# Geometry

Previous months:
2010 - 1003(11) - 1004(4) - 1005(2) - 1006(7) - 1007(2) - 1008(5) - 1009(7) - 1010(5) - 1011(1)
2011 - 1101(4) - 1102(3) - 1103(6) - 1104(18) - 1106(2) - 1107(4) - 1108(1) - 1110(2) - 1111(1)
2012 - 1201(2) - 1202(1) - 1203(2) - 1204(2) - 1205(3) - 1208(1) - 1209(1) - 1211(4)
2013 - 1301(1) - 1303(3) - 1304(1) - 1305(2) - 1306(9) - 1307(2) - 1308(2) - 1309(2) - 1310(2) - 1311(2) - 1312(5)
2014 - 1401(2) - 1404(3) - 1405(5) - 1406(2) - 1407(3) - 1408(4) - 1409(3) - 1410(2) - 1411(3)
2015 - 1502(5) - 1504(2) - 1507(2) - 1508(8) - 1510(1) - 1511(4) - 1512(4)
2016 - 1601(1) - 1602(6) - 1604(1) - 1605(4) - 1606(2) - 1607(27) - 1608(4) - 1609(2) - 1610(3) - 1611(2) - 1612(1)
2017 - 1701(1) - 1702(1) - 1703(3) - 1704(3) - 1706(4)

## Recent submissions

Any replacements are listed further down

[252] viXra:1706.0506 [pdf] submitted on 2017-06-26 13:21:15

### Introduction to Differential Geometry of Space Curves and Surfaces

Authors: Taha Sochi

This book, which consists of 260 pages, is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with an index, extensive sets of exercises and many cross references, which are hyperlinked for the ebook users, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.
Category: Geometry

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

### An Application of Finite Affine Plane of Order $n$, in an Experiment Planning

Authors: Orgest ZAKA, Kristaq FILIPI

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

### Symplectic Dirac Operators-Lichnerowicz Formula

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

### General Trigonometry of Rounded Geometric Figures

Authors: Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

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

### Moduli Space of Compact Lagrangian Submanifold

Authors: Giordano Colò

We describe the deformations of 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. By constructing Banach spaces on bundle sections and by elliptical operators, we are able to use Hodge theory to study the topology of the manifold. Starting from McLean results, for which moduli spaces of compact special Lagrangian submanifolds is smooth and its tangent space can be identified with harmonic 1-forms on the special Lagrangian submanifolds, we can analyze deformation theory. 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

### Apollonius Circles of Rank -1

Authors: Ion Patrascu, Florentin Smarandache

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

### The Homology Groups of $G_{n,m}(\mathbb{R})$

Authors: Xu Chen

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

### Triangle Inscribed-Triangle Picking

Authors: Arman Maesumi

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

### Iterative Computation of Moment Forms for Subdivision Surfaces

Authors: Jan Hakenberg, Ulrich Reif

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

### On the Dimensional Characteristics and Interpretation of Vectors

Authors: Gaurav S. Biraris

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

### A Toroidal Approach to the Archimedian Quadrature

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

### On Inpolars

Authors: Dragan Turanyanin, Svetozar Jovičin

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

### On Hypersurfaces F3 in e4 with Nonzero Laplacian of the Second Fundamental Form

Authors: Irina I. Bodrenko

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

### On Hyperspiral R = A*exp(b/theta)

Authors: Dragan Turanyanin

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

### The Hebrew Theorem

Authors: Brian Ekanyu

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

### Thurals & Inpolars

Authors: Dragan Turanyanin

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

### A Solution to the Problem of Apollonius Using Vector Dot Products

Authors: James A. Smith

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

### Some Solution Strategies for Equations that Arise in Geometric (Clifford) Algebra

Authors: James A. Smith

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

### An Additional Brief Solution of the CPP Limiting Case of the Problem of Apollonius Via Geometric Algebra (Ga)

Authors: James A. Smith

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

### The Geometry of the MRB Constant

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

### Additional Solutions of the Special Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith

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

### Simplified Solutions of the CLP and CCP Limiting Cases of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith

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

### The Cartan Model for Equivariant Cohomology

Authors: Xu Chen

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

Category: Geometry

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

### Using Geometry to Make Stellar Discovery

Authors: Jeffrey Joseph Wolynski

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

### Lucas’s Inner Circles

Authors: Ion Patrascu, Florentin Smarandache

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

### Magic Squares

Authors: Florentin Smarandache

We present here the magic square of order n.
Category: Geometry

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

### Negari ale Postulatului V al lui Euclid

Authors: Florentin Smarandache

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

### Neuberg’s Orthogonal Circles

Authors: Ion Patrascu, Florentin Smarandache

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

### Non-Geometry

Authors: Florentin Smarandache

It is a lot easier to deny the Euclids five postulates, than Hilberts twenty thorough axiom.
Category: Geometry

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

Authors: Florentin Smarandache

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

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

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

### Radical Axis of Lemoine’s Circles

Authors: Ion Patrascu, Florentin Smarandache

Category: Geometry

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

### Regarding the First Droz- Farny’s Circle

Authors: Ion Patrascu, Florentin Smarandache

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

### Regarding the Second Droz- Farny’s Circle

Authors: Ion Patrascu, Florentin Smarandache

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

### Theorems with Parallels Taken Through a Triangle’s Vertices and Constructions Performed Only with the Ruler

Authors: Ion Patrascu, Florentin Smarandache

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

### The Polars of a Radical Center

Authors: Ion Patrascu, Florentin Smarandache

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

### The Radical Circle of Ex- Inscribed Circles of a Triangle

Authors: Ion Patrascu, Florentin Smarandache

Category: Geometry

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

### Anti-Geometry

Authors: Florentin Smarandache

It is possible to de-formatize entirely Hilberts 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

### Apollonius’s Circle of Second Rank

Authors: Ion Patrascu, Florentin Smarandache

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

### A Sufficient Condition for the Circle of the 6 Points to Become Euler’s Circle

Authors: Ion Patrascu, Florentin Smarandache

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

### Counter-Projective Geometry

Authors: Florentin Smarandache

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

### Generalisations du Théorème de Ceva

Authors: Florentin Smarandache

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

### Generalizations of Desargues Theorem

Authors: Florentin Smarandache

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

### Geometric Conjecture

Authors: Florentin Smarandache

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

### Lemoine’s Circles

Authors: Ion Patrascu, Florentin Smarandache

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

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

### Solution of the LLP Limiting case of the Problem of Apollonius via Geometric Algebra, Using Reflections and Rotations

Authors: James A. Smith

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

### On Riemannian Model Theory

Authors: Kermit Ohlrabi

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

### Lost in an Isosceles Triangle

Authors: Philip Gibbs

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

### A Very Brief Introduction to Reflections in 2D Geometric Algebra, and their Use in Solving Construction" Problems

Authors: James A. Smith

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

### Bellman’s Escape Problem for Convex Polygons

Authors: Philip Gibbs

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

### Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith

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

### The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith

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

### Rotations of Vectors Via Geometric Algebra: Explanation, and Usage in Solving Classic Geometric "Construction" Problems (Version of 11 February 2016)

Authors: James A. Smith

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

### Complements to Classic Topics of Circles Geometry

Authors: Ion Patrascu, Florentin Smarandache

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

### Optimisation of Dynamical Systems Subject to Meta-Rules

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

### Unique in Mathematics

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

### Unique in Mathematics 2

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

### Double Conformal Geometric Algebras

Authors: Robert B. Easter

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

### One Construction of an Affine Plane Over a Corps

Authors: Orgest ZAKA, Kristaq FILIPI

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

### The Impossible is Possible! Squaring the Circle in Euclidean Space Time

Authors: Espen Gaarder Haug

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

### Techno-Art of Selariu Supermathematics Functions, 2nd Volume

Authors: editor Florentin Smarandache

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

### Create Polygon Through Fans Suitable for Parellel Calculations

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

### Modulated Chaos as a Source of Images for Numbers

Authors: Jose Carlos Tiago de Oliveira

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

### Overview over Reflection Formulas

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

### Differential Operators in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

### The Geodesics of the Tori

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

### The Right Scalene Triangle Problem

Authors: Rodolfo A. Frino

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

### Note de Géométrie Différentielle Application de la Méthode du Repère Mobile à L'Ellipsoïde de Référence

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

### A Note of Differentail Geometry: Peterson Operator: an Application to the Sphere and the Ellipsoïd

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

### Conic and Cyclidic Sections in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

[179] viXra:1510.0328 [pdf] submitted on 2015-10-19 12:12:59

### The Special Problems of Euclidean Geometry , and Relativity

Authors: Markos Georgallides

The Special Problems of E-geometry consist the Quantization Moulds of Euclidean Geometry in it , to become → The Basic monad , through mould of Space –Anti-space in itself , which is the material dipole in inner monad Structure as it is Electromagnetic cycloidal field → Linearly , through mould of Parallel Theorem , which are the equal distances between the points of parallel and line → In Plane , through mould of Squaring the circle , where the two equal and perpendicular monads consist a Plane acquiring the common Plane-meter → and in Space (volume) , through mould of the Duplication of the Cube , where any two Un-equal and perpendicular monads acquire the common Space-meter to be twice each other , as this in the analytical methods explained . The article consist also a provocation to all scarce today Geometers and to mathematicians in order to give an answer to this article and its content . All Geometrical solutions of the Old-age standing Unsolved Problems are now solved and are clearly Exposed , and reveal the pass-over-faults of Relativity .
Category: Geometry

[178] viXra:1508.0264 [pdf] submitted on 2015-08-27 02:36:17

### Axa Radicală a Cercurilor Lemoine

Authors: Ion Pătrașcu, Florentin Smarandache

În acest articol scoatem în evidență axa radicală a cercurilor Lemoine.
Category: Geometry

[177] viXra:1508.0262 [pdf] submitted on 2015-08-27 02:39:07

### Calculul Razelor Cercurilor Lemoine

Authors: Ion Pătrașcu, Florentin Smarandache

Laturile unui triunghi sunt împărțite de primul cerc Lemoine în segmente proporționale cu pătratele laturilor triunghiului.
Category: Geometry

[176] viXra:1508.0260 [pdf] submitted on 2015-08-27 02:43:09

### 255 Compiled and Solved Problems in Geometry and Trigonometry

Authors: Florentin Smarandache

The degree of difficulties of the problems is from easy and medium to hard. The solutions of the problems are at the end of each chapter. One can navigate back and forth from the text of the problem to its solution using bookmarks. The book is especially a didactical material for the mathematical students and instructors.
Category: Geometry

[175] viXra:1508.0245 [pdf] submitted on 2015-08-27 03:08:22

Authors: Ion Pătrașcu, Florentin Smarandache

The first Lemoine circle divides the sides of a triangle in segments proportional to the squares of the triangle’s sides.
Category: Geometry

[174] viXra:1508.0222 [pdf] submitted on 2015-08-27 03:47:56

### Rough Neutrosophic Multi-Attribute Decision-Making Based on Grey Relational Analysis

Authors: Kalyan Mondal, Surapati Pramanik

This paper presents rough netrosophic multiattribute decision making based on grey relational analysis. While the concept of neutrosophic sets is a powerful logic to deal with indeterminate and inconsistent data, the theory of rough neutrosophic sets is also a powerful mathematical tool to deal with incompleteness.
Category: Geometry

[173] viXra:1508.0188 [pdf] submitted on 2015-08-22 21:37:45

### The Geometry of Polychronous Wavefront Computation

Authors: Joseph I. Thomas

This paper consolidates all the salient geometrical aspects of the principle of Polychronous Wavefront Computation. A novel set of simple and closed planar curves are constructed based on this principle, using MATLAB. The algebraic and geometric properties of these curves are then elucidated as theorems, propositions and conjectures.
Category: Geometry

[172] viXra:1508.0154 [pdf] submitted on 2015-08-19 20:44:14

### The Sums of the Odd Number Series Are Squares

Authors: Ben Steber

The author will demonstrate that the sums of odd numbers to an nth value equals that nth value squared. A geometric proof will be provided to demonstrate the principle of sum odds equaling squares.
Category: Geometry

[171] viXra:1508.0086 [pdf] submitted on 2015-08-11 10:39:32

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and adds geometrical entities for all 3D quadric surfaces and a toroid entity. All entities, representing various geometric surfaces and points, can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying homogeneous polynomial equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[170] viXra:1507.0218 [pdf] submitted on 2015-07-29 23:15:12

### Three Conjectures in Euclidean Geometry

Authors: Dao Thanh Oai

In this note, I introduce three conjectures of generalization of the Lester circle theorem, the Parry circle theorem, the Zeeman-Gossard perspector theorem respectively
Category: Geometry

[169] viXra:1507.0216 [pdf] submitted on 2015-07-29 05:10:57

### Two Conjectures of Generalization of Feuerbach-Luchterhand Theorem

Authors: Dao Thanh Oai

In Euclidean geometry, Feuerbach-Luchterhand theorem is a generalization of Pythagorean theorem, Stewart theorem and the British Flag theorem.....In this note, I propose two conjectures of generalization of Feuerbach-Luchterhand theorem.
Category: Geometry

[168] viXra:1504.0189 [pdf] submitted on 2015-04-24 03:39:05

### On Kurt Gödel and Hyperbolic math.

Authors: S.Kalimuthu
Comments: 4 Pages. If there is a flaw in the proof, I welcome it.Thank you.

Reputed Austrian American mathematician Kurt Gödel formulated two extraordinary propositions in mathematical lo0gic.Accepted by all mathematicians they have revolutionized mathematics, showing that mathematical truth is more than logic and computation. These two ground breaking theorems changed mathematics, logic, and even the way we look at our Universe. The cognitive scientist Douglas Hofstadter described Gödel’s first incompleteness theorem as that in a formal axiomatic mathematical system there are propositions that can neither be proven nor disproven. The logician and mathematician Jean van Heijenoort summarizes that there are formulas that are neither provable nor disprovable. According to Peter Suber, inn a formal mathematical system, there are un decidable statements. S. M. Srivatsava formulates that formulations of number theory include undecidable propositions. And Miles Mathis describes Gödel’s first incompleteness theorem as that in a formal axiomatic mathematical system we can construct a statement which is neither true nor false. [Mathematical variance of liar’s paradox]In this short work, the author attempts to show these equivalent propositions to Gödel’s incompleteness theorems by applying elementary arithmetic operations, algebra and hyperbolic geometry. [1 – 6 ]
Category: Geometry

[167] viXra:1504.0085 [pdf] submitted on 2015-04-10 10:51:29

### La Conexión Afín. Aplicación a la Teoría Clásica de Campo

Authors: Wenceslao Segura González

Category: Geometry

## Replacements of recent Submissions

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

### The Symplectic Dirac Operator

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

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

### General Trigonometry of Rounded Geometric Figures

Authors: Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

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

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

### Additional Solutions of the Limiting Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith

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

### Three Solutions of the LLP Limiting Case of the Problem of Apollonius via Geometric Algebra, Using Reflections and Rotations

Authors: James A. Smith

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

### Solution of the Special Case "CLP" of the Problem of Apollonius via Vector Rotations using Geometric Algebra

Authors: James A. Smith

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

### The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith

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

### The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith

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

### The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via Geometric (Clifford) Algebra

Authors: James A. Smith

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

### The Problem of Apollonius as an Opportunity for Teaching Students to Use Reflections and Rotations to Solve Geometry Problems via GA

Authors: James A. Smith

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

### Squaring the Circle and Doubling the Cube

Authors: Espen Gaarder Haug

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

### The Impossible is Possible! Squaring the Circle and Doubling the Cube

Authors: Espen Gaarder Haug

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

### The Impossible is Possible! Squaring the Circle and Doubling the Cube

Authors: Espen Gaarder Haug

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

### Differential Operators in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

### Differential Operators in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

### Differential Operators in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

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

### Conic and Cyclidic Sections in the G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

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

[49] viXra:1508.0086 [pdf] replaced on 2015-10-01 17:57:21

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[48] viXra:1508.0086 [pdf] replaced on 2015-09-30 22:19:24

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[47] viXra:1508.0086 [pdf] replaced on 2015-09-24 13:06:54

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[46] viXra:1508.0086 [pdf] replaced on 2015-09-20 08:27:38

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general Darboux cyclide surfaces in Euclidean 3D space. The general Darboux cyclide is a quartic surface. Darboux cyclides include circular tori and all quadrics, and also all surfaces formed by their inversions in spheres. Dupin cyclide surfaces can be formed as inversions in spheres of circular toroid, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversion spheres centered on other surfaces. All DCGA entities can be conformally transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. All entities can be inversed in general spheres and reflected in general planes. Entities representing the intersections of surfaces can be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[45] viXra:1508.0086 [pdf] replaced on 2015-09-02 15:26:57

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA that adds geometrical entities for all 3D quadric surfaces and a torus surface. More generally, DCGA has an entity for general cyclide surfaces in 3D, which is a class of quartic surfaces that includes the quadric surfaces and toroid and also their inversions in spheres known as Dupin cyclides. All entities representing various geometric surfaces and points can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. All entities can be intersected with spheres, planes, lines, and circles. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[44] viXra:1508.0086 [pdf] replaced on 2015-08-21 22:21:34

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and adds geometrical entities for all 3D quadric surfaces and a toroid entity. All entities, representing various geometric surfaces and points, can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying implicit surface equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[43] viXra:1508.0086 [pdf] replaced on 2015-08-17 10:53:51

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and adds geometrical entities for all 3D quadric surfaces and a toroid entity. All entities, representing various geometric surfaces and points, can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying homogeneous polynomial equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[42] viXra:1508.0086 [pdf] replaced on 2015-08-13 13:34:52

### G8,2 Geometric Algebra, DCGA

Authors: Robert B. Easter

This paper introduces the Double Conformal Geometric Algebra (DCGA), based in the G8,2 Clifford geometric algebra. DCGA is an extension of CGA and adds geometrical entities for all 3D quadric surfaces and a toroid entity. All entities, representing various geometric surfaces and points, can be transformed in 3D space by rotors, dilators, translators, and motors, which are all types of versors. Versors provide an algebra of spatial transformations that are different than linear algebra transformations. Entities representing the intersections of geometric surfaces can also be created by wedge products. DCGA provides a higher-level algebra for working with 3D geometry in an object/entity-oriented system of mathematics above the level of the underlying homogeneous polynomial equations of algebraic geometry. DCGA could be used in the study of geometry in 3D, and also for some applications.
Category: Geometry

[41] viXra:1507.0218 [pdf] replaced on 2015-07-30 03:34:38

### Three Conjectures in Euclidean Geometry

Authors: Dao Thanh Oai