The Foundations of Geometry and the Non-Euclidean PlaneSpringer Science & Business Media, 19. des. 1997 - 512 sider This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary. |
Inni boken
Resultat 1-5 av 83
Side xxii
... definition from the theory , the Index will help you find the definition . The author hopes that you enjoy your study of the theory of parallels . INTRODUCTION The Introduction contains the prerequisites to our study of XVI FOREWORD TO ...
... definition from the theory , the Index will help you find the definition . The author hopes that you enjoy your study of the theory of parallels . INTRODUCTION The Introduction contains the prerequisites to our study of XVI FOREWORD TO ...
Side 7
Beklager, innholdet på denne siden er tilgangsbegrenset..
Beklager, innholdet på denne siden er tilgangsbegrenset..
Side 13
Beklager, innholdet på denne siden er tilgangsbegrenset..
Beklager, innholdet på denne siden er tilgangsbegrenset..
Side 15
Beklager, innholdet på denne siden er tilgangsbegrenset..
Beklager, innholdet på denne siden er tilgangsbegrenset..
Side 16
Beklager, innholdet på denne siden er tilgangsbegrenset..
Beklager, innholdet på denne siden er tilgangsbegrenset..
Innhold
Equivalence Relations 11 LOGIC | 2 |
12 SETS | 4 |
13 RELATIONS | 5 |
14 EXERCISES | 8 |
GRAFFITI | 9 |
Mappings 21 ONETOONE AND ONTO | 10 |
22 COMPOSITION OF MAPPINGS | 15 |
23 EXERCISES | 17 |
Reflections 191 INTRODUCING ISOMETRIES | 216 |
192 REFLECTION IN A LINE | 219 |
193 EXERCISES | 223 |
GRAFFITI | 225 |
Circles 201 INTRODUCING CIRCLES | 226 |
202 THE TWOCIRCLE THEOREM | 230 |
203 EXERCISES | 236 |
GRAFFITI | 238 |
GRAFFITI | 19 |
The Real Numbers 31 BINARY OPERATIONS | 20 |
32 PROPERTIES OF THE REALS | 26 |
33 EXERCISES | 31 |
GRAFFITI | 33 |
Axiom Systems 41 AXIOM SYSTEMS | 34 |
42 INCIDENCE PLANES | 36 |
43 EXERCISES | 45 |
GRAFFITI | 47 |
Models 51 MODELS OF THE EUCLIDEAN PLANE | 50 |
52 MODELS OF INCIDENCE PLANES | 55 |
53 EXERClSES | 61 |
GRAFFITI | 64 |
Incidence Axiom and Ruler Postulate 61 OUR OBJECTIVES | 65 |
THE INCIDENCE AXIOM | 66 |
THE RULER POSTULATE | 68 |
64 EXERCISES | 70 |
GRAFFITI | 72 |
Betweenness 71 ORDERING THE POINTS ON A LINE | 73 |
72 TAXICAB GEOMETRY | 77 |
73 EXERCISES | 81 |
GRAFFITI | 82 |
Segments Rays and Convex Sets 81 SEGMENTS AND RAYS | 84 |
82 CONVEX SETS | 89 |
83 EXERCISES | 92 |
GRAFFITI | 93 |
Angles and Triangles 91 ANGLES AND TRIANGLES | 95 |
92 MORE MODELS | 100 |
93 EXERCISES | 109 |
GRAFFITI | 110 |
The Golden Age of Greek Mathematics 101 ALEXANDRIA | 111 |
102 EXERCISES | 119 |
Euclids Elements 111 THE ELEMENTS | 121 |
112 EXERCISES | 129 |
GRAFFITI | 130 |
Paschs Postulate and Plane Separation Postulate 121 AXIOM 3 PSP | 131 |
122 PASCH PEANO PIERI AND HILBERT | 137 |
123 EXERCISES | 140 |
GRAFFITI | 142 |
Crossbar and Quadrilaterals 131 MORE INCIDENCE THEOREMS | 144 |
132 QUADRILATERALS | 149 |
133 EXERCISES | 152 |
GRAFFITI | 153 |
Measuring Angles and the Protractor Postulate 141 AXIOM 4 THE PROTRACTOR POSTULATE | 155 |
142 PECULIAR PROTRACTORS | 166 |
143 EXERCISES | 169 |
Alternative Axiom Systems 151 HILBERTS AXIOMS | 172 |
152 PIERIS POSTULATES | 175 |
153 EXERCISES | 180 |
Mirrors 161 RULERS AND PROTRACTORS | 182 |
162 MIRROR AND SAS | 184 |
163 EXERCISES | 189 |
GRAFFITI | 191 |
Congruence and the Penultimate Postulate 171 CONGRUENCE FOR TRIANGLES | 192 |
SAS | 195 |
173 CONGRUENCE THEOREMS | 198 |
174 EXERCISES | 201 |
GRAFFITI | 202 |
Perpendiculars and Inequalities 181 A THEOREM ON PARALLELS | 204 |
182 INEQUALITIES | 207 |
183 RIGHT TRIANGLES | 211 |
184 EXERCISES | 213 |
GRAFFITI | 215 |
Absolute Geometry and Saccheri Quadrilaterals 211 EUCLIDS ABSOLUTE GEOMETRY | 239 |
212 GIORDANOS THEOREM | 248 |
213 EXERCISES | 252 |
GRAFFITI | 253 |
Saccheris Three Hypotheses 221 OMAR KHAYYAMS THEOREM | 255 |
222 SACCHERIS THEOREM | 260 |
223 EXERCISES | 266 |
GRAFFITI | 267 |
Euclids Parallel Postulate 231 EQUIVALENT STATEMENTS | 269 |
232 INDEPENDENCE | 281 |
233 EXERCISES | 286 |
GRAFFITI | 289 |
Biangles 241 CLOSED BIANGLES | 292 |
242 CRITICAL ANGLES AND ABSOLUTE LENGTHS | 295 |
243 THE INVENTION OF NONEUCLIDEAN GEOMETRY | 302 |
244 EXERCISES | 314 |
GRAFFITI | 316 |
Excursions 251 PROSPECTUS | 317 |
252 EUCLIDEAN GEOMETRY | 320 |
253 HIGHER DIMENSIONS | 323 |
254 EXERCISES | 328 |
GRAFFITI | 330 |
Parallels and the Ultimate Axiom 261 AXIOM 6 HPP | 334 |
262 PARALLEL LINES | 338 |
263 EXERCISES | 344 |
GRAFFITI | 346 |
Brushes and Cycles 271 BRUSHES | 347 |
272 CYCLES | 351 |
273 EXERCISES | 356 |
GRAFFITI | 358 |
Rotations Translations and Horolations 281 PRODUCTS OF TWO REFLECTIONS | 360 |
282 REFLECTIONS IN LINES OF A BRUSH | 365 |
283 EXERCISES | 368 |
GRAFFITI | 370 |
The Classification of Isometries 291 INVOLUTIONS | 371 |
292 THE CLASSIFICATION THEOREM | 378 |
293 EXERCISES | 382 |
GRAFFITI | 384 |
Symmetry 301 LEONARDOS THEOREM | 386 |
302 FRIEZE PATTERNS | 392 |
303 EXERCISES | 397 |
GRAFFITI | 400 |
Horocircles 311 LENGTH OF ARC | 402 |
312 HYPERBOLIC FUNCTIONS | 415 |
313 EXERCISES | 417 |
GRAFFITI | 419 |
The Fundamental Formula 321 TRIGONOMETRY | 421 |
322 COMPLEMENTARY SEGMENTS | 434 |
323 EXERCISES | 439 |
GRAFFITI | 443 |
Categoricalness and Area 331 ANALYTIC GEOMETRY | 444 |
332 AREA | 450 |
333 EXERCISES | 459 |
GRAFFITI | 463 |
Quadrature of the Circle 341 CLASSICAL THEOREMS | 464 |
342 CALCULUS | 474 |
343 CONSTRUCTIONS | 479 |
344 EXERCISES | 490 |
Hints and Answers | 494 |
503 | |
504 | |
Andre utgaver - Vis alle
The Foundations of Geometry and the Non-Euclidean Plane G.E. Martin Ingen forhåndsvisning tilgjengelig - 2011 |
Vanlige uttrykk og setninger
AABC ABCD absolute geometry Acute Angle ADEF angle measure axiom system Beltrami coordinates biangle bijection Bolyai Bolyai-Lobachevsky plane brush Cartesian plane circle with center collinear congruent construction contains convex set Corollary cosh defined definition distance scale distinct points elements equation equidistant equivalence relation Euclid's Parallel Postulate Euclid's Proposition Euclidean geometry Euclidean plane exactly Exercise Figure fixes follows Gauss glide reflection GRAFFITI H₂ halfplane halfturn Hence horocircle horoparallel horopencil hyperbolic hyperbolic geometry hyperparallel Hypothesis implies Incidence Axiom Incidence Plane integer interior angles isometry isomorphic LAVB line intersects Lobachevsky mapping mathematics midpoint non-Euclidean geometry obtuse Parallel Postulate perpendicular bisector polygonal region Proof Let Proof Suppose prove radius real numbers right angle rotation Ruler Postulate SABCD Saccheri quadrilateral segment set of points sinh statement straight line symmetry tanh Taxicab Geometry three-space tion unique line vertex