Geometry: Euclid and Beyond

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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
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