Geometry: Euclid and Beyond

Springer Science & Business Media, Nov 11, 2013 - Mathematics - 528 pages
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.

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Contents

 Euclids Geometry 7 Hilberts Axioms 65 Geometry over Fields 117 Congruence of Segments and Angles 140 Rigid Motions and SAS 148 NonArchimedean Geometry 158 Segment Arithmetic 165 Similar Triangles 175
 Finite Field Extensions 280 NonEuclidean Geometry 295 History of the Parallel Postulate 296 Neutral Geometry 304 Archimedean Neutral Geometry 319 NonEuclidean Area 326 Circular Inversion 334 Circles Determined by Three Conditions 346

 Introduction of Coordinates 186 Area 195 Area in Euclids Geometry 196 Measure of Area Functions 205 Dissection 212 Quadratura Circuli 221 Euclids Theory of Volume 226 Hilberts Third Problem 231 Construction Problems and Field Extensions 241 Three Famous Problems 242 The Regular 17Sided Polygon 250 Constructions with Compass and Marked Ruler 259 Cubic and Quartic Equations 270
 The Poincaré Model 355 Hyperbolic Geometry 373 Hilberts Arithmetic of Ends 388 Hyperbolic Trigonometry 403 Characterization of Hilbert Planes 415 Polyhedra 435 The Five Regular Solids 436 Eulers and Cauchys Theorems 448 Brief Euclid 481 References 495 73 499 Copyright