41] viXra:1201.0090 [pdf] replaced on 2012-03-15 00:29:59
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
[40] viXra:1112.0073 [pdf] replaced on 2013-01-03 19:57:22
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
[39] viXra:1112.0073 [pdf] replaced on 2012-07-24 23:28:28
Authors: Morio Kikuchi
Comments: 13 Pages.
In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry
[38] viXra:1112.0073 [pdf] replaced on 2012-04-04 04:06:42
Authors: Morio Kikuchi
Comments: 13 Pages.
In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry
[37] viXra:1112.0073 [pdf] replaced on 2012-03-15 00:25:22
Authors: Morio Kikuchi
Comments: 13 Pages.
In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry
[36] viXra:1112.0073 [pdf] replaced on 2012-01-24 22:31:57
Authors: Morio Kikuchi
Comments: 13 Pages.
In hyperboloid model, a metric by use of equidistant curve coordinate system is obtained by parametric representation.
Category: Geometry
[35] viXra:1111.0113 [pdf] replaced on 2012-07-24 23:23:58
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
[34] viXra:1111.0113 [pdf] replaced on 2012-04-04 04:00:37
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
[33] viXra:1111.0113 [pdf] replaced on 2012-03-15 00:17:49
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 an infinity sphere smaller limitlessly and making a constant of length larger limitlessly.
Category: Geometry
[32] viXra:1111.0113 [pdf] replaced on 2011-12-09 03:01:20
Authors: Morio Kikuchi
Comments: 7 Pages.
The differential forms of two kinds in equidistant curve coordinate system by making a radius of an infinity sphere smaller limitlessly and making a constant of length larger limitlessly.
Category: Geometry
[31] viXra:1111.0005 [pdf] replaced on 2012-05-23 01:12:23
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
[30] viXra:1111.0005 [pdf] replaced on 2012-04-04 03:54:05
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
[29] viXra:1111.0005 [pdf] replaced on 2012-03-15 00:08:57
Authors: Morio Kikuchi
Comments: 16 Pages.
In 3-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
[28] viXra:1111.0005 [pdf] replaced on 2012-01-24 22:25:38
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 coodinate system by making a radius of an infinity sphere smaller limitlessly and making a constant of length larger limitlessly.
Category: Geometry
[27] viXra:1111.0005 [pdf] replaced on 2011-12-09 02:54:50
Authors: Morio Kikuchi
Comments: 16 Pages.
In 3-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
[26] viXra:1109.0061 [pdf] replaced on 2012-05-23 01:05:10
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
[25] viXra:1109.0061 [pdf] replaced on 2012-04-04 03:46:28
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
[24] viXra:1109.0061 [pdf] replaced on 2012-03-15 00:00:19
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
[23] viXra:1109.0061 [pdf] replaced on 2012-01-24 22:16:42
Authors: Morio Kikuchi
Comments: 7 Pages.
Two points on Poincare 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
[22] viXra:1109.0061 [pdf] replaced on 2011-12-09 02:47:16
Authors: Morio Kikuchi
Comments: 7 Pages.
Two points on Poincare 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
[21] viXra:1108.0023 [pdf] replaced on 2012-05-23 00:58:20
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
[20] viXra:1108.0023 [pdf] replaced on 2012-04-04 03:39:21
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
[19] viXra:1108.0023 [pdf] replaced on 2012-03-14 23:52:36
Authors: Morio Kikuchi
Comments: 14 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
[18] viXra:1108.0023 [pdf] replaced on 2012-01-24 22:09:22
Authors: Morio Kikuchi
Comments: 14 Pages.
An isometry is realized between Poincare disk of which radius is not limited to 1 and upper half-plane. Poincare metrics are the same in both regions when equidistant curve coordinate system is used.
Category: Geometry
[17] viXra:1108.0023 [pdf] replaced on 2011-12-09 02:40:26
Authors: Morio Kikuchi
Comments: 14 Pages.
An isometry is realized between Poincare disk of which radius is not limited to 1 and upper half-plane. Poincare metrics are the same in both regions when equidistant curve coordinate system is used.
Category: Geometry
[16] viXra:1106.0062 [pdf] replaced on 2013-02-09 21:07:35
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
[15] viXra:1106.0062 [pdf] replaced on 2012-04-04 03:32:45
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
[14] viXra:1106.0062 [pdf] replaced on 2012-03-14 23:45:25
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
[13] viXra:1106.0062 [pdf] replaced on 2011-12-09 02:32:22
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
[12] viXra:1103.0077 [pdf] replaced on 2012-04-04 03:24:03
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
[11] viXra:1103.0077 [pdf] replaced on 2012-03-14 23:36:11
Authors: Morio Kikuchi
Comments: 5 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, 3-dimensional orthogonal coordinates are transformed into new coordinates, namely 3-dimensional spherical orthogonal coordinates, however coordinates and so forth are constant.
Category: Geometry
[10] viXra:1103.0077 [pdf] replaced on 2011-12-09 02:24:26
Authors: Morio Kikuchi
Comments: 5 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, 3-dimensional orthogonal coordinates are transformed into new coordinates, namely 3-dimensional spherical orthogonal coordinates, however coordinates and so forth are constant.
Category: Geometry
[9] viXra:1103.0077 [pdf] replaced on 29 May 2011
Authors: Morio Kikuchi
Comments: 5 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, 3-dimensional
orthogonal coordinates are transformed into new coordinates, namely 3-dimensional
spherical orthogonal coordinstes, however coordinates and so forth are constant.
Category: Geometry
[8] viXra:1103.0077 [pdf] replaced on 20 Mar 2011
Authors: Morio Kikuchi
Comments: 3 pages, v1 in Japanese, v2 in English.
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, 3-dimensional
orthogonal coordinates are transformed into new coordinates, namely 3-dimensional
spherical orthogonal coordinstes, however coordinates and so forth are constant.
Category: Geometry
[7] viXra:1103.0043 [pdf] replaced on 23 May 2011
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
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
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
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
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
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
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