Geometry. Our cultural heritage (Q5957130)

This book contains selected topics from the history of geometry, with ``modern'' proofs of some of the results, as well as a fully modern treatment of selected basic issues in geometry. The book is divided into two parts. Part 1, called \textit{A Cultural Heritage}, contains material which is normally not included into a mathematical text. For example, the author relates some of the stories told by the Greek historian, informed public, interested in making a new beginning in math. The excursions into the history of geometry do not represent an attempt at writing the history of geometry. Instead, the author wishes to seek out the roots of the themes to be treated in Part 2, \textit{Introduction to Geometry}. These roots include not only the geometric ideas and their development, but also the historical context. The aim of the Part 2 is to provide an invitation to mathematics, in general, and to give a flavor of the geometry, in particular, without going into the technical details at all. The book aims at future teachers of mathematics and informed public, interested in making a new beginning in math. From the Contents: \textit{A Cultural Heritage.} Early beginnings. The great river civilizations. Geometry in the Land of the Pharaoh. Babylonian geometry. Greek and hellenic geometry. Pythagoras and the Pythagoreans. The discovery of irrational numbers. Constructions by compass and straightedge. Squaring the circle. Doubling the cube. Trisecting any angle. Plato and the platonic solids. Archytas and the doubling of the cube. Geometry in the hellenistic era. Euclid and Euclid's Elements. The Roman Empire. Archimedes. Erathostenes and the duplication of the cube. Nicomedes and his conchoid. Apollonius of Perga and the conic sections. Caesar and the end of the Republic in Rome. Heron of Alexandria. Menelaus of Alexandria. Claudius Ptolemy. Pappus of Alexandria. The murder of Hypatia. The decline and fall of the Roman Empire. The dark middle ages. Elementary geometry and higher geometry. Desargues and the two Pascals. Descartes and analytic geometry. Geometry in the 18th century. Some features of modern geometry. Geometry and the real world. \textit{Introduction to Geometry.} Axiomatic geometry. Logic and intuitive set theory. Axiomatic theories and models. An unsolved geometric problem. Hyperbolic geometry. Elliptic geometry. Riemannian geometry. Making things precise. Relations and their uses. Projective space. Geometry in the affine and the projective plane. The theorem of Desargues. Naive definition and first examples of affine plane curves. Conic sections in the affine plane. The theorems of Pappus and Pascal. Algebraic curves of higher degrees in the affine plane. Singularities and multiplicities. Tangency. Higher geometry in the projective plane. Projective curves. Projective closure and affine restriction. The tangent of a projective curve. Asymptotes. General conchoids. The dual of Pappus' theorem. Pascal's mysterium hexagrammicum. Sharpening the sword of algebra. The Euclidean algorithm. Number fields and field extensions. Fractals and their dimensions. Fractal shapes in nature. The Sierpinski triangles. A Cantor set. Catastrophe theory. The cusp catastrophe: geometry of a cubic surface. Rudiments of control theory. References. Index.