Geometry

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Recent submissions

Any replacements are listed farther down

[364] viXra:1905.0353 [pdf] submitted on 2019-05-18 07:00:19

From Pythagorean Theorem to Cosine Theorem.

Authors: Jesús Álvarez Lobo
Comments: 2 Pages.

Easy and natural demonstration of the cosine theorem, based on the extension of the Pythagorean theorem.
Category: Geometry

[363] viXra:1905.0248 [pdf] submitted on 2019-05-16 19:31:52

Solution of a Vector-Triangle Problem Via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 3 Pages.

As a high-school-level application of Geometric Algebra (GA), we show how to solve a simple vector-triangle problem. Our method highlights the use of outer products and inverses of bivectors.
Category: Geometry

[362] viXra:1905.0219 [pdf] submitted on 2019-05-16 04:19:51

The Flow of Chern

Authors: Antoine Balan
Comments: 1 page, written in english

We propose a flow over a Kaehler manifold, called the Chern flow.
Category: Geometry

[361] viXra:1905.0217 [pdf] submitted on 2019-05-14 07:31:01

Three, Four and N-Dimensional Swastikas & their Projections

Authors: Sascha Vongehr
Comments: 10 pages, Six Figures, Keywords: Higher Dimensional Geometry; Hyper Swastika; Reclaiming of Symbols; Didactic Arts

Difficulties with generalizing the swastika shape for N dimensional spaces are discussed. While distilling the crucial general characteristics such as whether the number of arms is 2^N or 2N, a three dimensional (3D) swastika is introduced and then a construction algorithm for any natural number N so that it reproduces the 1D, 2D, and 3D shapes. The 4D hyper swastika and surfaces in its hypercube envelope are then presented for the first time.
Category: Geometry

[360] viXra:1905.0088 [pdf] submitted on 2019-05-05 17:04:39

Via Geometric (Clifford) Algebra: Equation for Line of Intersection of Two Planes

Authors: James A. Smith
Comments: 6 Pages.

As a high-school-level example of solving a problem via Geometric Algebra (GA), we show how to derive an equation for the line of intersection between two given planes. The solution method that we use emphasizes GA's capabilities for expressing and manipulating projections and rotations of vectors.
Category: Geometry

[359] viXra:1905.0030 [pdf] submitted on 2019-05-02 22:25:09

Foundations of Conic Conformal Geometric Algebra and Simplified Versors for Rotation, Translation and Scaling

Authors: Eckhard Hitzer, Stephen J. Sangwine
Comments: 15 Pages. submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of FTHD 2018, 21 Feb. 2019, 1 table, 1 figure.

This paper explains in algebraic detail how two-dimensional conics can be defined by the outer products of conformal geometric algebra (CGA) points in higher dimensions. These multivector expressions code all types of conics in arbitrary scale, location and orientation. Conformal geometric algebra of two-dimensional Euclidean geometry is fully embedded as an algebraic subset. With small model preserving modifications, it is possible to consistently define in conic CGA versors for rotation, translation and scaling, similar to [https://doi.org/10.1007/s00006-018-0879-2], but simpler, especially for translations. Keywords: Clifford algebra, conformal geometric algebra, conics, versors. Mathematics Subject Classification (2010). Primary 15A66; Secondary 11E88, 15A15, 15A09.
Category: Geometry

[358] viXra:1905.0026 [pdf] submitted on 2019-05-03 05:23:22

Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer, Dietmar Hildenbrand
Comments: 11 Pages. accepted for M. Gavrilova et al (eds.), Proceedings of Workshop ENGAGE 2019 at CGI 2019 with Springer LNCS, April 2019, 1 table.

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9,7) over the real sixteen dimensional vector space R^{9,7}. These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19,16). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections
Category: Geometry

[357] viXra:1904.0494 [pdf] submitted on 2019-04-25 11:47:18

The Flow of Hermite-Ricci

Authors: Antoine Balan
Comments: 1 page, written in french

We define a flow for hermitian manifolds. We call it the Hermite-Ricci flow.
Category: Geometry

[356] viXra:1904.0418 [pdf] submitted on 2019-04-21 08:23:26

Expanding Polynomials with Regular Polygons

Authors: Timothy W. Jones
Comments: 7 Pages.

Expanding the root form of a polynomial for large numbers of roots can be complicated. Such polynomials can be used to prove the irrationality of powers of pi, so a technique for arriving at expanded forms is needed. We show here how roots of polynomials can generate regular polygons whose vertices considered as roots form known expanded polynomials. The product of these polynomials can be simple enough to yield the desired expanded form.
Category: Geometry

[355] viXra:1904.0398 [pdf] submitted on 2019-04-20 11:06:10

Computing a Well-Connected Midsurface

Authors: Yogesh H. Kulkarni, Anil D. Sahasrabudhe, Muknd S. Kale
Comments: 4 Pages.

Computer-aided Design (CAD) models of thin-walled parts such as sheet metal or plastics are often reduced dimensionally to their corresponding midsurfaces for quicker and fairly accurate results of Computer-aided Engineering (CAE) analysis. Generation of the midsurface is still a time-consuming and mostly, a manual task due to lack of robust and automated techniques. Midsurface failures manifest in the form of gaps, overlaps, not-lying-halfway, etc., which can take hours or even days to correct. Most of the existing techniques work on the complex final shape of the model forcing the usage of hard-coded heuristic rules, developed on a case-by-case basis. The research presented here proposes to address these problems by leveraging feature-parameters, made available by the modern feature-based CAD applications, and by effectively leveraging them for sub-processes such as simplification, abstraction and decomposition. In the proposed system, at first, features which are not part of the gross shape are removed from the input sheet metal feature-based CAD model. Features of the gross-shape model are then transformed into their corresponding generic feature equivalents, each having a profile and a guide curve. The abstracted model is then decomposed into non-overlapping cellular bodies. The cells are classified into midsurface-patch generating cells, called ‘solid cells’ and patch-connecting cells, called ‘interface cells’. In solid cells, midsurface patches are generated either by offset or by sweeping the midcurve generated from the owner-feature’s profile. Interface cells join all the midsurface patches incident upon them. Output midsurface is then validated for correctness. At the end, real-life parts are used to demonstrate the efficacy of the approach.
Category: Geometry

[354] viXra:1904.0359 [pdf] submitted on 2019-04-18 15:46:35

The Ricci Flow for Connections

Authors: Antoine Balan
Comments: 1 page, written in english

We define a natural Ricci flow for connections over a Riemannian manifold.
Category: Geometry

[353] viXra:1904.0328 [pdf] submitted on 2019-04-16 10:22:17

A Special Geometry - and its Consequences

Authors: Ulrich E. Bruchholz
Comments: 5 Pages.

It is explained why the geometry of space-time, first found by Rainich, is generally valid. The equations of this geometry, the known Einstein-Maxwell equations, are discussed, and results are listed. We shall see how these tensor equations can be solved. As well, neutrosophics is more supported than dialectics. We shall find even more categories than described in neutrosophics.
Category: Geometry

[352] viXra:1904.0123 [pdf] submitted on 2019-04-06 21:12:45

A Note on the Arbelos in Wasan Geometry, Satoh's Problem and a Circle Pattern

Authors: Hiroshi Okumura
Comments: 3 Pages.

We generalize a problem in Wasan geometry involving an arbelos, and construct a self-similar circle pattern.
Category: Geometry

[351] viXra:1904.0023 [pdf] submitted on 2019-04-01 07:30:13

Dos Cuestiones de Geometría

Authors: Edgar Valdebenito
Comments: 3 Pages.

En esta nota mostramos dos cuestiones elementales de geometría.
Category: Geometry

[350] viXra:1904.0019 [pdf] submitted on 2019-04-01 09:08:00

A Closed 3-Form in Spinorial Geometry

Authors: Antoine Balan
Comments: 2 pages, written in english

We define a 3-form over a spinorial manifold by mean of the curvature tensor and the Clifford multiplication.
Category: Geometry

[349] viXra:1903.0566 [pdf] submitted on 2019-03-31 15:59:03

Division by Zero Calculus in Trigonometric Functions

Authors: Saburou Saitoh
Comments: 19 Pages. In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.

In this paper, we will introduce the division by zero calculus in triangles and trigonometric functions as the first stage in order to see the elementary properties.
Category: Geometry

[348] viXra:1903.0433 [pdf] submitted on 2019-03-24 19:08:38

Refutation of Euclidean Geometry Embedded in Hyperbolic Geometry for Equal Consistency

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

The following conjecture is refuted: "[An] n-dimensional Euclidean geometry can be embedded into (n+1)-dimensional hyperbolic non Euclidean geometry. Therefore hyperbolic non Euclidean geometry and Euclidean geometry are equally consistent, that is, either both are consistent or both are inconsistent." Hence, the conjecture is a non tautologous fragment of the universal logic VŁ4.
Category: Geometry

[347] viXra:1903.0317 [pdf] submitted on 2019-03-17 21:21:17

Denial of Consistency for the Lobachevskii Non Euclidean Geometry

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

We prove two parallel lines are tautologous in Euclidean geometry. We next prove that non Euclidean geometry of Lobachevskii is not tautologous and hence not consistent. What follows is that Riemann geometry is the same, and non Euclidean geometry is a segment of Euclidean geometry, not the other way around. Therefore non Euclidean geometries are a non tautologous fragment of the universal logic VŁ4.
Category: Geometry

[346] viXra:1903.0244 [pdf] submitted on 2019-03-12 10:13:23

A New Tensor in Differential Geometry

Authors: Antoine Balan
Comments: 2 pages, written in english

We propose a 3-form in differential geometry which depends only of a connection over the tangent fiber bundle.
Category: Geometry

[345] viXra:1903.0241 [pdf] submitted on 2019-03-12 13:24:51

A Note of Differential Geometry

Authors: Abdelmajid Ben Hadj Salem
Comments: 10 Pages.

In this note, we give an application of the Method of the Repère Mobile to the Ellipsoid of Reference in Geodesy using a symplectic approach.
Category: Geometry

[344] viXra:1903.0126 [pdf] submitted on 2019-03-07 10:25:27

Refutation of Riemannian Geometry as Generalization of Euclidean Geometry

Authors: Colin James III
Comments: 2 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the classical logic section on set theory, we evaluate definitions of the atom and primitive set. None is tautologous. From the quantum logic and topology section on set theory, we evaluate the disjoint union (as equivalent to the XOR operator) and variances in equivalents for the AND and OR operators. None is tautologous. This reiterates that set theory and quantum logic are not bivalent, and hence non-tautologous segments of the universal logic VŁ4. The assertion of Riemannian geometry as generalization of Euclidean geometry is not supported.
Category: Geometry

[343] viXra:1903.0100 [pdf] submitted on 2019-03-07 05:46:58

Proceedings on Non Commutative Geometry.

Authors: Johan Noldus
Comments: 67 Pages.

Non commutative geometry is developed from the point of view of an extension of quantum logic. We provide for an example of a non-abelian simplex as well as a non-abelian curved Riemannian space.
Category: Geometry

[342] viXra:1903.0082 [pdf] submitted on 2019-03-05 20:48:15

Solution Proof of Bellman's Lost in the Forest Problem for Triangles

Authors: Colin James III
Comments: 3 Pages. © Copyright 2019 by Colin James III All rights reserved. Respond to author by email only: info@cec-services dot com. See updated abstract at ersatz-systems.com. (We warn troll Mikko at Disqus to read the article four times before hormonal typing.)

From the area and dimensions of an outer triangle, the height point of an inner triangle implies the minimum distance to the outer triangle. This proves the solution of Bellman's Lost in the forest problem for triangles. By extension, it is the general solution proof for other figures.
Category: Geometry

[341] viXra:1903.0023 [pdf] submitted on 2019-03-01 09:40:43

Cluster Packaging of Spheres Versus Linear Packaging of Spheres

Authors: Helmut Söllinger
Comments: 10 Pages. language: German

The paper analyses the issue of optimised packaging of spheres of the same size. The question is whether a linear packaging of spheres in the shape of a sausage or a spatial cluster of spheres can minimise the volume enveloping the spheres. There is an assumption that for less than 56 spheres the linear packaging is denser and for 56 spheres the cluster is denser, but the question remains how a cluster of 56 spheres could look like. The paper shows two possible ways to build such a cluster of 56 spheres. The author finds clusters of 59, 62, 65, 66, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79 and 80 spheres - using the same method - which are denser than a linear packaging of the same number and gets to the assumption that all convex clusters of spheres of sufficient size are denser than linear ones.
Category: Geometry

[340] viXra:1902.0444 [pdf] submitted on 2019-02-25 06:00:04

On the Equivalence of Closed Figures and Infinitely Extended Lines and the Conclusions Drawn from it

Authors: Madhur Sorout
Comments: 12 Pages.

The equivalence of closed figures and infinitely extended lines may lead us to understand the physical reality of infinities. This paper doesn’t include what infinities mean in the physical world, but the paper is mainly focused on the equivalence of closed figures and infinitely extended lines. Using this principle, some major conclusions can be drawn. The equivalence of closed figures and infinitely extended lines is mainly based on the idea that closed figures and infinitely extended lines are equivalent. One of the most significant conclusions drawn from this equivalency is that if any object moves along a straight infinitely extended line, it will return back to the point, where it started to move, after some definite time.
Category: Geometry

[339] viXra:1902.0401 [pdf] submitted on 2019-02-23 08:07:46

Three-Dimensional Quadrics in Conformal Geometric Algebras and Their Versor Transformations

Authors: Eckhard Hitzer
Comments: 15 Pages. Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore a newly modified (compared to Breuils et al, 2018, https://doi. org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.
Category: Geometry

[338] viXra:1902.0370 [pdf] submitted on 2019-02-21 09:48:48

The G-connections

Authors: Antoine Balan
Comments: 1 page, written in english

We define the notion of G-connections over vector fiber bundles with action of a Lie group G.
Category: Geometry

[337] viXra:1902.0283 [pdf] submitted on 2019-02-16 15:24:00

The Symplectic Laplacian

Authors: Antoine Balan
Comments: 1 page, written in english

We construct a symplectic Laplacian which is a differential operator of order 1 depending only on a connection and a symplectic form.
Category: Geometry

[336] viXra:1902.0028 [pdf] submitted on 2019-02-02 12:41:24

A Generalized Clifford Algebra

Authors: Antoine Balan
Comments: 1 page, written in english

We propose a generalization of the Clifford algebra. We give application to the Dirac operator.
Category: Geometry

[335] viXra:1901.0471 [pdf] submitted on 2019-01-31 08:47:29

Blow-up of Feuerbach's Theorem

Authors: Alexander Skutin
Comments: 5 Pages.

In this short note we introduce the blow-up of the Feuerbach’s theorem.
Category: Geometry

[334] viXra:1901.0195 [pdf] submitted on 2019-01-14 11:16:25

A Closed 2-Form in Spinorial Geometry

Authors: Antoine Balan
Comments: 1 pages, written in english

We define a closed 2-form for any spinorial manifold. We deduce characteristic classes.
Category: Geometry

[333] viXra:1901.0162 [pdf] submitted on 2019-01-11 17:27:46

Archimedean Incircle of a Triangle

Authors: Hiroshi Okumura
Comments: 3 Pages.

We generalize several Archimedean circles, which are the incircles of special triangles.
Category: Geometry

[332] viXra:1901.0152 [pdf] submitted on 2019-01-11 06:31:41

Fractales : La Geometría del Caos

Authors: Edgar Valdebenito
Comments: 63 Pages.

Esta nota muestra una colección de fractales.
Category: Geometry

[331] viXra:1812.0226 [pdf] submitted on 2018-12-12 06:35:09

Elementary Fractals : Part V

Authors: Edgar Valdebenito
Comments: 108 Pages.

This note presents a collection of elementary fractals.
Category: Geometry

[330] viXra:1812.0206 [pdf] submitted on 2018-12-11 21:37:57

Solution of a Sangaku ``Tangency" Problem via Geometric Algebra

Authors: James A. Smith
Comments: 5 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to solve one of the beautiful \emph{sangaku} problems from 19th-Century Japan. Among the GA operations that prove useful is the rotation of vectors via the unit bivector i.
Category: Geometry

[329] viXra:1812.0090 [pdf] submitted on 2018-12-05 19:29:23

Refutation of the Planar Eucleadian R-Geometry of Tarski

Authors: Colin James III
Comments: 3 Pages. © Copyright 2016-2018 by Colin James III All rights reserved. Updated abstract at ersatz-systems.com . Respond to the author by email at: info@ersatz-systems dot com.

We evaluate the axioms of the title. The axiom of identity of betweenness and axiom Euclid are tautologous, but the others are not. The commonplace expression of the axiom of Euclid does not match its other two variations which is troubling. This effectively refutes the planar R-geometry.
Category: Geometry

[328] viXra:1812.0085 [pdf] submitted on 2018-12-04 07:19:40

Geometries of O

Authors: Hannes Hutzelmeyer
Comments: 93 Pages.

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial-axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.
Category: Geometry

[327] viXra:1812.0061 [pdf] submitted on 2018-12-03 06:42:06

Elementary Fractals: Part IV

Authors: Edgar Valdebenito
Comments: 119 Pages.

This note presents a collection of elementary Fractals.
Category: Geometry

[326] viXra:1811.0435 [pdf] submitted on 2018-11-26 06:16:16

Elementary Fractals: Part III

Authors: Edgar Valdebenito
Comments: 101 Pages.

This note presents a collection of elementary Fractals.
Category: Geometry

[325] viXra:1811.0324 [pdf] submitted on 2018-11-20 06:38:27

Elementary Fractals: Part II

Authors: Edgar Valdebenito
Comments: 113 Pages.

This note presents a collection of elementary Fractals.
Category: Geometry

[324] viXra:1811.0214 [pdf] submitted on 2018-11-13 06:40:17

Elementary Fractals: Part I

Authors: Edgar Valdebenito
Comments: 109 Pages.

This note presents a collection of elementary fractals.
Category: Geometry

[323] viXra:1811.0132 [pdf] submitted on 2018-11-08 20:40:58

The Arbelos in Wasan Geometry, Problems of Izumiya and Nait\=o

Authors: Hiroshi Okumura
Comments: 5 Pages.

We generalize two sangaku problems involving an arbelos proposed by Izumiya and Nait\=o, and show the existence of six non-Archimedean congruent circles.
Category: Geometry

[322] viXra:1811.0103 [pdf] submitted on 2018-11-06 09:20:05

Possbile Cubic to Spherical Transfomaions

Authors: Adham Ahmed Mohamed Ahmed
Comments: 1 Page.

this paper talks about a hypothesis between the cube and the sphere which is inside the cube and the excess volume of the cube than the sphere and the excess volume of the sphere where the cube is inside of the sphere What If you spin a cube around an axis passing through its midpoint of the cube would the cylinder formed have an excess in volume than the sphere equal to the excess in volume of the cylinder than he cube?
Category: Geometry

[321] viXra:1810.0421 [pdf] submitted on 2018-10-26 02:32:17

Derivative-Based Numerical Method for Penalty-Barrier Nonlinear Programming

Authors: Martin Peter Neuenhofen
Comments: 16 Pages.

We present an NLP solver for nonlinear optimization with quadratic penalty terms and logarithmic barrier terms. The method is suitable for large sparse problems. Each iteration has a polynomial time-complexity. The method has global convergence and local quadratic convergence, with a convergence radius that depends little on our method but rather on the geometry of the problem.
Category: Geometry

[320] viXra:1810.0379 [pdf] submitted on 2018-10-24 03:13:11

От ошибки Гильберта к исчислению сфер

Authors: Франц Герман
Comments: 9 Pages.

Что такое след проективной плоскости и как можно его увидеть рассказывается в этой заметке.
Category: Geometry

[319] viXra:1810.0324 [pdf] submitted on 2018-10-21 03:08:17

Exact and Intuitive Geometry Explanation: Why Does a Half-angle-rotation in Spin or Quaternion Space Correspond to the Whole-angle-rotation in Normal 3D Space?

Authors: Hongbing Zhang
Comments: 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.
Category: Geometry

[318] viXra:1810.0295 [pdf] submitted on 2018-10-18 09:47:40

К вопросу о представлении дельта-функции Дирака

Authors: Франц Герман
Comments: Pages.

В данной заметке мы покажем представление дельта-функции Дирака, которое будем назвать естественным. Существующие способы представления дельта-функции Дирака носят в общем-то искусственный характер.
Category: Geometry

[317] viXra:1810.0283 [pdf] submitted on 2018-10-17 05:53:48

Smoothing using Geodesic Averages

Authors: Jan Hakenberg
Comments: 10 Pages.

Geodesic averages have been used to generalize curve subdivision and Bézier curves to Riemannian manifolds and Lie groups. We show that geodesic averages are suitable to perform smoothing of sequences of data in nonlinear spaces. In applications that produce temporal uniformly sampled manifold data, the smoothing removes high-frequency components from the signal. As a consequence, discrete differences computed from the smoothed sequence are more regular. Our method is therefore a simpler alternative to the extended Kalman filter. We apply the smoothing technique to noisy localization estimates of mobile robots.
Category: Geometry

[316] viXra:1810.0171 [pdf] submitted on 2018-10-10 13:28:08

The Seiber-Witten Equations for Vector Fields

Authors: Antoine Balan
Comments: 2 pages, written in english

By similarity with the Seiberg-Witten equations, we propose a set of two equations, depending of a spinor and a vector field.
Category: Geometry

[315] viXra:1810.0068 [pdf] submitted on 2018-10-06 02:12:13

An Introduction to Non-Commutative (Pseudo) Metric Geometry.

Authors: Johan Noldus
Comments: 11 Pages.

We introduce the reader to the problematic aspects of formulating in concreto a suitable notion of geometry. Here, we take the canonical approach and give some examples.
Category: Geometry

[314] viXra:1810.0057 [pdf] submitted on 2018-10-04 09:55:33

Теорема о поляре треугольника

Authors: Франц Герман
Comments: Pages.

Сформулирована и доказана теорема, ранее не встречавшаяся в литературе по проективной геометрии. На основании «теоремы о поляре трёхвершинника» открывается целый класс задач на построение. Теорема может быть полезна студентам математических факультетов педагогических вузов, а также учителям математики средней школы для проведения факультативных занятий.
Category: Geometry

[313] viXra:1809.0515 [pdf] submitted on 2018-09-24 07:45:56

Fractales Del Tipo Newton Asociados al Polinomio: P(z)=z^9+3z^6+3z^3-1,z Complejo.

Authors: Edgar Valdebenito
Comments: 15 Pages.

En esta nota mostramos algunos fractales del tipo Newton asociados al polinomio: p(z)=z^9+3z^6+3z^3-1,z complejo.
Category: Geometry

[312] viXra:1809.0472 [pdf] submitted on 2018-09-22 12:54:37

A Generalization of the Levi-Civita Connection

Authors: Antoine Balan
Comments: 2 pages, written in english

We define here a generalization of the well-know Levi-Civita connection. We choose an automorphism and define a connection with help of a (non-symmetric) bilinear form.
Category: Geometry

[311] viXra:1809.0323 [pdf] submitted on 2018-09-15 09:41:12

A Generalization of the Clifford Algebra

Authors: Antoine Balan
Comments: 2 pages, written in english

We propose here a generalization of the Clifford algebra by mean of two endomorphisms. We deduce a generalized Lichnerowicz formula for the space of modified spinors.
Category: Geometry

[310] viXra:1808.0595 [pdf] submitted on 2018-08-25 08:47:43

Definition of the Term "DIRECTION" Определение термина «НАПРАВЛЕНИЕ»

Authors: Somsikov A.I.
Comments: 2 Pages.

One of initial or primary concepts which is considered to the "protozoa" (who aren't expressed through other concepts) is considered. The structure of this concept is revealed. Algebraic and geometrical consequences are found. Рассмотрено одно из исходных или первичных понятий, считающееся «простейшим» (не выражаемым через другие понятия). Выявлена структура этого понятия. Найдены алгебраические и геометрические следствия.
Category: Geometry

[309] viXra:1808.0208 [pdf] submitted on 2018-08-15 11:10:29

(Ip-GP Version 1.0 15 Pages 14.08.2018) on the Intrinsic Paradox of the Geometric Point Definition (Solved Using the Included Middle Logic) as the Main Cause of Euclid’s Postulate “inaccuracy”, Allowing the Existence not Only of Non-Euclidean Geomet

Authors: Andrei Lucian Dragoi
Comments: 15 Pages.

This paper brings to attention the intrinsic paradox of the geometric point (GP) definition, a paradox solved in this paper by using Stéphane Lupasco’s Included Middle Logic (IML) (which was stated by Basarab Nicolescu as one of the three pillars of transdisciplinarity [TD]) and its extended form: based on IML, a new “t-metamathematics” (tMM) (including a t-metageometry[tMG]) is proposed, which may explain the main cause of Euclid’s parallel postulate (EPP) “inaccuracy”, allowing the existence not only of non-Euclidean geometries (nEGs), but also the existence of new EPP variants. tMM has far-reaching implications, including the help in redefining the basics of Einstein’s General relativity theory (GRT), quantum field theory (QFT), superstring theories (SSTs) and M-theory (MT). KEYWORDS (including a list of main abbreviations): geometric point (GP); Stéphane Lupasco’s Included Middle Logic (IML); Basarab Nicolescu, transdisciplinarity (TD); “t-metamathematics” (tMM); t-metageometry (tMG); Euclid’s parallel postulate (EPP); non-Euclidean geometries; new EPP variants; Einstein’s General relativity theory (GRT); quantum field theory (QFT); superstring theories (SSTs); M-theory (MT);
Category: Geometry

[308] viXra:1808.0206 [pdf] submitted on 2018-08-15 12:34:49

The Balan-Killing Manifolds

Authors: Antoine Balan
Comments: 2 pages, written in english

We define here the notion of Balan-Killing manifolds which are solutions of differential equations over the metrics of spin manifolds.
Category: Geometry

[307] viXra:1807.0463 [pdf] submitted on 2018-07-26 06:25:03

Curve Subdivision in SE(2)

Authors: Jan Hakenberg
Comments: 6 Pages.

We demonstrate that curve subdivision in the special Euclidean group SE(2) allows the design of planar curves with favorable curvature. We state the non-linear formula to position a point along a geodesic in SE(2). Curve subdivision in the Lie group consists of trigonometric functions. When projected to the plane, the refinement method reproduces circles and straight lines. The limit curves are designed by intuitive placement of control points in SE(2).
Category: Geometry

[306] viXra:1807.0298 [pdf] submitted on 2018-07-17 17:10:18

Study of Transformations

Authors: Yeray Cachón Santana
Comments: 10 Pages.

This paper covers a first approach study of the angles and modulo of vectors in spaces of Hilbert considering a riemannian metric where, instead of taking the usual scalar product on space of Hilbert, this will be extended by the tensor of the geometry g. As far as I know, there is no a study covering space of Hilbert with riemannian metric. It will be shown how to get the angle and modulo on Hilbert spaces with a tensor metric, as well as vector product, symmetry and rotations. A section of variationals shows a system of differential equations for a riemennian metric.
Category: Geometry

[305] viXra:1807.0234 [pdf] submitted on 2018-07-12 16:30:22

Making Sense of Bivector Addition

Authors: James A. Smith
Comments: 18 Pages.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Category: Geometry

[304] viXra:1806.0116 [pdf] submitted on 2018-06-09 15:36:41

The Dirac Operators for Lie Groups

Authors: Antoine Balan
Comments: 1 page, written in english

In the case of a manifold which is a Lie group, a Dirac operator can be defined acting over the vector fields of the Lie group instead of the spinors.
Category: Geometry

[303] viXra:1805.0030 [pdf] submitted on 2018-05-01 00:29:25

On Surface Measures on Convex Bodies and Generalizations of Known Tangential Identities

Authors: Johan Aspegren
Comments: 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.
Category: Geometry

[302] viXra:1804.0397 [pdf] submitted on 2018-04-27 03:22:51

Vortex Equation in Holomorphic Line Bundle Over Non-Compact Gauduchon Manifold

Authors: Zhenghan Shen, Wen Wang, Pan Zhang
Comments: 9 Pages.

In this paper, by the method of heat flow and the method of exhaustion, we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundle over a class of non-compact Gauduchon manifold.
Category: Geometry

[301] viXra:1804.0363 [pdf] submitted on 2018-04-24 20:34:14

Learning Geometric Algebra by Modeling Motions of the Earth and Shadows of Gnomons to Predict Solar Azimuths and Altitudes

Authors: James A. Smith
Comments: 54 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.
Category: Geometry

[300] viXra:1804.0360 [pdf] submitted on 2018-04-25 02:11:06

A Note on a Problem in Misho Sampo

Authors: Hiroshi Okumura
Comments: 2 Pages. This paper will be submitted to Sangaku Journal of Mathematics.

A problem involving an isosceles triangle with a square and three congruent circles is generalized.
Category: Geometry

[299] viXra:1804.0132 [pdf] submitted on 2018-04-10 11:52:50

Euclid's Geometry is Just in Our Mind, Rather Than Describing the Real World

Authors: Arturo Tozzi, James Peters
Comments: 6 Pages.

The first definition (prior to the well-known five postulates) of Euclid describes the point as “that of which there is no part”. Here we show how the Euclidean account of manifolds is untenable in our physical realm and that the concepts of points, lines, surfaces, volumes need to be revisited, in order to allow us to be able to describe the real world. Here we show that the basic object in a physical context is a traversal of spacetime via tiny subregions of spatial regions, rather than the Euclidean point. We also elucidate the psychological issues that lead our mind to think to points and lines as really existing in our surrounding environment.
Category: Geometry

[298] viXra:1804.0032 [pdf] submitted on 2018-04-03 04:56:49

The Radian's Coincidence Conjecture

Authors: Ryan Haddad
Comments: 1 Page.

This conjecture may be a tool in defining the indefinite tangent of 90 degrees, and is a (new) mathematical coincidence that is indeed strange; why would the tangent of angles near 90 degrees be equal to the angle of the radian multiplied by powers of 10? In fact, if there is no geometrical explanation in current mathematics, it may resides in metamathematics.
Category: Geometry

[297] viXra:1803.0463 [pdf] submitted on 2018-03-22 09:24:56

Toroidal Approach to the Doubling of the Cube

Authors: Gerasimos T. Soldatos
Comments: Published in: FORUM GEOMETRICORUM, VOL. 18, PAGES 93-97

A doubling of the cube is attempted as a problem equivalent to the doubling of a horn torus. Both doublings are attained through the circle of Apollonius.
Category: Geometry

[296] viXra:1803.0242 [pdf] submitted on 2018-03-16 18:19:17

Proof of Playfair’s Axiom Hits a Roadblock

Authors: Prashanth R. Rao
Comments: 3 Pages.

The Playfair’s axiom is considered an equivalent of Euclid’s fifth postulate or parallel postulate in Euclidean planar geometry. It states that in a given plane, with a line in the plane and a point outside the line that is also in the same plane, one and only one line passes through that point that is also parallel to the given line. Previous proofs of Euclid’s postulate or the Playfair’s axiom have unintentionally assumed parallel postulate to prove it. Also, these axioms have different results in hyperbolic and spherical geometries. We offer proof for the Playfair’s axiom for subset of cases in the context of plane Euclidean geometry and describe another subset of cases that cannot be proven by the same approach.
Category: Geometry

[295] viXra:1803.0119 [pdf] submitted on 2018-03-08 13:22:17

The Flow of Dirac-Ricci

Authors: Antoine Balan
Comments: 2 pages, written in french

Following the definition of the flow of Ricci, we construct a flow of hermitian metrics for the spinors fiber bundle.
Category: Geometry

[294] viXra:1803.0050 [pdf] submitted on 2018-03-04 11:55:15

The Flow of Ricci over an Hermitian Fiber Bundle

Authors: Antoine Balan
Comments: 1 page, written in french

The flow of Ricci is defined for the hermitian metric of a fiber bundle.
Category: Geometry

[293] viXra:1802.0196 [pdf] submitted on 2018-02-15 23:59:18

A Note on a Problem Invoving a Square in a Curvilinear Triangle

Authors: Hiroshi Okumura
Comments: 3 Pages. This is a paper considering a problem in Wasan geometry.

A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.
Category: Geometry

[292] viXra:1802.0123 [pdf] submitted on 2018-02-10 09:10:18

Extension of Proposition 23 from Euclid’s Elements

Authors: Prashanth R. Rao
Comments: 1 Page.

Proposition 23 states that two parallel lines in a plane never intersect. We use this definition with first and second postulate of Euclid to prove that two distinct lines through a single point cannot be parallel.
Category: Geometry

[291] viXra:1802.0092 [pdf] submitted on 2018-02-08 07:24:07

A Relationship Between a Cevian, Two Perpendicular Bisectors and a Median in an Isosceles Triangle.

Authors: Jesús Álvarez Lobo
Comments: 1 Page. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 18. Spanish.

A very simple solution to a geometric problem (proposed by Alex Sierra Cardenas, Medellin, Colombia) that involves a cevian, two perpendicular bisectors and a median in an isosceles triangle.
Category: Geometry

[290] viXra:1802.0091 [pdf] submitted on 2018-02-08 07:47:03

Geometry Beyond Algebra. the Theorem of Overlapped Polynomials (Top) Andits Application to the Sawa Masayoshi\'s Sangaku Problem. the Adventure Ofsolving a Mathematical Challenge Stated in 1821.

Authors: Jesús Álvarez Lobo
Comments: 47 Pages. https://arxiv.org/abs/1110.1299

This work presents for the first time a solution to the 1821 unsolved Sawa Masayoshi's problem, giving an explicit and algebraically exact solution for the symmetric case (particular case b = c, i.e., for ABC isosceles right-angled triangle), see (1.60) and (1.61). Despite the isosceles triangle restriction is not necessary, in view of the complexity of the explicit algebraic solution for the symmetric case, one can guessing the impossibility of achieving an explicit relationship for the asymmetric case (the more general case: ABC right-angled scalene triangle). For this case is given a proof of existence and uniqueness of solution and a proof of the impossibility of getting such a relationship, even implicitly, if the sextic equation (2.54) it isn't solvable. Nevertheless, in (2.56) - (2.58) it is shown the way to solve the asymmetric case under the condition that (2.54) be solvable. Furthermore, it is proved that with a slight modification in the final set of variables (F), it is still possible to establish a relation between them, see (2.59) and (2.61), which provides a bridge that connects the primitive relationship by means of numerical methods, for every given right-angled triangle ABC. And as the attempt to solve Fermat's conjecture (or Fermat's last theorem), culminated more than three centuries later by Andrew Wiles, led to the development of powerful theories of more general scope, the attempt to solve the Masayoshi's problem has led to the development of the Theory of Overlapping Polynomials (TOP), whose application to this problem reveals a great potential that might be extrapolated to other frameworks.
Category: Geometry

[289] viXra:1802.0079 [pdf] submitted on 2018-02-08 06:28:49

Wasan Geometry: 3-Isoincircles Problem. Trigonometric Analysis of a Hard Sangaku Challenge.

Authors: Jesús Álvarez Lobo
Comments: 11 Pages.

Sacred Mathematics: Japanese Temple Geometry. Fukagawa Hidetoshi - Tony Rothman. Still Harder Temple Geometry Problems: Chapter 6 - Problem 3.
Category: Geometry

[288] viXra:1802.0071 [pdf] submitted on 2018-02-07 07:09:05

La Représentation de Bonne

Authors: Abdelmajid Ben Hadj Salem
Comments: 13 Pages. In French.

This paper gives the elements of definition of the Bonne's map projection. It was used for the ancient cartography at 1/50000 scale in Tunisia and Algeria.
Category: Geometry

[287] viXra:1802.0047 [pdf] submitted on 2018-02-05 08:51:37

Calculating the Angle Between Projections of Vectors Via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 14 Pages.

We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.
Category: Geometry

[286] viXra:1802.0036 [pdf] submitted on 2018-02-03 13:17:15

The Flow of HyperKaehler-Ricci

Authors: Antoine Balan
Comments: 2 pages, written in french

Studying the flow of Kaehler-Ricci, a flow is defined for a manifold which is HyperKaehler.
Category: Geometry

[285] viXra:1801.0347 [pdf] submitted on 2018-01-25 14:50:11

The flow of Ricci-Schrödinger

Authors: Antoine Balan
Comments: 1 page, written in french

The flow of Ricci-Schrödinger is defined from the flow of Ricci, like the Schrödinger equation is a twist of the heat equation.
Category: Geometry

[284] viXra:1801.0309 [pdf] submitted on 2018-01-23 19:52:26

Remarks on Liouville-Type Theorems on Complete Noncompact Finsler Manifolds

Authors: Songting Yin, Pan Zhang
Comments: 9 Pages.

In this paper, we give a gradient estimate of positive solution to the equation $$\Delta u=-\lambda^2u, \ \ \lambda\geq 0$$ on a complete non-compact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete non-compact Finsler manifold we also prove a Liouville-type theorem for a $C^2$-nonegative function $f$ satisfying $$\Delta f\geq cf^d, c>0, d>1, $$ which improves a result obtained by Yin and He.
Category: Geometry

[283] viXra:1801.0292 [pdf] submitted on 2018-01-22 15:19:04

An Upper Bound for Lebesgue’s Universal Covering Problem

Authors: Philip Gibbs
Comments: 21 Pages.

The universal covering problem as posed by Henri Lebesgue in 1914 seeks to find the convex planar shape of smallest area that contains a subset congruent to any point set of unit diameter in the Euclidean plane. Methods used previously to construct such a cover can be refined and extended to provide an improved upper bound for the optimal area. An upper bound of 0.8440935944 is found.
Category: Geometry

[282] viXra:1801.0156 [pdf] submitted on 2018-01-13 21:02:25

A Remark on the Localization Formulas About Two Killing Vector Fields

Authors: Xu Chen
Comments: 13 Pages.

In this article, we will discuss a localization formulas of equlvariant cohomology about two Killing vector fields on the set of zero points ${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold.
Category: Geometry

[281] viXra:1801.0155 [pdf] submitted on 2018-01-13 21:07:54

A Poincaré-Hopf Type Formula for A Pair of Vector Fields

Authors: Xu Chen
Comments: 7 Pages.

We extend the reslut about Poincar\'e-Hopf type formula for the difference of the Chern character numbers (cf.[3]) to the non-isolated singularities, and establish a Poincar\'e-Hopf type formula for a pair of vector field with the function $h^{T_{\mathbb{C}}M}(\cdot,\cdot)$ has non-isolated zero points over a closed, oriented smooth manifold of dimension $2n$.
Category: Geometry

[280] viXra:1801.0146 [pdf] submitted on 2018-01-13 01:53:56

Sennimalai Kalimuthu Publications

Authors: Sennimalai Kalimuthu
Comments: 03 Pages. Interested people may contact k,me at any time. Tha

The 5th Euclidean postulate is 2300 years old mathematical impossibility. I have worked on this problem for nearly b35 years and found a number of consistent solutions. My findings have been appeared in international peer reviewed research journals. Generation of power freely from space, space Bombs, Lion’s Tonic and Lemurian Yoga are my ambitious scientific projects. Interested researchers and people may contact me at +91 8508991577. My email is math.kalimuthu@gmail.com and arutperunjothi@outlook.com
Category: Geometry

[279] viXra:1801.0056 [pdf] submitted on 2018-01-05 09:56:59

Poliedros Fórmulas Indemostradas

Authors: Carlos Alejandro Chiappini
Comments: 7 Pages.

Leonhard Euler demostró que en un poliedro regular convexo hay tres números ue cumplen una ley, expresada en una ecuación conocida como fórmula de Euler. Son el número de caras, el número de vértices y el número de aristas. Este documento presenta algunas fórmulas más, obtenidas por ensayo y error a partir de una tabla que contiene los datos de los 5 poliedros regulares convexos. Estas fórmulas indemostradas tienen rasgos verosímiles. Buscar el modo de demostrar la invalidez o la validez de esas fórmulas podría ser, para las personas amantes de la topología, una tarea interesante.
Category: Geometry

[278] viXra:1712.0524 [pdf] submitted on 2017-12-19 13:56:07

Projection of a Vector upon a Plane from an Arbitrary Angle, via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 8 Pages.

We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.
Category: Geometry

[277] viXra:1712.0393 [pdf] submitted on 2017-12-11 16:46:30

Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vectors and Bivectors in Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 29 Pages. Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vectors and Bivectors in Geometric (Clifford) Algebra

We show how to express the representations of single, composite, and ``rotated" rotations in GA terms that allow rotations to be calculated conveniently via spreadsheets. Worked examples include rotation of a single vector by a bivector angle; rotation of a vector about an axis; composite rotation of a vector; rotation of a bivector; and the ``rotation of a rotation". Spreadsheets for doing the calculations are made available via live links.
Category: Geometry

[276] viXra:1711.0317 [pdf] submitted on 2017-11-14 17:32:57

Special Lagrangian Submanifolds with Singularities

Authors: Giordano Colò
Comments: 22 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture on mirror symmetry by studying the singularities of special Lagrangian submanifolds of 3-dimensional Calabi-Yau manifolds. In this paper we’ll give the description of the boundary of the moduli space of special Lagrangian manifolds. We do this by introducing special Lagrangian cones in the more general Kähler manifolds. Then we can focus on the almost Calabi- Yau manifolds. We consider the behaviour of the Lagrangian manifolds near the conical singular points to classify them according to the way they are approximated from the asymptotic cone. Then we analyze their deformations in Calabi-Yau manifolds.
Category: Geometry

[275] viXra:1711.0306 [pdf] submitted on 2017-11-14 06:43:32

Question 410 :Some Fractals

Authors: Edgar Valdebenito
Comments: 8 Pages.

This note presents some fractals.
Category: Geometry

[274] viXra:1710.0264 [pdf] submitted on 2017-10-23 07:56:01

Question 403: Newton Fractal for F(z)=((1-Z^5)^2 / (1+z^10)) Z

Authors: Edgar Valdebenito
Comments: 7 Pages.

This note presents some fractals related with the function: f(z)=((1-z^5)^2/(1+z^10))-z
Category: Geometry

[273] viXra:1710.0241 [pdf] submitted on 2017-10-22 16:40:29

Mathematical Geometry and Calculus of Numbers

Authors: Paris Samuel Miles-Brenden
Comments: 5 Pages. None.

None.
Category: Geometry

[272] viXra:1710.0147 [pdf] submitted on 2017-10-14 08:51:55

How to Effect a Composite Rotation of a Vector via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 13 Pages.

We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.
Category: Geometry

[271] viXra:1710.0137 [pdf] submitted on 2017-10-13 01:17:37

The Modified Dirac Operator

Authors: Antoine Balan
Comments: 5 pages, written in french

The Dirac operator is twisted by a symmetric automorphism, the Dirac-Lichnerowicz formula is proved. An application for the Seiberg-Witten equations is proposed.
Category: Geometry

[270] viXra:1710.0131 [pdf] submitted on 2017-10-11 07:44:42

Question 201: a Fractal Image

Authors: Edgar Valdebenito
Comments: 8 Pages.

This note presents a fractal image for f(z)=ln(1+g(z)).
Category: Geometry

[269] viXra:1710.0127 [pdf] submitted on 2017-10-11 20:27:18

Proof of Happy Ending Problem

Authors: Choe ryujin
Comments: 4 Pages.

Proof of happy ending problem
Category: Geometry

[268] viXra:1710.0110 [pdf] submitted on 2017-10-10 11:31:30

The 4th Spatial Dimension W

Authors: Mauro Bernardini
Comments: 4 Pages.

This paper attempts to provide a new vision on the 4th spatial dimension starting on the known symmetries of the Euclidean geometry. It results that, the points of the 4th dimensional complex space are circumferences of variable ray. While the axis of the 4th spatial dimension, to be orthogonal to all the three 3d cartesinan axes, is a complex line made of two specular cones surfaces symmetrical on their vertexes corresponding to the common origin of both the real and complex cartesian systems.
Category: Geometry

[267] viXra:1709.0439 [pdf] submitted on 2017-09-30 10:53:14

Locally flat Manifolds after Boyom-Giraldo-Medina-Saldarriaga-Villabon

Authors: Bouetou Bouetou Thomas
Comments: 10 Pages. non

I have noticed a situation of plagiarist and want to draw the attention of the authors and readers.
Category: Geometry

[266] viXra:1709.0109 [pdf] submitted on 2017-09-10 05:11:19

A Special Hexagon with Two-Fold Symmetry Must Have an Inscribed Square Satisfying the Toeplitz Conjecture

Authors: Prashanth R. Rao
Comments: 3 Pages.

In this paper, we generate a special hexagon with two-fold symmetry by diagonally juxtaposing two squares of different dimensions so that they share exactly one common vertex and their adjacent sides are perpendicular to one another. We connect in specific pairs, the vertices adjacent to common vertex of both squares to generate a hexagon that is symmetrical about a line connecting the unconnected vertices. We show that this special hexagon must have one square whose points lie on its sides. With suitable modifications, it may be possible to use this technique to prove the Toeplitz conjecture for a simple closed curve generated by connecting the same six vertices of this special hexagon.
Category: Geometry

[265] viXra:1709.0026 [pdf] submitted on 2017-09-02 14:50:05

Solution for a Special Case of the Toeplitz Conjecture

Authors: Prashanth R. Rao
Comments: 2 Pages.

According to Toeplitz conjecture or the inscribed square conjecture, every simple closed curve in a plane must have atleast one set of four points on it that belong to a square. This conjecture remains unsolved for a general case although it has been proven for some special cases of simple closed curves. In this paper, we prove the conjecture for a special case of a simple closed curve derived from two simple closed curves, each of which have exactly only one set of points defining a square. The Toeplitz solution squares of two parent simple closed curves have the same dimensions and share exactly one common vertex and the adjacent sides of the two squares form a right angle. The derived simple closed curve is formed by eliminating this common vertex (that belonged to the two solutions squares to begin with) and connecting other available points on the parent curves. We show that this derived simple closed curve has atleast one solution square satisfying the Toeplitz conjecture.
Category: Geometry

[264] viXra:1708.0462 [pdf] submitted on 2017-08-29 17:16:30

How to Effect a Desired Rotation of a Vector about a Given Axis via Geometric (Clifford) Algebra

Authors: James A. Smith
Comments: 9 Pages.

We show how to transform a "rotate a vector around a given axis" problem into one that may be solved via GA, which rotates objects with respect to bivectors. A sample problem is worked to show how to calculate the result of such a rotation conveniently via an Excel spreadsheet, to which a link is provided.
Category: Geometry

Replacements of recent Submissions

[101] viXra:1905.0026 [pdf] replaced on 2019-05-11 06:44:00

Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer, Dietmar Hildenbrand
Comments: 11 Pages. accepted for M. Gavrilova et al (eds.), Proceedings of Workshop ENGAGE 2019 at CGI 2019 with Springer LNCS, April 2019, 1 table, corrections: 03+11 May 2019.

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9,7) over the real sixteen dimensional vector space R^{9,7}. These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19,16). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections
Category: Geometry

[100] viXra:1905.0026 [pdf] replaced on 2019-05-03 10:07:54

Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

Authors: Eckhard Hitzer, Dietmar Hildenbrand
Comments: 11 Pages. accepted for M. Gavrilova et al (eds.), Proceedings of Workshop ENGAGE 2019 at CGI 2019 with Springer LNCS, April 2019, 1 table, correction: 03 May 2019.

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9,7) over the real sixteen dimensional vector space R^{9,7}. These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19,16). Keywords: Clifford algebra, conformal geometric algebra, cubic curves, cubic surfaces, intersections
Category: Geometry

[99] viXra:1903.0560 [pdf] replaced on 2019-05-06 19:59:05

Direct Sum Decomposition of a Linear Vector Space

Authors: Anamitra Palit
Comments: 5 Pages.

The direct sum decomposition of a vector space has been explored to bring out a conflicting feature in the theory. We decompose a vector space using two subspaces. Keeping one subspace fixed we endeavor to replace the other by one which is not equal to the replaced subspace. Proceeding from such an effort we bring out the conflict. From certain considerations it is not possible to work out the replacement with an unequal subspace. From alternative considerations an unequal replacement is possible.
Category: Geometry

[98] viXra:1903.0317 [pdf] replaced on 2019-03-18 18:09:25

Denial of Consistency for the Lobachevskii Non Euclidean Geometry

Authors: Colin James III
Comments: 2 Pages.

We prove two parallel lines are tautologous in Euclidean geometry. We next prove that non Euclidean geometry of Lobachevskii is not tautologous and hence not consistent. What follows is that Riemann geometry is the same, and non Euclidean geometry is a segment of Euclidean geometry, not the other way around. Therefore non Euclidean geometries are a non tautologous fragment of the universal logic VŁ4.
Category: Geometry

[97] viXra:1902.0444 [pdf] replaced on 2019-02-26 08:55:44

On the Equivalence of Closed Figures and Infinitely Extended Lines and the Conclusions Drawn from it

Authors: Madhur Sorout
Comments: 13 Pages.

This paper is mainly focused on the equivalence of closed figures and infinitely extended lines. Using this principle, some major conclusions can be drawn. The equivalence of closed figures and infinitely extended lines is mainly based on the idea that closed figures and infinitely extended lines are equivalent. One of the most significant conclusions drawn from this equivalency is that if any object moves along a straight infinitely extended line, it will return back to the point, where it started to move, after some definite time. This principle of equivalence of closed figures and infinitely extended lines may lead us to understand the physical reality of infinities.
Category: Geometry

[96] viXra:1902.0401 [pdf] replaced on 2019-04-16 08:32:36

Three-Dimensional Quadrics in Conformal Geometric Algebras and Their Versor Transformations

Authors: Eckhard Hitzer
Comments: 16 Pages. published in Adv. of App. Cliff. Algs., 29:46, pp. 1-16, 2019. DOI: 10.1007/s00006-019-0964-1, 1 table.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore, a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.
Category: Geometry

[95] viXra:1902.0401 [pdf] replaced on 2019-03-02 02:42:11

Three-Dimensional Quadrics in Conformal Geometric Algebras and Their Versor Transformations

Authors: Eckhard Hitzer
Comments: Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019, 15 pages. 4 errors corrected: 25 Feb. 2019. Proposition 4.1 corrected: 02 Mar. 2019.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore, a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.
Category: Geometry

[94] viXra:1902.0401 [pdf] replaced on 2019-02-25 06:01:09

Three-Dimensional Quadrics in Conformal Geometric Algebras and Their Versor Transformations

Authors: Eckhard Hitzer
Comments: Submitted to Topical Collection of Adv. in Appl. Clifford Algebras, for Proceedings of AGACSE 2018, 23 Feb. 2019, 15 pages. 4 errors corrected: 25 Feb. 2019.

This work explains how three dimensional quadrics can be defined by the outer products of conformal geometric algebra points in higher dimensions. These multivector expressions code all types of quadrics in arbitrary scale, location and orientation. Furthermore a newly modified (compared to Breuils et al, 2018, https://doi.org/10.1007/s00006-018-0851-1.) approach now allows not only the use of the standard intersection operations, but also of versor operators (scaling, rotation, translation). The new algebraic form of the theory will be explained in detail.
Category: Geometry

[93] viXra:1812.0423 [pdf] replaced on 2019-01-11 20:21:50

The Curvature and Dimension of Non-Differentiable Surfaces

Authors: Shawn Halayka
Comments: 5 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered. Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.
Category: Geometry

[92] viXra:1812.0423 [pdf] replaced on 2019-01-08 10:44:26

The Curvature and Dimension of Non-Differentiable Surfaces

Authors: Shawn Halayka
Comments: 5 Pages.

The curvature of a surface can lead to fractional dimension. In this paper, the properties of the 2-sphere surface of a 3D ball and the 2.x-surface of a 3D fractal set are considered. Tessellation is used to approximate each surface, primarily because the 2.x-surface of a 3D fractal set is otherwise non-differentiable.
Category: Geometry

[91] viXra:1812.0085 [pdf] replaced on 2019-01-24 08:52:30

Geometries of O

Authors: Hannes Hutzelmeyer
Comments: 93 Pages.

Geometries of O adhere to Ockham's principle of simplest possible ontology: the only individuals are points, there are no straight lines, circles, angles etc. , just as it was was laid down by Tarski in the 1920s, when he put forward a set of axioms that only contain two relations, quaternary congruence and ternary betweenness. However, relations are not as intuitive as functions when constructions are concerned. Therefore the planar geometries of O contain only functions and no relations to start with. Essentially three quaternary functions occur: appension for line-joining of two pairs of points, linisection representing intersection of straight lines and circulation corresponding to intersection of circles. Functions are strictly defined by composition of given ones only. Both, Euclid and Lobachevsky planar geometries are developed using a precise notation for object-language and metalanguage, that allows for a very broad area of mathematical systems up to theory of types. Some astonishing results are obtained, among them: (A) Based on a special triangle construction Euclid planar geometry can start with a less powerful ontological basis than Lobachevsky geometry. (B) Usual Lobachevsky planar geometry is not complete, there are nonstandard planar Lobachevsky geometries. One needs a further axiom, the 'smallest' system is produced by the proto-octomidial- axiom. (C) Real numbers can be abandoned in connection with planar geometry. A very promising conjecture is put forward stating that the Euclidean Klein-model of Lobachevsky planar geometry does not contain all points of the constructive Euclidean unit-circle.
Category: Geometry

[90] viXra:1810.0324 [pdf] replaced on 2018-10-23 10:40:48

Exact and Intuitive Geometry Explanation: Why Does a Half-angle-rotation in Spin or Quaternion Space Correspond to the Whole-angle-rotation in Normal 3D Space?

Authors: Hongbing Zhang
Comments: 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.
Category: Geometry

[89] viXra:1810.0324 [pdf] replaced on 2018-10-22 05:26:02

Exact and Intuitive Geometry Explanation: Why Does a Half-angle-rotation in Spin or Quaternion Space Correspond to the Whole-angle-rotation in Normal 3D Space?

Authors: Hongbing Zhang
Comments: 18 Pages. Please Indicate This Source From Hongbing Zhang When Cite the Contents in Works of Sience or Popular Sience

Why does a half-angle-rotation in quaternion space or spin space correspond to a whole-angle-rotation in normal 3D space? The question is equivalent to why a half angle in the representation of SU(2) corresponds to a whole angle in the representation of SO(3). Usually we use the computation of the abstract mathematics to answer the question. But now I will give an exact and intuitive geometry-explanation on it in this paper.
Category: Geometry

[88] viXra:1810.0171 [pdf] replaced on 2018-10-14 11:00:59

The Seiberg-Witten Equations for Vector Fields

Authors: Antoine Balan
Comments: 2 pages, written in english

By similarity with the Seiberg-Witten equations, we propose two differential equations, depending of a spinor and a vector field, instead of a connection. Good moduli spaces are espected as a consequence of commutativity.
Category: Geometry

[87] viXra:1810.0171 [pdf] replaced on 2018-10-13 14:13:21

The Seiberg-Witten Equations for Vector Fields

Authors: Antoine Balan
Comments: 2 pages, written in english

By similarity with the Seiberg-Witten equations, we propose two differential equations, depending of a spinor and a vector field, instead of a connection. Good moduli spaces are espected as a consequence of commutativity.
Category: Geometry

[86] viXra:1808.0206 [pdf] replaced on 2018-08-18 16:33:06

The Balan-Killing Manifolds

Authors: Antoine Balan
Comments: 2 pages, written in english

We define here the notion of Balan-Killing manifolds which are spin manifolds whose metrics verify a certain differential equation. We take our inspiration from the notion of Killing spinors.
Category: Geometry

[85] viXra:1808.0206 [pdf] replaced on 2018-08-18 05:09:12

The Balan-Killing Manifolds

Authors: Antoine Balan
Comments: 2 pages, written in english

We define here the notion of Balan-Killing manifolds which are spin manifolds whose metrics verify a certain differential equation. We take our inspiration from the notion of Killing spinors.
Category: Geometry

[84] viXra:1807.0234 [pdf] replaced on 2018-07-14 06:23:07

Making Sense of Bivector Addition

Authors: James A. Smith
Comments: 18 Pages.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
Category: Geometry

[83] viXra:1805.0030 [pdf] replaced on 2018-05-30 16:22:45

On Surface Measures on Convex Bodies and Generalizations of Known Tangential Identities

Authors: Johan Aspegren
Comments: 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.
Category: Geometry

[82] viXra:1805.0030 [pdf] replaced on 2018-05-22 11:28:23

On Surface Measures on Convex Bodies and Generalizations of Known Tangential Identities

Authors: Johan Aspegren
Comments: 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.
Category: Geometry

[81] viXra:1805.0030 [pdf] replaced on 2018-05-20 22:09:51

On Surface Measures on Convex Bodies and Generalizations of Known Tangential Identities

Authors: Johan Aspegren
Comments: 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.
Category: Geometry

[80] viXra:1805.0030 [pdf] replaced on 2018-05-06 01:48:26

On Surface Measures on Convex Bodies and Generalizations of Known Tangential Identities

Authors: Johan Aspegren
Comments: 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.
Category: Geometry

[79] viXra:1804.0003 [pdf] replaced on 2018-06-05 10:27:07

The Quaternionic Seiberg-Witten Equations

Authors: Antoine Balan
Comments: 6 pages, written in french

We define here the Seiberg-Witten equations in the quaternionic case. We formulate some algebra of the Hamilton numbers and study geometric applications of the quaternions.
Category: Geometry

[78] viXra:1803.0119 [pdf] replaced on 2018-03-09 09:44:09

The Flow of Dirac-Ricci

Authors: Antoine Balan
Comments: 2 pages, written in french

Following the definition of the flow of Ricci and with help of the Dirac operator, we construct a flow of hermitian metrics for the spinors fiber bundle.
Category: Geometry

[77] viXra:1803.0050 [pdf] replaced on 2018-03-05 11:16:01

The Flow of Ricci over an Hermitian Fiber Bundle

Authors: Antoine Balan
Comments: 1 page, written in french

The flow of Ricci is defined for the hermitian metrics of a complex fiber bundle.
Category: Geometry

[76] viXra:1802.0196 [pdf] replaced on 2018-02-16 15:58:29

A Note on a Problem Invoving a Square in a Curvilinear Triangle

Authors: Hiroshi Okumura
Comments: 3 Pages. The paper is considering a problem in Wasan geometry.

A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.
Category: Geometry

[75] viXra:1801.0347 [pdf] replaced on 2018-01-27 07:26:09

The Flow of Ricci-Schrödinger

Authors: Antoine Balan
Comments: 2 pages, written in french

The flow of Ricci-Schrödinger is defined from the Ricci flow, like the Schrödinger equation with respect to the heat equation.
Category: Geometry

[74] viXra:1712.0642 [pdf] replaced on 2018-04-23 22:13:50

Estimation of the Earth's "Unperturbed" Perihelion from Times of Solstices and Equinoxes

Authors: James A. Smith
Comments: 13 Pages.

Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the Earth would follow if unaffected by other bodies such as the Moon. To estimate the timing of that ``unperturbed" perihelion, we fit an unperturbed Kepler orbit to the timings of the year 2017's equinoxes and solstices. We find that the unperturbed 2017 perihelion, defined in that way, would occur 12.93 days after the December 2016 solstice. Using that result, calculated times of the year 2017's solstices and equinoxes differ from published values by less than five minutes. That degree of accuracy is sufficient for the intended use of the result.
Category: Geometry

[73] viXra:1711.0317 [pdf] replaced on 2017-12-17 12:03:53

Moduli Spaces of Special Lagrangian Submanifolds with Singularities

Authors: Giordano Colò
Comments: 22 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture on mirror symmetry by studying the singularities of special Lagrangian submanifolds of 3-dimensional Calabi-Yau manifolds. In this paper we’ll give the description of the boundary of the moduli space of special Lagrangian manifolds. We do this by introducing special Lagrangian cones in the more general Kähler manifolds. Then we can focus on the almost Calabi-Yau manifolds. We consider the behaviour of the Lagrangian manifolds near the conical singular points to classify them according to the way they are approximated from the asymptotic cone. Then we analyze their deformations in Calabi-Yau manifolds.
Category: Geometry

[72] viXra:1711.0317 [pdf] replaced on 2017-11-16 11:38:05

Moduli Spaces of Special Lagrangian Submanifolds with Singularities

Authors: Giordano Colò
Comments: 21 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture on mirror symmetry by studying the singularities of special Lagrangian submanifolds of 3-dimensional Calabi-Yau manifolds. In this paper we’ll give the description of the boundary of the moduli space of special Lagrangian manifolds. We do this by introducing special Lagrangian cones in the more general Kähler manifolds. Then we can focus on the textitalmost Calabi-Yau manifolds. We consider the behaviour of the Lagrangian manifolds near the conical singular points to classify them according to the way they are approximated from the asymptotic cone. Then we analyze their deformations in Calabi-Yau manifolds.
Category: Geometry

[71] viXra:1710.0110 [pdf] replaced on 2017-10-13 16:26:37

The 4th Spatial Dimension W

Authors: Mauro Bernardini
Comments: 4 Pages. this version corrects some accidental writing errors of the previous loaded version.

This paper attempts to provide a new vision on the 4th spatial dimension starting on the known symmetries of the Euclidean geometry. It results that, the points of the 4th dimensional complex space are circumferences of variable ray. While the axis of the 4th spatial dimension, to be orthogonal to all the three 3d cartesinan axes, is a complex line made of two specular cones surfaces symmetrical on their vertexes corresponding to the common origin of both the real and complex cartesian systems.
Category: Geometry

[70] viXra:1709.0144 [pdf] replaced on 2018-12-29 04:20:55

Convergence of the Ratio of Perimeter of a Regular Polygon to the Length of Its Longest Diagonal as the Number of Sides of Polygon Approaches to ∞

Authors: Pawan Kumar B.K.
Comments: 15 Pages.

Regular polygons are planar geometric structures that are used to a great extent in mathematics, engineering and physics. For all size of a regular polygon, the ratio of perimeter to the longest diagonal length is always constant and converges to the value of π as the number of sides of the polygon approaches to ∞. The purpose of this paper is to introduce Bishwakarma Ratio Formulas through mathematical explanations. The Bishwakarma Ratio Formulae calculate the ratio of perimeter of regular polygon to the longest diagonal length for all possible regular polygons. These ratios are called Bishwakarma Ratios- often denoted by short term BK ratios- as they have been obtained via Bishwakarma Ratio Formulae. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. Computational calculations of the ratios have also been presented upto 30 and 50 significant figures to validate the convergence.
Category: Geometry