Geometry: Euclid and BeyondIn 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. |
Contents
Introduction | 1 |
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 |
Cubic and Quartic Equations | 270 |
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 |
Similar Triangles | 175 |
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 |
Circles Determined by Three Conditions | 346 |
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 |
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Common terms and phrases
algebraic altitudes angle bisectors Archimedes Cartesian plane circle with center circular inversion congruent convex cross-ratio cube define definition dihedral angles dissection draw equal content equation equidecomposable equilateral triangle equivalent Euclid's Elements Euclidean plane example Exercise exists field extension field F figure finite number follows Galois group given line Hence Hilbert plane Hilbert's axioms hyperbolic plane icosahedron inscribed intersection isomorphic isosceles lemma Let ABC limiting parallel line segments marked ruler meet midpoint non-Euclidean geometry notion ordered field parallel axiom parallel postulate perpendicular Poincaré model polygon polyhedra polyhedron polynomial problem proof Proposition prove radius real Cartesian plane real numbers rectangle result right angles right triangle rigid motion rotation ruler and compass Saccheri quadrilateral satisfying Section sides splitting field square roots steps subgroup suppose symmetry tangent tetrahedron theorem theory of area triangle ABC unique vertex vertices