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Any replacements are listed further down

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

**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

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

**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

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

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

This note presents some fractals.

**Category:** Geometry

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

**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

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

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 5 Pages. None.

None.

**Category:** Geometry

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

**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

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

**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

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

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

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

**Category:** Geometry

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

**Authors:** Choe ryujin

**Comments:** 4 Pages.

Proof of happy ending problem

**Category:** Geometry

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

**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

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

**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

[269] **viXra:1709.0144 [pdf]**
*submitted on 2017-09-11 16:58:23*

**Authors:** Pawan Kumar Bishwakarma, Sumit Khadka

**Comments:** 16 Pages.

A regular polygon is a planar geometrical structure with all sides of equal length and all angles of equal magnitude. The ratio of perimeter of any regular polygon to the length of its longest diagonal is a constant term and the ratio converges to the value of as the number of sides of the polygon increases. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. A computational calculation of the ratio has also been presented to validate the convergence. The values have been calculated up to 30 significant digits.

**Category:** Geometry

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

**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

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

**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

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

**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

[265] **viXra:1708.0459 [pdf]**
*submitted on 2017-08-30 06:57:06*

**Authors:** Adham ahmed mohamed ahmed

**Comments:** 1 Page.

This is a method to calculate pi using a computer

**Category:** Geometry

[264] **viXra:1708.0420 [pdf]**
*submitted on 2017-08-28 08:14:53*

**Authors:** Edgar Valdebenito

**Comments:** 56 Pages.

This note presents some examples of Newton-Raphson fractals.

**Category:** Geometry

[263] **viXra:1708.0236 [pdf]**
*submitted on 2017-08-19 17:45:13*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

Abstract: In this paper we prove that any two points A and B in space that are at different distances from a third point C, when connected by any curve in three dimensional space, must contain points such as D that are at intermediate distances from the third point C (length DC is intermediate to length AC and length BC).

**Category:** Geometry

[262] **viXra:1708.0229 [pdf]**
*submitted on 2017-08-19 12:49:36*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper, we give a simple proof that if there are two points A and B that are at distinct linear distances from a third point C (AC is not equal to BC), then any curve connecting the points A and B (this curve lies within the same plane containing A,B,C) must contain points such as D that lie at an intermediate distance from C, (DC is of length intermediate to AC and BC).

**Category:** Geometry

[261] **viXra:1708.0191 [pdf]**
*submitted on 2017-08-17 03:54:59*

**Authors:** Vu B Ho

**Comments:** 6 Pages.

In this work we show that by restricting the coordinate transformations to the group of time-independent coordinate transformations it is possible to derive the Ricci flow from the contracted Bianchi identities.

**Category:** Geometry

[260] **viXra:1708.0027 [pdf]**
*submitted on 2017-08-02 13:40:27*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The pseudo-quaternionic manifolds are studied. Several tensors are defined.

**Category:** Geometry

[259] **viXra:1707.0374 [pdf]**
*submitted on 2017-07-28 10:50:06*

**Authors:** Liu Ran

**Comments:** 3 Pages.

因为圆周率是一个比值，圆周长和直径都是测量而来，只要去测量圆周长和直径，就不可避免要在空间去度量，而空间，全都是非欧空间。

**Category:** Geometry

[258] **viXra:1707.0249 [pdf]**
*submitted on 2017-07-18 12:52:38*

**Authors:** Ulrich E. Bruchholz, Horst Eckardt

**Comments:** 19 Pages.

The well known numerical method of approximating differential
quotients by quotients of differences is used in a novel context.
This method is commonly underestimated, wrongly.
The method is explained by an ordinary differential equation first.
Then it is demonstrated how this simple method
proves successful for non-linear field equations with chaotic
behaviour. Using certain discrete values of their integration constants,
a behaviour comparable with Mandelbrot sets is obtained.
Instead of solving the
differential equations directly, their convergence behaviour is analyzed.
As an example the Einstein-Maxwell equations are investigated,
where discrete
particle quantities are obtained from a continuous theory, which is possible
only by this method.
The special set of integration constants contains values identical with
particle characteristics.
Known particle values are confirmed, and unknown values can be predicted.
In this paper, supposed neutrino masses are presented.

**Category:** Geometry

[257] **viXra:1707.0242 [pdf]**
*submitted on 2017-07-17 13:20:25*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

In this note we briefly examine the curve: x^3+y^3=sqrt(2)*x*y

**Category:** Geometry

[256] **viXra:1707.0181 [pdf]**
*submitted on 2017-07-12 20:58:33*

**Authors:** Pan Zhang

**Comments:** 14 Pages.

Let $V$ be an asymptotically cylindrical K\"{a}hler manifold with asymptotic
cross-section $\mathfrak{D}$. Let $E_\mathfrak{D}$ be a stable Higgs bundle over $\mathfrak{D}$, and $E$ a Higgs bundle over $V$ which is asymptotic to
$E_\mathfrak{D}$. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant Hermitian projectively Hermitian Yang-Mills metric on $E$.

**Category:** Geometry

[255] **viXra:1707.0046 [pdf]**
*submitted on 2017-07-04 11:58:48*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

A Laplacian operator is defined bound to a riemannian manifold with a pseudo-complex structure.

**Category:** Geometry

[254] **viXra:1707.0011 [pdf]**
*submitted on 2017-07-01 08:14:34*

**Authors:** John Atwell Moody

**Comments:** 17 Pages. written September 2014

When a meromorphic vector field is given on the projective plane, a complete holomorphic limit cycle, because it is a closed singular submanifold of projective space, is defined by algebraic equations. Also the meromorphic vector field is an algebraic object. Poincare had asked, is there just an algebraic calculation leading from the vector field to the defining equations of the solution, without the
mysterious intermediary of the dynamical system.
The answer is yes, that there is nothing more mysterious or wonderful that happens when a complete holomorphic limit cycle is formed than could have been defined using algebra.

**Category:** Geometry

[253] **viXra:1707.0010 [pdf]**
*submitted on 2017-07-01 08:17:31*

**Authors:** John Atwell Moody

**Comments:** Pages.

Contents:

Analytic primality testing 1

Lefshetz numbers of modular curves 23

Grothendieck sections and rational points of modular curves 29

Rational points of modular curves 33

Conclusion about modular forms 41

Outline geometric proof of Mordell’s conjecture 48

Example:the Fermat curves

63 The residue calculation 69

The meaning of positive and negative 81

**Category:** Geometry

[252] **viXra:1707.0009 [pdf]**
*submitted on 2017-07-01 08:21:26*

**Authors:** John Atwell Moody

**Comments:** Pages.

Contents

Projective toric varieties

Divisors

Maps

Chow ring

Fourier series and generalizations

**Category:** Geometry

[251] **viXra:1706.0497 [pdf]**
*submitted on 2017-06-27 02:58:55*

**Authors:** Orgest ZAKA, Kristaq FILIPI

**Comments:** 4 Pages. https://www.ijsr.net/archive/v6i6/ART20174592.pdf

In this paper we present an application possibility of the affine plane of order $n$, in the planning experiment, taking samples as his point. In this case are needed $n^2$ samples. The usefulness of the support of experimental planning in a finite affine plane consists in avoiding the partial repetition combinations within a proof. Reviewed when planning cannot directly drawn over an affine plane. In this case indicated how the problem can be completed, and when completed can he, with intent to drawn on an affine plane.

**Category:** Geometry

[250] **viXra:1706.0409 [pdf]**
*submitted on 2017-06-21 01:40:42*

**Authors:** Antoine Balan

**Comments:** 5 pages, written in french

A symplectic Dirac operator is defined for a spin Kaehler manifold. The corresponding Schrödinger-Lichnerowicz formula is proved.

**Category:** Geometry

[249] **viXra:1706.0021 [pdf]**
*submitted on 2017-06-02 19:22:09*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 24 Pages.

The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.
The understanding of these equations has taken us very far in a universe so familiar to mathematicians, the universe of periodic functions, and that of geometric forms with rounded ends revealing an infinity of new mathematical constants associated with them.

**Category:** Geometry

[248] **viXra:1704.0343 [pdf]**
*submitted on 2017-04-25 13:47:47*

**Authors:** Giordano Colò

**Comments:** 28 Pages.

We describe the deformations of the moduli space M of Special Lagrangian submanifolds in the compact case and we give a characterization of the topology of M by using McLean theorem. We consider Banach spaces on bundle sections and elliptical operators and we use Hodge theory to study the topology of the manifold. Starting from McLean results, for which the moduli space of compact special Lagrangian submanifolds is smooth and its tangent space can be identified with harmonic 1-forms on these submanifolds, we can analyze their deformations. Then we introduce a Riemannian metric on M, from which we obtain other important properties.

**Category:** Geometry

[247] **viXra:1704.0328 [pdf]**
*submitted on 2017-04-25 03:20:29*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 3 Pages.

In this article, we highlight some properties of the Apollonius circles of rank - 1 associated with a triangle.

**Category:** Geometry

[246] **viXra:1704.0134 [pdf]**
*submitted on 2017-04-11 04:15:20*

**Authors:** Xu Chen

**Comments:** 23 Pages.

In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.

**Category:** Geometry

[245] **viXra:1703.0273 [pdf]**
*submitted on 2017-03-28 19:09:13*

**Authors:** Arman Maesumi

**Comments:** 5 Pages.

Given a triangle ABC, the average area of an inscribed triangle RST whose vertices are uniformly distributed on BC, CA and AB, is proven to be one-fourth of the area of ABC. The average of the square of the area of RST is shown to be one-twelfth of the square of the area of ABC, and the average of the cube of the ratio of the areas is 5/144. A Monte Carlo simulation confirms the theoretical results, as well as a Maxima program which computes the exact averages.

**Category:** Geometry

[244] **viXra:1703.0267 [pdf]**
*submitted on 2017-03-28 08:30:43*

**Authors:** Jan Hakenberg, Ulrich Reif

**Comments:** 5 Pages.

The derivation of multilinear forms used to compute the moments of sets bounded by subdivision surfaces requires solving a number of systems of linear equations. As the support of the subdivision mask or the degree of the moment grows, the corresponding linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration and the geometric series are feasible methods to approximate the multilinear forms. The tensor iterations investigated in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, and have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroid, and inertia of spatial domains bounded by Catmull-Clark and Loop subdivision surfaces.

**Category:** Geometry

[243] **viXra:1703.0080 [pdf]**
*submitted on 2017-03-09 03:29:56*

**Authors:** Gaurav S. Biraris

**Comments:** 20 Pages.

The paper proposes a generalization of geometric notion of vectors concerning dimensionality of the configuration space. In certain dimensional spaces, certain types of ordered directions exist along which elements of vector spaces can be interpreted. Scalars along the ordered directions form Banach spaces. Different types of geometrical vectors are algebraically identical, the difference arises in the configuration space geometrically. In the universe four types of vectors exists. Thus any physical quantity in the universe comes in four types of vectors. Though All the types of vectors belong to different Banach spaces (& their directions can’t be compared), their magnitudes can be compared. A gross comparison between the magnitudes of the different typed geometric vectors is obtained at end of the paper.

**Category:** Geometry

[242] **viXra:1702.0049 [pdf]**
*submitted on 2017-02-03 12:34:37*

**Authors:** Gerasimos T. Soldatos

**Comments:** 3 Pages. Published in: Forum Geometricorum, 2017, Volume 17, pp. 13-15

An “Archimedean” quadrature is attempted “borrowing” π from the 3-dimensional space of a horn torus

**Category:** Geometry

[241] **viXra:1701.0576 [pdf]**
*submitted on 2017-01-23 09:11:18*

**Authors:** Dragan Turanyanin, Svetozar Jovičin

**Comments:** 14 Pages.

The aim of this article would be to show planar (whether polar, Cartesian or parametric) functions from a different, implicit viewpoint, hence the term inpolars (inpolar curves). The whole set of brand new planar curves can be seen from that perspective. Their generic mechanism is the so called inpolar transformation as well as its inpolar inversion. One entirely new geometric system is defined this way.

**Category:** Geometry

[240] **viXra:1612.0398 [pdf]**
*submitted on 2016-12-29 12:32:03*

**Authors:** Irina I. Bodrenko

**Comments:** 3 Pages.

The some properties of hypersurfaces F3 in Euclidean space E4 of nonzero
Laplacian of the second fundamental form b are studied in this report.

**Category:** Geometry

[239] **viXra:1611.0035 [pdf]**
*submitted on 2016-11-02 18:56:08*

**Authors:** Dragan Turanyanin

**Comments:** 3 Pages.

This spiral (given in polar coordinates r, theta) can be seen as a missing member of the set of known spirals.

**Category:** Geometry

[238] **viXra:1611.0003 [pdf]**
*submitted on 2016-11-01 01:37:29*

**Authors:** Brian Ekanyu

**Comments:** 3 Pages.

This paper proves a geometric theorem about the Hebrew nation that is the Patriarchs Abraham, Isaac and Jacob and the Twelve tribes of Israel. The circle represents the state of Israel.

**Category:** Geometry

[237] **viXra:1610.0367 [pdf]**
*submitted on 2016-10-30 14:48:27*

**Authors:** Dragan Turanyanin

**Comments:** 6 Pages.

The aim of this review is firstly, to present again a new family of polar curves (e.g. thurals [1]) and secondly, to introduce their so called inpolars as main objects of one original geometrical transformation [2]. Addendum is completely new with a brief analysis of s-thural.

**Category:** Geometry

[236] **viXra:1610.0076 [pdf]**
*submitted on 2016-10-06 19:47:24*

**Authors:** James A. Smith

**Comments:** 5 Pages.

To the collections of problems solved via Geometric Algebra (GA) in References 1-13, this document adds a solution, using only dot products, to the Problem of Apollonius. The solution is provided for completeness and for contrast with the GA solutions presented in Reference 3.

**Category:** Geometry

[235] **viXra:1610.0054 [pdf]**
*submitted on 2016-10-04 18:46:06*

**Authors:** James A. Smith

**Comments:** 15 Pages.

Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.

**Category:** Geometry

[234] **viXra:1609.0365 [pdf]**
*submitted on 2016-09-25 18:17:04*

**Authors:** James A. Smith

**Comments:** 5 Pages.

This document adds to the collection of GA solutions to plane-geometry problems, most of them dealing with tangency, that are presented in References 1-7. Reference 1 presented several ways of solving the CPP limiting case of the Problem of Apollonius. Here, we use ideas from Reference 6 to solve that case in yet another way.

**Category:** Geometry

[233] **viXra:1609.0082 [pdf]**
*submitted on 2016-09-06 19:08:36*

**Authors:** Marvin Ray Burns

**Comments:** 8 Pages. This classic paper shows the utter simplicity of the geometric description of the MRB constant (oeis.org/A037077).

The MRB constant is the upper limit point of the sequence of partial sums defined by S(x)=sum((-
1)^n*n^(1/n),n=1..x). The goal of this paper is to show that the MRB constant is geometrically
quantifiable. To “measure” the MRB constant, we will consider a set, sequence and alternating series of
the nth roots of n. Then we will compare the length of the edges of a special set of hypercubes or ncubes
which have a content of n. (The two words hypercubes and n-cubes will be used synonymously.)
Finally, we will look at the value of the MRB constant as a representation of that comparison, of the length of the edges of a special set of hypercubes, in units of dimension 1/ (units of dimension 2 times
units of dimension 3 times units of dimension 4 times etc.). For an arbitrary example we will use units of
length/ (time*mass* density*…).

**Category:** Geometry

[232] **viXra:1608.0328 [pdf]**
*submitted on 2016-08-24 18:28:40*

**Authors:** James A. Smith

**Comments:** 38 Pages.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.

**Category:** Geometry

[231] **viXra:1608.0217 [pdf]**
*submitted on 2016-08-19 21:41:57*

**Authors:** James A. Smith

**Comments:** 10 Pages.

The new solutions presented herein for the CLP and CCP limiting cases of the Problem of Apollonius are much shorter and more easily understood than those provided by the same author in References 1 and 2. These improvements result from (1) a better selection of angle relationships as a starting point for the solution process; and (2) better use of GA identities to avoid forming troublesome combinations of terms within the resulting equations.

**Category:** Geometry

[230] **viXra:1608.0153 [pdf]**
*submitted on 2016-08-15 04:33:03*

**Authors:** Xu Chen

**Comments:** 8 Pages.

In this article, we will discuss a new operator $d_{C}$ on $W(\mathfrak{g})\otimes\Omega^{*}(M)$ and to construct a new Cartan model for equivariant cohomology. We use the new Cartan model to construct the corresponding BRST model and Weil model, and discuss the relations between them.

**Category:** Geometry

[229] **viXra:1608.0103 [pdf]**
*submitted on 2016-08-09 15:55:41*

**Authors:** Robert B. Easter

**Comments:** 4 Pages.

This note on quadrics and pseudoquadrics inversions in hyperpseudospheres shows that the inversions produce different results in a three-dimensional spacetime. Using Geometric Algebra, all quadric and pseudoquadric entities and operations are in the G(4,8) Double Conformal Space-Time Algebra (DCSTA). Quadrics at zero velocity are purely spatial entities in x y z-space that are hypercylinders in w x y z-spacetime. Pseudoquadrics represent quadrics in a three-dimensional (3D) x y w, y z w, or z x w spacetime with the pseudospatial w-axis that is associated with time w=c t. The inversion of a quadric in a hyperpseudosphere can produce a Darboux pseudocyclide in a 3D spacetime that is a quartic hyperbolic (infinite) surface, which does not include the point at infinity. The inversion of a pseudoquadric in a hyperpseudosphere can produce a Darboux pseudocyclide in a 3D spacetime that is a quartic finite surface. A quadric and pseudoquadric can represent the same quadric surface in space, and their two different inversions in a hyperpseudosphere represent the two types of reflections of the quadric surface in a hyperboloid.

**Category:** Geometry

[228] **viXra:1607.0369 [pdf]**
*submitted on 2016-07-19 15:02:18*

**Authors:** Jeffrey Joseph Wolynski

**Comments:** 2 Pages. 3 illustrations

It is shown that simple geometry could have been used to make the discovery that planet formation is stellar evolution.

**Category:** Geometry

[227] **viXra:1607.0356 [pdf]**
*submitted on 2016-07-18 07:10:10*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 11 Pages.

In this article, we define the Lucas’s inner
circles and we highlight some of their properties.

**Category:** Geometry

[226] **viXra:1607.0355 [pdf]**
*submitted on 2016-07-18 07:11:14*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

We present here the magic square of order n.

**Category:** Geometry

[225] **viXra:1607.0349 [pdf]**
*submitted on 2016-07-18 07:18:12*

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

Postulatul V al lui Euclid se enunta sub forma: daca o dreapta, care intersecteaza doua drepte, formeaza unghiuri interioare de aceeasi parte
mai mici decat doua unghiuri drepte, aceste drepte, prelungite la infinit, se intalnesc in parte a unde unghiurile interioare sunt mai mici decal doua unghiuri drepte.

**Category:** Geometry

[224] **viXra:1607.0348 [pdf]**
*submitted on 2016-07-18 07:18:58*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we highlight some metric
properties in connection with Neuberg's circles and triangle. We recall some results that are necessary.

**Category:** Geometry

[223] **viXra:1607.0346 [pdf]**
*submitted on 2016-07-18 07:21:25*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

It is a lot easier to deny the Euclid`s five postulates, than Hilbert`s twenty thorough axiom.

**Category:** Geometry

[222] **viXra:1607.0341 [pdf]**
*submitted on 2016-07-18 07:27:33*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

In 1969, fascinat de geometrie, am construit un spatiu partial eueliadian si partial neeuclidian
in acelasi timp, inlocuind postulatul V al lui Euclid (axioma paralelelor) prin urmatoarea
propozitie stranie continand cinci asertiuni.

**Category:** Geometry

[221] **viXra:1607.0331 [pdf]**
*submitted on 2016-07-18 07:38:11*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Les axes radicals de n cercles d'un même plan, pris deux à deux, dont les centres ne sont pas alignes, sont concourants.

**Category:** Geometry

[220] **viXra:1607.0330 [pdf]**
*submitted on 2016-07-18 07:40:58*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

In 1969, intrigued by geometry I constructed a partially euclidean and partially non-Euclidean space.

**Category:** Geometry

[219] **viXra:1607.0325 [pdf]**
*submitted on 2016-07-18 07:53:17*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 11 Pages.

In this article, we emphasize the radical axis of the Lemoine’s circles.

**Category:** Geometry

[218] **viXra:1607.0323 [pdf]**
*submitted on 2016-07-18 07:55:17*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we define the first Droz-Farny’s circle, we establish a connection between it and a concyclicity theorem, then we generalize this theorem, leading to the generalization of Droz-Farny’s circle.

**Category:** Geometry

[217] **viXra:1607.0322 [pdf]**
*submitted on 2016-07-18 07:56:54*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove the theorem
relative to the second Droz-Farny’s circle, and a sentence that generalizes it.

**Category:** Geometry

[216] **viXra:1607.0304 [pdf]**
*submitted on 2016-07-18 08:23:01*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 25 Pages.

In this article, we solve problems of geometric constructions only with the ruler, using known theorems.

**Category:** Geometry

[215] **viXra:1607.0303 [pdf]**
*submitted on 2016-07-18 08:23:58*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 6 Pages.

The late mathematician Cezar Cosnita, using the barycenter coordinates, proves two theorems which are the subject of this article.

**Category:** Geometry

[214] **viXra:1607.0302 [pdf]**
*submitted on 2016-07-18 08:24:44*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove several theorems about the radical center and the radical circle of ex-inscribed circles of a triangle and calculate the radius of the circle from vectorial considerations.

**Category:** Geometry

[213] **viXra:1607.0260 [pdf]**
*submitted on 2016-07-18 05:32:50*

**Authors:** Florentin Smarandache

**Comments:** 14 Pages.

It is possible to de-formatize entirely Hilbert`s group of axioms of the Euclidean Geometry, and to construct a model such that none of this fixed axiom holds.

**Category:** Geometry

[212] **viXra:1607.0258 [pdf]**
*submitted on 2016-07-18 05:34:44*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

This article highlights some properties of
Apollonius’s circle of second rank in connection with the adjoint circles and the second Brocard’s triangle.

**Category:** Geometry

[211] **viXra:1607.0253 [pdf]**
*submitted on 2016-07-18 05:40:32*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 9 Pages.

In this article, we prove the theorem relative to the circle of the 6 points and, requiring on this circle to have three other remarkable triangle’s points, we obtain the circle of 9 points (the Euler’s Circle).

**Category:** Geometry

[210] **viXra:1607.0240 [pdf]**
*submitted on 2016-07-18 06:07:02*

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

Let P, L be two sets, and r a relation included in PxL.

**Category:** Geometry

[209] **viXra:1607.0229 [pdf]**
*submitted on 2016-07-18 06:36:40*

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

Dans ces paragraphss on présente "trois généralisations du célèbre théorème de Ceva.

**Category:** Geometry

[208] **viXra:1607.0227 [pdf]**
*submitted on 2016-07-18 06:39:23*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Let’s consider the points A1,...,An situated on the same plane, and B1,...,Bn situated on another plane, such that the lines A1B1 are concurrent. Let’s prove that if the lines AiAj and BiBj are concurrent, then their intersecting points are collinear.

**Category:** Geometry

[207] **viXra:1607.0225 [pdf]**
*submitted on 2016-07-18 06:41:46*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

What happens in 3-space when the poiygon is replaced by a polyhedron?

**Category:** Geometry

[206] **viXra:1607.0209 [pdf]**
*submitted on 2016-07-18 07:05:18*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 7 Pages.

In this article, we get to Lemoine's circles
in a different manner than the known one.

**Category:** Geometry

[205] **viXra:1607.0208 [pdf]**
*submitted on 2016-07-18 07:06:38*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 8 Pages.

For the calculus of the first Lemoine’s circle, we will first prove.

**Category:** Geometry

[204] **viXra:1607.0166 [pdf]**
*submitted on 2016-07-13 22:10:37*

**Authors:** James A. Smith

**Comments:** 11 Pages.

This document adds to the collection of solved problems presented in [1]-[4]. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in two ways.

**Category:** Geometry

[203] **viXra:1607.0086 [pdf]**
*submitted on 2016-07-07 07:28:29*

**Authors:** Kermit Ohlrabi

**Comments:** 14 pages

Let ${j_{\mathscr{{Y}}}}$ be a M\"obius homomorphism. Every student is aware that there exists a freely Gaussian meager, globally tangential polytope. We show that every continuous triangle is finitely Desargues. The work in \cite{cite:0} did not consider the almost Torricelli, co-locally $\varphi$-standard case. Now this leaves open the question of convergence.

**Category:** Geometry

[202] **viXra:1607.0015 [pdf]**
*submitted on 2016-07-02 02:33:06*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.13171.12325

Sixty years ago Richard Bellman issued a difficult challenge to his fellow mathematicians. If a rambler is lost in a forest of known shape and size, how can she find the best path to follow in order to escape as quickly as possible? So far solutions are only known for a handful of simple cases and the general problem has therefore been described as “unapproachable.” In this work a computational “random paths” method to search for optimal escape paths inside convex polygonal forests is described. In particular likely solutions covering all cases of isosceles triangles are given. Each conjectured solution provides a potentisl upper-bound for Moser’s worm problem. Surprisingly there are two cases of triangles which would provide improvements on the best known proven upper bounds.

**Category:** Geometry

[201] **viXra:1606.0253 [pdf]**
*submitted on 2016-06-24 06:43:12*

**Authors:** James A. Smith

**Comments:** 8 Pages.

This document is intended to be a convenient collection of explanations and techniques given elsewhere in the course of solving tangency problems via Geometric Algebra.

**Category:** Geometry

[200] **viXra:1606.0050 [pdf]**
*submitted on 2016-06-05 13:16:17*

**Authors:** Philip Gibbs

**Comments:** Pages. DOI: 10.13140/RG.2.2.28270.61767

Bellman’s challenge to find the shortest path to escape
from a forest of known shape is notoriously difficult. Apart from a
few of the simplest cases, there are not even many conjectures for
likely solutions let alone proofs. In this work it is shown that when
the forest is a convex polygon then at least one shortest escape path
is a piecewise curve made from segments taking the form of either
straight lines or circular arcs. The circular arcs are formed from the
envelope of three sides of the polygon touching the escape path at
three points. It is hoped that in future work these results could lead
to a practical computational algorithm for finding the shortest escape
path for any convex polygon.

**Category:** Geometry

[199] **viXra:1605.0314 [pdf]**
*submitted on 2016-05-31 21:49:03*

**Authors:** James A. Smith

**Comments:** 19 Pages.

The famous "Problem of Apollonius", in plane geometry, is to construct all of the circles that are tangent, simultaneously, to three given circles. In one variant of that problem, one of the circles has innite radius (i.e., it's a line). The Wikipedia article that's current as of this writing has an extensive description of the problem's history, and of methods that have been used to solve it. As described in that article, one of the methods reduces the "two circles and a line" variant to the so-called "Circle-Line-Point" (CLP) special case: Given a circle C, a line L, and a point P, construct the circles that are tangent to C and L, and pass through P. This document has been prepared for two very different audiences: for my fellow students of GA, and for experts who are preparing materials for us, and need to know which GA concepts we understand and apply readily, and which ones we do not.

**Category:** Geometry

[198] **viXra:1605.0233 [pdf]**
*submitted on 2016-05-22 20:06:09*

**Authors:** James A. Smith

**Comments:** 6 Pages.

The beautiful Problem of Apollonius from classical geometry (``\textit{Construct all of the circles that are tangent, simultaneously, to three given coplanar circles}") does not appear to have been solved previously by vector methods. It is solved here via GA to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors.

**Category:** Geometry

[197] **viXra:1605.0232 [pdf]**
*submitted on 2016-05-22 20:17:30*

**Authors:** James A. Smith

**Comments:** 76 Pages.

Written as somewhat of a "Schaums Outline" on the subject, which is especially useful in robotics and mechatronics. Geometric Algebra (GA) was invented in the 1800s, but was largely ignored until it was revived and expanded beginning in the 1960s. It promises to become a "universal mathematical language" for many scientific and mathematical disciplines. This document begins with a review of the geometry of angles and circles, then treats rotations in plane geometry before showing how to formulate problems in GA terms, then solve the resulting equations. The six problems treated in the document, most of which are solved in more than one way, include the special cases that Viete used to solve the general Problem of Apollonius.

**Category:** Geometry

[196] **viXra:1605.0024 [pdf]**
*submitted on 2016-05-03 01:13:07*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 180 Pages.

We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors aspiration and attraction, as a poet writes lyrics about spring according to his emotions.

**Category:** Geometry

[195] **viXra:1604.0148 [pdf]**
*submitted on 2016-04-09 00:13:17*

**Authors:** Christopher Goddard

**Comments:** Presentation / Slidedeck, 41 pages

This slide-deck was used as a vehicle for delivery of a talk received at the Mathematics of Planet Earth conference 2013 in Melbourne. In the contents provided herein, I sketch an approach to understand and control dynamical systems that may be subject to tipping points, vis a vis catastrophe theory.

**Category:** Geometry

[194] **viXra:1602.0270 [pdf]**
*submitted on 2016-02-21 13:29:33*

**Authors:** Bogdan Szenkaryk "Pinopa"

**Comments:** 1 Page. Contact the Author - ratunek.nauki(at)onet.pl

The article comprises equation even more beautiful than Euler's identity, which is considered the most beautiful math equation. The equation is even more beautiful, because from it is derived Euler's identity. Besides, there can be derived from it many other no less beautiful mathematical identities as Euler's.

**Category:** Geometry

[193] **viXra:1602.0269 [pdf]**
*submitted on 2016-02-21 13:31:46*

**Authors:** Bogdan Szenkaryk "Pinopa"

**Comments:** 1 Page. Contact the Author - ratunek.nauki(at)onet.pl

Novelty which earlier - before it appears - no one saw; the beautiful equation.

**Category:** Geometry

[192] **viXra:1602.0252 [pdf]**
*submitted on 2016-02-20 11:33:33*

**Authors:** Robert B. Easter

**Comments:** 12 Pages.

This paper gives an overview of two different, but closely related, double conformal geometric algebras. The first is the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), and the second is the G(4,8) Double Conformal Space-Time Algebra (DCSTA). DCSTA is a straightforward extension of DCGA. The double conformal geometric algebras that are presented in this paper have a large set of operations that are valid on general quadric surface entities. These operations include rotation, translation, isotropic dilation, spacetime boost, anisotropic dilation, differentiation, reflection in standard entities, projection onto standard entities, and intersection with standard entities. However, the quadric surface entities and other "non-standard entities" cannot be intersected with each other.

**Category:** Geometry

[191] **viXra:1602.0249 [pdf]**
*submitted on 2016-02-20 04:32:07*

**Authors:** Orgest ZAKA, Kristaq FILIPI

**Comments:** 9 Pages.

In this paper, based on several meanings and statements discussed in the literature, we intend
constuction a affine plane about a of whatsoever corps (K,+,*). His points conceive as
ordered pairs (α,β), where α and β are elements of corps (K,+,*). Whereas straight-line in
corps, the conceptualize by equations of the type x*a+y*b=c, where a≠0K or b≠0K the
variables and coefficients are elements of that body. To achieve this construction we prove
some theorems which show that the incidence structure A=(P, L, I) connected to the corps
K satisfies axioms A1, A2, A3 definition of affine plane. In all proofs rely on the sense of the
corps as his ring and properties derived from that definition.

**Category:** Geometry

[190] **viXra:1602.0234 [pdf]**
*submitted on 2016-02-19 06:49:44*

**Authors:** Espen Gaarder Haug

**Comments:** 14 Pages.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proved that ⇡ was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in Euclidean space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in Euclidean space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space the Impossible is possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well.

**Category:** Geometry

[189] **viXra:1602.0074 [pdf]**
*submitted on 2016-02-06 11:14:56*

**Authors:** editor Florentin Smarandache

**Comments:** 156 Pages.

In the new Techno-Art of Selariu SuperMathematics Functions ALBUM (the second book of Selariu SuperMathematics Functions), one contemplates a unique COMPOSITION, INTER-, INTRA- and TRANS-DISCIPLINARY. A comprehensive and savant INTRODUCTION explains the genesis of the inserted "figures", the addressees being, without discriminating criteria, equally engineers, mathematicians, artists, graphic designers, architects, and all lovers of beauty – as the love of beauty is the supreme form of love. If I should put a label on the "content" of this ALBUM, I would concoct the word NEO-BEAUTY!
The new complements of mathematics, reunited under the name of ex-centric mathematics (EM), extend (theoretically, endless) their scope; in this respect, Selariu SuperMathematics Functions are undeniable arguments! The author has labored (especially in the last three decades) extensively and fruitfully in the elitist field of the domain.
To mention some 'milestones' in this ALBUM, I choose specific mathematical elements, supermatematically hybridated: quadrilobic cubes, sphericubes, conopyramids, ex-centric spirals, severed toroids. There are also those that would qualify as "utilitarian": clepsydras, vases, baskets, lampions, or those suggestively “baptized” (by the author): butterflies, octopuses, flying saucers, jellies, roundabouts, ribands, and so on – all superlatively designed in shapes and colors! Striking phrases, such as “staggering multiplication of the dimensions of the Universe”, “integration through differential division” etc., become plausible (and explained) by replacing the time (of Einstein's four-dimensional space) with ex-centricity. Consequently, classical geometrical bodies (for e=0): the sphere, the cylinder, the cone, undergo metamorphosis (for e = +/-1), respectively into a cube, a prism, a pyramid. Inevitably and invariably, it is confirmed again that science is a finite space that grows in the infinite space; each new "expansion" does include a new area of unknown, but the unknown is inexhaustible...
Just browsing the ALBUM pages, you feel induced by the sensation of pleasure, of love at first sight; the variety of "exhibits", most of them unusual, the elegance, the symmetry of the layout, the chromatic, and so one, delight the eye, but equally incites to catchy intellectual exploration.

**Category:** Geometry

[188] **viXra:1601.0127 [pdf]**
*submitted on 2016-01-12 09:19:21*

**Authors:** Kang Yang, Kevin yang, Shuang-ren Ren Zhao

**Comments:** 13 Pages. This is one of best method to create a polygon or to solve the problem "inside the polygon"

There are many method for nding whether a point is inside a polygon or not. The congregation
of all points inside a polygon can be referred point congregation of polygon. Assume on a plane
there are N points. Assume the polygon have M vertexes. There are O(NM) calculations to create
the point congregation of polygon. Assume N>>M, we oer a parallel calculation method which
is suitable for GPU programming. Our method consider a polygon is consist of many fan regions.
The fan region can be positive and negative.
We wold like to extended this method to 3 D problem where a polyhedron instead of polygon should be drawn using cones.

**Category:** Geometry

[187] **viXra:1512.0414 [pdf]**
*submitted on 2015-12-23 16:24:57*

**Authors:** Jose Carlos Tiago de Oliveira

**Comments:** 6 Pages.

ABSTRACT
Mario Markus, a Chilean scientist and artist from Dortmund Max Planck Institute, has exposed a large set of
images of Lyapunoff exponents for the logistic equation modulated through rhythmic oscillation of parameters. The
pictures display features like foreground/background contrast, visualizing superstability, structural instability and, above
all, multistability, in a way visually analogous to three-dimensional representation.
See, for instance, http://www.mariomarkus.com/hp4.html.
The present papers aims to classify, through codification of numbers in the unit interval, the ensemble of images
thus generated. The above is intended as a part of a still unfulfilled work in progress, the classification of style in visual
fractal images-a common endeavour to Art and Science.

**Category:** Geometry

[186] **viXra:1512.0403 [pdf]**
*submitted on 2015-12-23 03:32:54*

**Authors:** Martin Erik Horn

**Comments:** 1 Page. The complete paper can be found at http://www.phydid.de (Wuppertal 2015)

An overview over all possible elementary reflections is given. It shows that a quarter of all reflections are negative.

**Category:** Geometry

[185] **viXra:1512.0303 [pdf]**
*submitted on 2015-12-13 02:52:45*

**Authors:** Robert B. Easter

**Comments:** 25 Pages.

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

**Category:** Geometry

[184] **viXra:1512.0233 [pdf]**
*submitted on 2015-12-06 05:52:26*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. In French.

In this paper, we give the equations of the geodesics of the tori and the integration of it.

**Category:** Geometry

[183] **viXra:1511.0245 [pdf]**
*submitted on 2015-11-24 16:21:26*

**Authors:** Rodolfo A. Frino

**Comments:** 7 Pages.

This paper solves the problem of the right scalene triangle through a general sequential solution. A simplified solution is also presented.

**Category:** Geometry

[182] **viXra:1511.0212 [pdf]**
*submitted on 2015-11-22 07:13:44*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. In French.

This note of differential geometry concerns the formulas of Elie Cartan about the differntial forms on a surface. We calculate these formulas for an ellipsoïd of revolution used in geodesy.

**Category:** Geometry

[181] **viXra:1511.0202 [pdf]**
*submitted on 2015-11-21 05:45:24*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 9 Pages. In French.

The paper concerns the Peterson operator in differentail geometry. The case of the sphere is presented as an example.

**Category:** Geometry

[180] **viXra:1511.0182 [pdf]**
*submitted on 2015-11-19 20:01:11*

**Authors:** Robert B. Easter

**Comments:** 16 Pages.

The G(8,2) Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include reflection, projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.

**Category:** Geometry

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

**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

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

**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

[67] **viXra:1707.0374 [pdf]**
*replaced on 2017-07-29 09:16:08*

**Authors:** Liu Ran

**Comments:** 3 Pages.

因为圆周率是一个比值，圆周长和直径都是测量而来，只要去测量圆周长和直径，就不可避免要在空间去度量，而空间，全都是非欧空间。

**Category:** Geometry

[66] **viXra:1706.0409 [pdf]**
*replaced on 2017-06-28 01:37:22*

**Authors:** Antoine Balan

**Comments:** 4 pages, written in french

The symplectic Dirac operator is defined over a spin Kaehler manifold. The corresponding Schrödinger-Lichnerowicz formula is proved.

**Category:** Geometry

[65] **viXra:1706.0021 [pdf]**
*replaced on 2017-06-04 16:51:38*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 25 Pages.

The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.
The understanding of these equations has taken us very far in a universe so familiar to mathematicians, the universe of periodic functions, and that of geometric forms with rounded ends revealing an infinity of new mathematical constants associated with them.

**Category:** Geometry

[64] **viXra:1703.0080 [pdf]**
*replaced on 2017-11-27 07:14:25*

**Authors:** Gaurav Biraris

**Comments:** 20 Pages. A citation is published/changed

The paper proposes a generalization of geometric notion of vectors concerning dimensionality of the configuration space. In certain dimensional spaces, certain types of ordered directions exist along which elements of vector spaces can be interpreted. Scalars along the ordered directions form Banach spaces. Different types of geometrical vectors are algebraically identical, the difference arises in the configuration space geometrically. In the universe four types of vectors exists. Thus any physical quantity in the universe comes in four types of vectors. Though All the types of vectors belong to different Banach spaces (& their directions can’t be compared), their magnitudes can be compared. A gross comparison between the magnitudes of the different typed geometric vectors is obtained at end of the paper.

**Category:** Geometry

[63] **viXra:1608.0328 [pdf]**
*replaced on 2016-08-27 23:19:26*

**Authors:** James A. Smith

**Comments:** 38 Pages.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.

**Category:** Geometry

[62] **viXra:1607.0166 [pdf]**
*replaced on 2016-07-26 09:29:46*

**Authors:** James A. Smith

**Comments:** Pages.

This document adds to the collection of solved problems presented in References 1-4. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in three ways.

**Category:** Geometry

[61] **viXra:1605.0314 [pdf]**
*replaced on 2016-08-20 22:58:10*

**Authors:** James A. Smith

**Comments:** 22 Pages.

NOTE: A new Appendix presents alternative solutions.
The famous "Problem of Apollonius", in plane geometry, is to construct all of the circles that are tangent, simultaneously, to three given circles. In one variant of that problem, one of the circles has innite radius (i.e., it's a line). The Wikipedia article that's current as of this writing has an extensive description of the problem's history, and of methods that have been used to solve it. As described in that article, one of the methods reduces the "two circles and a line" variant to the so-called "Circle-Line-Point" (CLP) special case: Given a circle C, a line L, and a point P, construct the circles that are tangent to C and L, and pass through P. This document has been prepared for two very different audiences: for my fellow students of GA, and for experts who are preparing materials for us, and need to know which GA concepts we understand and apply readily, and which ones we do not.

**Category:** Geometry

[60] **viXra:1605.0233 [pdf]**
*replaced on 2016-08-20 21:29:55*

**Authors:** James A. Smith

**Comments:** 18 Pages.

Note: The Appendix to this new version gives an alternate--and much simpler--solution that does not use reflections.
The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors.
As an aid to students, the author has prepared a dynamic-geometry construction to accompany this article.

**Category:** Geometry

[59] **viXra:1605.0233 [pdf]**
*replaced on 2016-06-11 07:43:18*

**Authors:** James A. Smith

**Comments:** 15 Pages.

The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors.

**Category:** Geometry

[58] **viXra:1605.0233 [pdf]**
*replaced on 2016-06-05 15:58:39*

**Authors:** James A. Smith

**Comments:** 15 Pages.

The beautiful Problem of Apollonius from classical geometry ("Construct all of the circles that are tangent, simultaneously, to three given coplanar circles") does not appear to have been solved previously by vector methods. It is solved here via Geometric Algebra (GA, also known as Clifford Algebra) to show students how they can make use of GA's capabilities for expressing and manipulating rotations and reflections. As Viète did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving the transformed problem, guidance is provided to help students ``see" geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3) recognizing complex algebraic expressions that reduce to simple rotations of multivectors.

**Category:** Geometry

[57] **viXra:1605.0233 [pdf]**
*replaced on 2016-06-03 19:48:48*

**Authors:** James A. Smith

**Comments:** 14 Pages.

The beautiful Problem of Apollonius from classical geometry (“Construct all of the circles that are tangent, simultaneously, to three given coplanar circles”) does not appear to have been solved previously by vector methods. It is solved here via GA to show students how they can make use of GA’s capabilities for expressing and manipulating rotations and reflections. As Viete did when deriving his ruler-and-compass solution, we first transform the problem by shrinking one of the given circles to a point. In the course of solving
the transformed problem, guidance is provided to help students “see” geometric content in GA terms. Examples of the guidance that is given include (1) recognizing and formulating useful reflections and rotations that are present in diagrams; (2) using postulates on the equality of multivectors to obtain solvable equations; and (3)recognizing complex algebraic expressions that reduce to simple rotations of multivectors.

**Category:** Geometry

[56] **viXra:1602.0234 [pdf]**
*replaced on 2016-03-08 16:32:26*

**Authors:** Espen Gaarder Haug

**Comments:** 19 Pages.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proved that pi was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in Euclidean space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in Euclidean space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space the Impossible is possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus we have added a solution to the impossibility of Doubling the Cube. As a double bonus we also have also boxed the sphere! As one will see one can claim we simply have bent the rules and moved a problem from one place to another. One of the main essences of this paper is that we can move challenging space problems out from space and into time, and vice versa.

**Category:** Geometry

[55] **viXra:1602.0234 [pdf]**
*replaced on 2016-02-24 04:24:44*

**Authors:** Espen Gaarder Haug

**Comments:** 17 Pages.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proved that pi was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in Euclidean space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in Euclidean space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space the Impossible is possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus we have added a solution to the impossibility of Doubling the Cube. As a double bonus we also have also boxed the sphere! As one will see one can claim we simply have bent the rules and moved a problem from one place to another. One of the main essences of this paper is that we can move challenging space problems out from space and into time, and vice versa.

**Category:** Geometry

[54] **viXra:1602.0234 [pdf]**
*replaced on 2016-02-22 17:51:04*

**Authors:** Espen Gaarder Haug

**Comments:** 16 Pages.

Squaring the Circle is a famous geometry problem going all the way back to the ancient Greeks. It is the great quest of constructing a square with the same area as a circle using a compass and straightedge in a finite number of steps. Since it was proved that pi was a transcendental number in 1882, the task of Squaring the Circle has been considered impossible. Here, we will show it is possible to Square the Circle in Euclidean space-time. It is not possible to Square the Circle in Euclidean space alone, but it is fully possible in Euclidean space-time, and after all we live in a world with not only space, but also time. By drawing the circle from one reference frame and drawing the square from another reference frame, we can indeed Square the Circle. By taking into account space-time rather than just space the Impossible is possible! However, it is not enough simply to understand math in order to Square the Circle, one must understand some “basic” space-time physics as well. As a bonus we have added a solution to the impossibility of Doubling the Cube. As a double bonus we also have tried to box the sphere!

**Category:** Geometry

[53] **viXra:1512.0303 [pdf]**
*replaced on 2015-12-18 20:37:40*

**Authors:** Robert B. Easter

**Comments:** 30 Pages.

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

**Category:** Geometry

[52] **viXra:1512.0303 [pdf]**
*replaced on 2015-12-18 01:51:12*

**Authors:** Robert B. Easter

**Comments:** 28 Pages.

This paper introduces the differential operators in the G(8,2) Geometric Algebra, called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). The differential operators are three x, y, and z-direction bivector-valued differential elements and either the commutator product or the anti-commutator product for multiplication into a geometric entity that represents the function to be differentiated. The general form of a function is limited to a Darboux cyclide implicit surface function. Using the commutator product, entities representing 1st, 2nd, or 3rd order partial derivatives in x, y, and z can be produced. Using the anti-commutator product, entities representing the anti-derivation can be produced from 2-vector quadric surface and 4-vector conic section entities. An operator called the pseudo-integral is defined and has the property of raising the x, y, or z degree of a function represented by an entity, but it does not produce a true integral. The paper concludes by offering some basic relations to limited forms of vector calculus and differential equations that are limited to using Darboux cyclide implicit surface functions. An example is given of entity analysis for extracting the parameters of an ellipsoid entity using the differential operators.

**Category:** Geometry

[51] **viXra:1512.0303 [pdf]**
*replaced on 2015-12-16 23:13:56*

**Authors:** Robert B. Easter

**Comments:** 28 Pages.

**Category:** Geometry

[50] **viXra:1511.0182 [pdf]**
*replaced on 2016-07-22 09:22:40*

**Authors:** Robert B. Easter

**Comments:** 16 Pages.

The G(8,2) Geometric Algebra, also called the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), has entities that represent conic sections. DCGA also has entities that represent planar sections of Darboux cyclides, which are called cyclidic sections in this paper. This paper presents these entities and many operations on them. Operations include reflection, projection, rejection, and intersection with respect to spheres and planes. Other operations include rotation, translation, and dilation. Possible applications are introduced that include orthographic and perspective projections of conic sections onto view planes, which may be of interest in computer graphics or other computational geometry subjects.

**Category:** Geometry