Desargues Theorem Research Articles

Simpler foundations for the hyperbolic plane

Abstract H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Identities Generalizing the Theorems of Pappus and Desargues

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

Two-valenced association schemes and the Desargues theorem

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition, the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique, we prove a new result: given a prime p, any {1,p}-scheme with thin residue isomorphic to an elementary abelian p-group of rank greater than two, is schurian and separable.

ФОРМУВАННЯ КРИТИЧНОГО МИСЛЕННЯ МАЙБУТНІХ УЧИТЕЛІВ МАТЕМАТИКИ ЗАСОБАМИ ГЕОМЕТРІЇ

The essence of the concept of critical thinking as a modern pedagogical phenomenon is disclosed in the article. It is revealed that among the most important competencies, which are relevant to the work of a modern teacher, teachers often refer to critical thinking as the ability to critically process various information, the ability to produce ideas, identify strategic issues and solve them, find convincing arguments, take responsibility, defend your position and correct it under the reasonable influence of the opponents. The theoretical foundations of the process of formation of critical thinking of future teachers in the process of professional training at the level of principles, properties, methods, stages are identified. The analysis of the project "The Higher Education Standard of Ukraine" in the field of knowledge 01 "Education" for specialty 014 "Secondary Education", subject specialization 014.04 "Secondary Education (Mathematics)", which provides for the formation of a number of competencies among students is carried out. It is found that the basis of the above listed competencies and learning outcomes is critical thinking. It is proved that geometry (in particular, constructive and projective), its methods and tasks have inexhaustible possibilities and means that can be used as an effective tool for critical thinking formation in the process of professional training of future mathematics teachers. For an example, we solve two problems of Euclidean geometry (a construction task and a proof) with the use of the Desargues theorem- one of the key statements of projective geometry.

Sphericity of a real hypersurface via projective geometry

In this work, we obtain an unexpected geometric characterization of sphericity of a real-analytic Levi-nondegenerate hypersurface [Formula: see text]. We prove that [Formula: see text] is spherical if and only if its Segre(-Webster) varieties satisfy an elementary combinatorial property, identical to a property of straight lines on the plane and known in Projective Geometry as the Desargues Theorem.

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.

The projective plane, J-holomorphic curves and Desarguesʼ theorem

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys.

A case study in formalizing projective geometry in Coq: Desargues theorem

Formalizing geometry theorems in a proof assistant like Coq is challenging. As emphasized in the literature, the non-degeneracy conditions lead to long technical proofs. In addition, when considering higher-dimensions, the amount of incidence relations (e.g. point–line, point–plane, line–plane) induce numerous technical lemmas. In this article, we investigate formalizing projective plane geometry as well as projective space geometry. We mainly focus on one of the fundamental properties of the projective space, namely Desargues property. We formally prove that it is independent of projective plane geometry axioms but can be derived from Pappus property in a two-dimensional setting. Regarding at least three-dimensional projective geometry, we present an original approach based on the notion of rank which allows to describe incidence and non-incidence relations such as equality, collinearity and coplanarity homogeneously. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize Desargues theorem in Coq. This illustrates the power and efficiency of our approach (using only ranks) to prove properties of the projective space.

Clifford lattices and a conformal generalization of Desargues’ theorem

Lattices composed of Clifford point–circle configurations provide a geometric representation of the discrete Schwarzian KP (dSKP) equation. Based on an An perspective on such lattices, it is shown that their integrability, and hence that of the dSKP equation, is a consequence of a conformal generalization of the classical Desargues theorem of projective geometry.

Classification of Integrable Discrete Equations of Octahedron Type

We use the consistency approach to classify discrete integrable 3D equations of the octahedron type. They are naturally treated on the root lattice $Q(A_3)$ and are consistent on the multidimensional lattice $Q(A_N)$. Our list includes the most prominent representatives of this class, the discrete KP equation and its Schwarzian (multi-ratio) version, as well as three further equations. The combinatorics and geometry of the octahedron type equations are explained. In particular, the consistency on the 4-dimensional Delaunay cells has its origin in the classical Desargues theorem of projective geometry. The main technical tool used for the classification is the so called tripodal form of the octahedron type equations.

Revolutions of geometry

Preface. Acknowledgments. PART I FOUNDATIONS. 1 The First Geometers. 1.1 Egypt. 1.2 Babylon. 1.3 China. 2 Thales. 2.1 The Axiomatic System. 2.2 Deductive Logic. 2.3 Proof Writing. 3 Plato and Aristotle. 3.1 Form. 3.2 Categorical Propositions. 3.3 Categorical Syllogisms. 3.4 Figures. PART II THE GOLDEN AGE. 4 Pythagoras. 4.1 Number Theory. 4.2 The Pythagorean Theorem. 4.3 Archytas. 4.4 The Golden Ratio. 5 Euclid. 5.1 The Elements. 5.2 Constructions. 5.3 Triangles. 5.4 Parallel Lines. 5.5 Circles. 5.6 The Pythagorean Theorem Revisited. 6 Archimedes. 6.1 The Archimedean Library. 6.2 The Method of Exhaustion. 6.3 The Method. 6.4 Preliminaries to the Proof. 6.5 The Volume of a Sphere. PART III ENLIGHTENMENT. 7 Francois Viete. 7.1 The Analytic Art. 7.2 Three Problems. 7.3 Conic Sections. 7.4 The Analytic Art in Two Variables. 8 Rene Descartes. 8.1 Compasses. 8.2 Method. 8.3 Analytic Geometry. 9 Gerard Desargues. 9.1 Projections. 9.2 Points at Infinity. 9.3 Theorems of Desargues and Menelaus. 9.4 Involutions. PART IV A STRANGE NEW WORLD. 10 Giovanni Saccheri. 10.1 The Question of Parallels. 10.2 The Three Hypotheses. 10.3 Conclusions for Two Hypotheses. 10.4 Properties of Parallel Lines. 10.5 Parallelism Redefined. 11 Johann Lambert. 11.1 The Three Hypotheses Revisited. 11.2 Polygons. 11.3 Omega Triangles. 11.4 Pure Reason. 12 Nicolai Lobachevski and Janos Bolyai. 12.1 Parallel Fundamentals. 12.2 Horocycles. 12.3 The Surface of a Sphere. 12.4 Horospheres. 12.5 Evaluating the Pi Function. PART V NEW DIRECTIONS. 13 Bernhard Riemann. 13.1 Metric Spaces. 13.2 Topological Spaces. 13.3 Stereographic Projection. 13.4 Consistency of Non-Euclidean Geometry. 14 Jean-Victor Poncelet. 14.1 The Projective Plane. 14.2 Duality. 14.3 Perspectivity. 14.4 Homogeneous Coordinates. 15 Felix Klein. 15.1 Group Theory. 15.2 Transformation Groups. 15.3 The Principal Group. 15.4 Isometries of the Plane. 15.5 Consistency of Euclidean Geometry. References. Index.

“Theoric Transformations” and a New Classification of Abductive Inferences

Based on a definition of "abductive insight" and a critical discussion of G. Schurz's (2008) distinction of eleven "patterns of abduction" that he organizes in four groups, I suggest an even more comprehensive classification that distinguishes 15 forms in an alternative structure. These forms are organized, on the one hand, with regard to what is abductively inferred—singular facts, types, laws, theoretical models, or representation systems—and, on the other, with regard to the question whether the abductive procedure is selective or creative (including a distinction between "psychologically creative," as in school learning, or "historically creative"). Moreover, I argue that theoretical-model abduction—which seems to be the most important form of abduction—depends on two preconditions: first on the availability of an adequate system of representation, and second on finding a new "perspective" on a given problem, as Peirce described it with the notion of a "theoric transformation." To understand the significance of theoric transformations—especially in mathematics—it is necessary to analyze in some detail Peirce's main example for a theoric transformation: the proof of Desargues's theorem.

Desargues maps and the Hirota–Miwa equation

We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is equivalent to the Desargues theorem and its higher dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

GENERALIZATION OF THE DESARGUES THEOREM FOR SPARSE 3D RECONSTRUCTION

Visual perception for walking machines needs to handle more degrees of freedom than for wheeled robots. For humanoid, four- or six-legged robots, camera motion is 6D instead of 3D or planar motion. Classical 3D reconstruction methods cannot be applied directly, because explicit sensor motion is needed. In this paper, we propose an algorithm for 3D reconstruction of an unstructured environment using motion-free uncalibrated single camera. Computer vision techniques are employed to obtain an incremental geometrical reconstruction of the environment, therefore using vision as a sensor for robot control tasks like navigation, obstacle avoidance, manipulation, tracking, etc. and 3D model acquisition. The main contribution is that the offline 3D reconstruction problem is considered as a point trajectory search through the video stream. The algorithm takes into account the temporal aspect of the sequence of images in order to have an analytical expression of the geometrical locus of the point trajectories through the sequence of images. The approach is a generalization of the Desargues theorem applied to multiple views taken from nearby viewpoints. Experiments on both synthetic and real image sequences show the simplicity and efficiency of the proposed method. This method provides an alternative technical solution easy to use, flexible in the context of robotic applications and can significantly improve the 3D estimation accuracy.

Projectivities between bundles of plane higher order curves

In [5] we give a theorem about projectivities between three line bundles in the projective plane with an almost trivial proof, and with a lot of special cases among which the theorems of Desargues, Pappus-Pascal, Pascal and many other lesser known results. In this paper we retake this idea, but now for projectivities between bundles of higher order curves in the plane.

Geometry: our cultural heritage

Part I. A Cultural Heritage: Chapter 1. Early Beginnings 1.1 Prehistory 1.2 Geometry in the New Stone Age 1.3 Early Mathematics and Ethnomathematics Chapter 2. The Great River Civilizations 2.1 Civilizations long dead -- and yet alive 2.2 Birth of Geometry as we know it 2.3 Geometry in the Land of the Pharaoh 2.4 Babylonian Geometry Chapter 3. Greek and Hellenic Geometry 3.1 Early Greek Geometry. Thales of Miletus 3.2 The story of Pythagoras and the Pythagoreans 3.3 The Geometry of the Pythagoreans 3.4 The Discovery of Irrational Numbers 3.5 Origin of the Classical Problems 3.6 Constructions by Compass and Straightedge 3.7 Squaring the Circle 3.8 Doubling the Cube 3.9 Trisecting any Angle 3.10 Plato and the Platonic Solids 3.11 Archytas and the Doubling of the Cube Chapter 4. Geometry in the Hellenistic Era 4.1 Euclid and Euclids Elements 4.2 The Books of Euclids Elements 4.3 The Roman Empire 4.4 Archimedes 4.5 Erathostenes and the Duplication of the Cube 4.6 Nicomedes and his Conchoid 4.7 Apollonius and the Conic Sections 4.8 Caesar and the End of the Republic in Rome 4.9 The Murder of Hypatia 4.10 The Decline and Fall of the Roman Empire Chapter 5. The Geometry of Yesterday and Today 5.1 The Dark Middle Ages 5.2 Geometry Reawakening: A new Dawn in Europe 5.3 Elementary Geometry and Higher Geometry 5.4 Desargues and the two Pascals 5.5 Descartes and Analytic Geometry 5.6 Geometry in the 18th. Century 5.6.1 Cramers theorem 5.7 Some Features of Modern Geometry Chapter 6. Geometry and the Real World 6.1 Mathematics and Predicting Catastrophes 6.2 Catastrophe Theory 6.3 Geometric Shapes in Nature 6.4 Fractal Structures in Nature Part II. Introduction to Geometry Chapter 7. Axiomatic Geometry 7.1 The Postulates of Euclid and Hilbert's Explanation 7.2 Non-Euclidian Geometry 7.3 Logic and Intuitive Set Theory 7.4 Axioms, Axiomatic Theories and Models 7.5 General Theory of Axiomatic Systems Chapter 8. Axiomatic Projective Geometry 8.1 Plane Projective Geometry 8.2 An Unsolved Geometric Problem 8.3 The Real Projective Plane Chapter 9. Models for non-Euclidian Geometry 9.1 Three Types of Geometry 9.2 Hyperbolic Geometry 9.3 Elliptic Geometry 9.4 Euclidian and non-Euclidian Geometry in Space 9.5 Riemannian Geometry Chapter 10. Making Things Precise 10.1 Relations and Their Uses 10.2 Identification of Points 10.3 Our Number System 10.3.1 The integers 10.3.2 The rational numbers 10.3.3 The real numbers 10.3.4 The complex numbers Chapter 11. Projective Space 11.1 Coordinates in the Projective Plane 11.2 Projective n-Space 11.3 Affine and Projective Coordinate Systems Chapter 12. Geometry in the Affine and the Projective plane 12.1 The Theorem of Desargues 12.2 Duality for the projective plane 12.3 Naive Definition and First Examples of Affine Plane Curves 12.4 Straight Lines 12.5 Conic Sections in the Affine plane 12.6 Constructing Points on Conic Sections by Compass and Straightedge 12.7 Further Properties of Conic Sections 12.8 Conic Sections in the Projective Plane 12.9 The Theorems of Pappus and Pascal Chapter 13. Algebraic Curves of Higher Degrees in the Affine Plane 13.1 The Cubical Curves in the affine plane 13.1.1 Cubical parabolas 13.1.2 Semi cubical parabolas 13.1.3 Degenerate curves and degeneration of a family of curves 13.1.4 Folium of Descartes 13.1.5 Elliptic curves 13.1.6 Trisectrix of MacLaurin and the Clover leaf curve 13.2 Curves of Degree Higher than Three 13.3 Affine Algebraic Curves 13.4 Singularities and Multiplicities 13.5 Tangency Chapter 14. Higher Geometry in the Projective plane 14.1 Projective Curves 14.2 Projective Closure and Affine Restriction 14.3 Smooth and Singular Points on Affine and Projective Curves 14.4 The Tangent of a Projective Curve 14.5 Projective Equivalence 14.6 Asymptotes 14.7 General Conchoids 14.8 The Dual Curve 14.9 The Dual of Pappus'Theorem 14.10 Pascals Mysterium Hexagrammicum Chapter 15. Sharpening the Sword of Algebra 15.1 On Rational Polynomials 15.2 The Minimal Polynomial 15.3 The Euclidian Algorithm 15.4 Number Fields and Field Extensions 15.5 More on Field Extensions Chapter 16. Constructions with Straightedge and Compass 16.1 Review of Legal Constructions 16.2 Constructible Points 16.3 What is Possible? 16.4 Trisecting an Angle 16.5 Doubling the Cube 16.6 Squaring the Circle 16.7 Regular Polygons 16.8 Constructions by Folding 16.9 Concluding Remarks on Constructions Chapter 17. Fractal Geometry 17.1 Fractals and their Dimensions 17.2 The von Koch Snowflake Curve 17.3 Fractal Shapes in Nature 17.4 The Sierpinski Triangles 17.5 A Cantor Set Chapter 18. Catastrophe Theory 18.1 The Cusp Catastrophe: Geometry of a Cubic Surface 18.2 Rudiments of Control Theory

On collinear Griffiths points

J. Tabov has proved [1] that four Griffiths points are collinear if the vertices of a given quadrangle are on a circle. In this article we prove some generalization of this result in a very simple geometrical way (based on Desargues theorem).

Curves of genus 2 and Desargues configurations

A Desargues configuration is the configuration of 10 points and 10 lines of the classical theorem of Desargues in the complex projective plane. For a precise definition see Section 2. The Greek mathematician C. Stephanos showed in 1883 (see [10]) that one can associate to every Desargues configuration a curve of genus 2 in a canonical way. Moreover he proved that the induced map from the moduli space of Desargues configurations to the moduli space of curves of genus 2 is birational. Our main motivation for writing this paper was to understand the last result. Stephanos needed about a hundred pages of classical invariant theory to prove it. We apply instead a simple argument of Schubert calculus to prove a slightly more precise version of his result. Let MD denote the (coarse) moduli space of Desargues configurations. It is a threedimensional quasiprojective variety. On the other hand, let M 6 denote the moduli space of stable binary sextics. There is a canonical isomorphism H2nD1 GM 6 (see [1]). Here H2 denotes the moduli space of stable curves of genus 2 in the sense of Deligne–Mumford and D1 the boundary divisor parametrizing two curves of genus 1 intersecting transversely in one point. So instead of curves of genus 2, we may speak of binary sextics. The main result of the paper consists of the following two statements:

Desargues Theorem, Dynamics, and Hyperplane Arrangements

The Desargues theorem is a basic theorem in classical projective geometry. In this paper we generalize Desargues theorem in the direction of dynamical systems. Our result comprises an infinite family of configurations, having unbounded complexity. The proof of the result involves constructing special kinds of hyperplane arrangements and then projecting subsets of them into the plane.

蛭子井氏の定理と (15<SUB>4</SUB>, 20<SUB>3</SUB>) コンフィグレイション

A simple geometric proof of Ebisui's theorem, if two triangles A1A2A3 and B1B2B3 are perspective and C3 = A1B2∩A2B1, C1=A2B3∩A3B2, C2=A3B1∩A1B3 then A1A2 A3 and C1C2C3 are also perspective, is given, which is using Desargues's theorem and its converse. With the theorem and an additional theorem, a (154, 203) configuration can be constructed, which is transitive both on points and lines.