Geometry

Our Cultural Heritage

Audun Holme (Autor)

Buch | Softcover

XVI, 378 Seiten

2010 | 1., st Edition. Softcover version of original hardcover edition 2002
Springer Berlin (Verlag)
978-3-642-07546-9 (ISBN)

Infos

Mathematics is more important than ever, but phrases like "math avoidance" and "math anxiety" are very much in the public vocabulary. In addition to providing an invitation to mathematics in general, this book emphasizes the dynamic character of geometry and its role as part of the foundation for our cultural heritage. Aimed at an informed public and future teachers of mathematics, it seeks to heal the ills of math phobia in society. The book aims at future teachers of mathematics. All too often the geometry which goes into the syllabus for teacher-students presents the material as pedantic and formalistic, suppressing its dynamic character and its role as part of the foundation for our common cultural heritage. The motivation for the book is to open up these aspects of the field. Another motivation is to provide an invitation to mathematics in general. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics are more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. An important task is seriously attempting to heal these ills. Ills perhaps inflicted on students at an early age, through deficient or even harmful teaching practices. Thus the book also aims at an informed public, interested in making a new beginning in math. And in doing so, learning more about this part of our cultural heritage. The book is divided into two parts. Part 1 is called A Cultural Heritage. The section constitutes an excursion into the history of geometry. It also contains material which is normally not included into a mathematical text. For example, we relate some of the stories told by the Greek historian, Herodotus. Apart from being good reading, some of the early geometers were very much part of these events.

Inhalt

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 T

Produkt-Details

• Sprache: englisch
• Maße: 155 x 235 mm
• Gewicht: 598 g
• Einbandart: Paperback
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: Algebraic Geometry • Geometry • history of geometry • teaching mathematics
• ISBN-10: 3-642-07546-0 / 3642075460
• ISBN-13: 978-3-642-07546-9 / 9783642075469
• Zustand: Neuware