DetallesThis book presents plane geometry following Hilbert\'s axiomatic system. It is inspired on R. Hartshome\'s fantastic book Geometry: Euclid and Beyond, and can be considered as a careful exposition of most of chapter II of that book.It must be remarked that non-euclidean geometries, i.e. geometries not satisfying the fifth postulate of Euclid, enter the scene, in a natural way, from the very beginning.The vast majority of plane geometry texts are only concerned with euclidean geometry, loosing the oportunity of not only teaching rigorous reasoning to freshmen, but at the same time exposing them to the mind expanding experience of contemplating the strange geometries conceived, in the first decades of the XIX century, by Gauss, Lobachevsky and Bolyai.
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Editor / Marca U. EAFIT Editor Facultad Escuela de Ciencias Año de Edición 2019 Número de Páginas 330 Idioma(s) Español Terminado Tapa Rústica Alto y ancho 16.5 x 24 cm Peso 0.5300 Tipo Producto libro Colección Colección Académica PDF URL
Carlos Alberto Cadavid Moreno
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- Tabla de Contenido
- ContentsPrefaceAcknowledgments1 Introduction1.1 A Short History of Geometry1.2 What you will and will not learn in this book1.3 Audience prerequisites and style of explanation1.4 Book plan1.5 How to study this book2 Preliminaries2.1 Proof methods2.1.1 Methods for proving conditional statements2.1.2 Methods for proving other types of statements2.1.3 Symbolic representation2.1.4 More examples of proofs2.1.5 Exercises2.2 Elementary theory of sets2.2.1 Set operations2.2.2 Relations2.2.3 Equivalence relations3 Incidence geometry3.1 The notion of incidence geometry3.2 Lines and collinearity3.3 Examples of incidence geometries3.3.1 Some basic examples of incidence geometries3.3.2 The main incidence geometries3.3.3 Generalizing the real cartesian plane3.4 Parallelism3.5 Behavior of parallelism in our examples4 Betweenness4.1 Betweenness structures, segments, triangles, and convexity4.2 Separation of the plane by a line4.3 Separation of a line by one of its points4.4 Rays4.5 Angles4.6 Betweenness structure for the real cartesian plane4.7 Betweenness structure for the hyperbolic plane5 Congruence of segments5.1 Congruence of segments structure and segment comparison5.2 The usual congruence of segments structure for the real cartesian plane5.3 The usual congruence of segments structure for the hyperbolic plane6 Congruence of angles6.1 Congruence of angles structure and angle comparison6.2 Angle congruence in our main examples6.2.1 Congruence of angles in the real cartesian plane6.2.2 Congruence of angles in the hyperbolic plane7 Hilbert planes7.1 Circ1es7.2 Book 1 of The ElementsReferencesIndex