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Geometry:

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Bibliographische Detailangaben
Hauptverfasser:	Lang, Serge 1927-2005 (VerfasserIn), Murrow, Gene (VerfasserIn)
Format:	Buch
Sprache:	German
Veröffentlicht:	New York [u.a.] Springer 2010
Ausgabe:	2. ed.
Schlagworte:	Geometry Geometrie Einführung Lehrbuch
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XII, 394 S. graph. Darst.
ISBN:	9781441930842

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Datensatz im Suchindex

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adam_text	CONTENTS PART I A CULTURAL HERITAGE 1 EARLY BEGINNINGS.............................................................. 3 1.1 PREHISTORY.............................................................. 3 1.2 GEOMETRY IN THE NEW STONE AGE ..................................... 4 1.3 EARLY MATHEMATICS AND ETHNOMATHEMATICS .......................... 6 2 THE GREAT RIVER CIVILIZATIONS ............................................... 7 2.1 CIVILIZATIONS LONG DEAD: AND YET ALIVE ............................ 7 2.2 BIRTH OF GEOMETRY AS WE KNOW IT ................................... 11 2.3 GEOMETRY IN THE LAND OF THE PHARAOH ................................ 12 2.4 BABYLONIAN GEOMETRY ................................................ 15 2.5 THE (U,V) EXPLANATION OF PLIMPTON 322 .............................. 22 2.6 REGULAR RECIPROCAL PAIRS, BABYLONIAN NUMBER-WORK ANDPLIMPTON322 ..................................................... 23 2.7 PARAMETRIZATION OF PYTHAGOREAN TRIPLES ............................. 25 3 GREEK AND HELLENIC GEOMETRY............................................... 31 3.1 EARLY GREEK GEOMETRY: THALES OF MILETUS ........................... 31 3.2 THE STORY OF PYTHAGORAS AND THE PYTHAGOREANS...................... 34 3.3 THE GEOMETRY OF THE PYTHAGOREANS .................................. 44 3.4 THE DISCOVERY OF IRRATIONAL NUMBERS ................................ 46 3.5 ORIGIN OF THE CLASSICAL PROBLEMS ..................................... 50 3.6 CONSTRUCTIONS BY COMPASS AND STRAIGHTEDGE ........................ 54 3.7 SQUARING THE CIRCLE ................................................... 56 3.8 DOUBLING THE CUBE .................................................... 56 3.9 TRISECTING ANY ANGLE ................................................. 60 3.10 PLATO AND THE PLATONIC SOLIDS ......................................... 61 3.11 ARCHYTAS AND DOUBLING THE CUBE..................................... 64 4 GEOMETRY IN THE HELLENISTIC ERA ............................................ 75 4.1 EUCLID AND EUCLID S ELEMENTS ........................................ 75 4.2 THE BOOKS OF EUCLID S ELEMENTS ..................................... 77 4.2.1 EUCLID S DEFINITIONS ......................................... 77 4.2.2 EUCLID S POSTULATES .......................................... 78 XIII XIV CONTENTS 4.2.3 ALTERNATIVE VERSIONS OF EUCLID S FIFTH POSTULATE ........... 80 4.2.4 EUCLID S COMMON NOTIONS OR AXIOMS ..................... 80 4.3 THE ROMAN EMPIRE ................................................... 85 4.4 ARCHIMEDES............................................................ 88 4.5 ERATOSTHENES AND DOUBLING THE CUBE ................................ 106 4.6 NICOMEDES AND HIS CONCHOID ........................................109 4.7 APOLLONIUS OF PERGA AND THE CONIC SECTIONS.........................116 4.8 THE END OF THE REPUBLIC IN ROME....................................120 4.9 THE FIRST EMPERORS ...................................................134 4.10 HERON OF ALEXANDRIA ..................................................135 4.11 NERO AND THE YEAR OF THE FOUR EMPERORS ............................138 4.12 FROM VESPASIAN TO MARCUS AURELIUS .................................139 4.13 MENELAUS OF ALEXANDRIA ..............................................143 4.14 CLAUDIUS PTOLEMY .....................................................144 4.15 THE RULE OF SINES AND THE LAW OF COSINES ..........................148 4.16 FROM COMMODUS TO THE END OF THE CRISIS OFTHETHIRDCENTURY ..................................................149 4.17 DIOPHANTUS OF ALEXANDRIA ............................................151 4.18 PAPPUS OF ALEXANDRIA .................................................153 4.19 THE LATE ROMAN EMPIRE..............................................155 4.20 THE MURDER OF HYPATIA ...............................................158 4.21 FALL OF THE ROMAN EMPIRE ............................................165 4.22 BYZANTIUM .............................................................165 4.23 PRESERVATION OF A HERITAGE ............................................167 5 ARABIC MATHEMATICS AND GEOMETRY ........................................173 5.1 THE ARAB EXPANSION ..................................................173 5.2 ARAB SCIENCE AND CULTURE ............................................177 5.3 THE FOUNDER OF THE HOUSE OF WISDOM IN BAGHDAD..................180 5.4 ABU JA FAR MUHAMMAD IBN MUSA AL KHWARIZMI ...................181 5.5 IBN QURRA AND AL-BATTANI ..............................................188 5.6 MUHAMMAD ABU AL WAFA AL-BUZJANI .................................189 5.7 ABU SAHL WIJAN BIN RUSTAM AL QUHI .................................190 5.8 YUSUF AL MUTAMAN IBN HUD AND HIS LIBRARY .........................193 5.9 OMAR AL-KHAYYAM ....................................................194 5.10 SHARAF AL-DIN ..........................................................201 5.11 NASIR AL-DIN AL-TUSI ...................................................207 6 THE GEOMETRY OF YESTERDAY AND TODAY ....................................211 6.1 THE DARK MIDDLE AGES ...............................................211 6.2 GEOMETRY REAWAKENING: A NEW DAWN IN EUROPE ..................216 6.3 ELEMENTARY GEOMETRY AND HIGHER GEOMETRY........................216 6.4 DESARGUES AND THE TWO PASCALS ......................................219 6.5 DESCARTES AND ANALYTIC GEOMETRY....................................221 6.6 NEWTON AND LEIBNIZ ..................................................222 CONTENTS XV 6.7 GEOMETRY IN THE EIGHTEENTH CENTURY .................................222 6.8 SOME FEATURES OF MODERN GEOMETRY ................................ 228 6.9 ARCHIMEDEAN POLYHEDRA AND TESSELLATIONS ..........................233 7 GEOMETRY AND THE REAL WORLD ..............................................241 7.1 MATHEMATICS AND PREDICTING CATASTROPHES ...........................241 7.2 CATASTROPHE THEORY ...................................................243 7.3 GEOMETRIC SHAPES IN NATURE ..........................................245 7.4 FRACTAL STRUCTURES IN NATURE ..........................................247 PART II INTRODUCTION TO GEOMETRY 8 AXIOMATIC GEOMETRY .........................................................253 8.1 THE POSTULATES OF EUCLID AND HILBERT S EXPLANATION .................253 8.2 NON-EUCLIDIAN GEOMETRY .............................................255 8.3 LOGIC AND INTUITIVE SET THEORY .......................................256 8.4 AXIOMS, AXIOMATIC THEORIES AND MODELS ...........................257 8.5 GENERAL THEORY OF AXIOMATIC SYSTEMS ..............................262 9 AXIOMATIC PROJECTIVE GEOMETRY .............................................265 9.1 PLANE PROJECTIVE GEOMETRY ...........................................265 9.2 AN UNSOLVED GEOMETRIC PROBLEM ....................................270 9.3 THE REAL PROJECTIVE PLANE ............................................273 10 MODELS FOR NON-EUCLIDIAN GEOMETRY .......................................283 10.1 THREE TYPES OF GEOMETRY ............................................283 10.2 HYPERBOLIC GEOMETRY .................................................283 10.3 ELLIPTIC GEOMETRY .....................................................287 10.4 EUCLIDIAN AND NON-EUCLIDIAN GEOMETRY IN SPACE ...................288 10.5 RIEMANNIAN GEOMETRY ................................................292 11 MAKING THINGS PRECISE .......................................................299 11.1 RELATIONS AND THEIR USES .............................................299 11.2 IDENTIFICATION OF POINTS................................................300 11.3 OUR NUMBER SYSTEM ..................................................302 11.4 COMPLEX NUMBERS AND TRIGONOMETRY ...............................307 12 PROJECTIVE SPACE ..............................................................313 12.1 COORDINATES IN THE PROJECTIVE PLANE ..................................313 12.2 PROJECTIVE N-SPACE ....................................................316 12.3 AFFINE AND PROJECTIVE COORDINATE SYSTEMS...........................317 XVI CONTENTS 13 GEOMETRY IN THE AFFINE AND THE PROJECTIVE PLANE .........................325 13.1 THE THEOREM OF DESARGUES ...........................................325 13.2 DUALITY FOR P 2 . R / .....................................................327 13.3 NAIVE DEFINITION AND FIRST EXAMPLES OF AFFINE PLANE CURVES ......328 13.4 STRAIGHT LINES .........................................................328 13.5 CONIC SECTIONS IN THE AFFINE PLANE R 2 ...............................329 13.6 CONSTRUCTING POINTS ON CONIC SECTIONS BYCOMPASSANDSTRAIGHTEDGE........................................ 336 13.7 FURTHER PROPERTIES OF CONIC SECTIONS.................................338 13.8 CONIC SECTIONS IN THE PROJECTIVE PLANE ...............................344 13.9 THE THEOREMS OF PAPPUS AND PASCAL .................................347 14 ALGEBRAIC CURVES OF HIGHER DEGREES IN THE AFFINE PLANE R 2 .............351 14.1 CURVES OF DEGREE 3 AND 4 IN R 2 ......................................351 14.2 AFFINE ALGEBRAIC CURVES ..............................................356 14.3 SINGULARITIES AND MULTIPLICITIES .......................................360 14.4 TANGENCY ..............................................................362 15 HIGHER GEOMETRY IN THE PROJECTIVE PLANE ..................................367 15.1 PROJECTIVE CURVES .....................................................367 15.2 PROJECTIVE CLOSURE AND AFFINE RESTRICTION ...........................368 15.3 SMOOTH AND SINGULAR POINTS ON AFFINE AND PROJECTIVE CURVES ......371 15.4 THE TANGENT TO A PROJECTIVE CURVE ...................................374 15.5 PROJECTIVE EQUIVALENCE................................................381 15.6 ASYMPTOTES ............................................................386 15.7 GENERAL CONCHOIDS ....................................................387 15.8 THE DUAL CURVE .......................................................390 15.9 THE DUAL OF PAPPUS THEOREM .......................................393 15.10 PASCAL S MYSTERIUM HEXAGRAMMICUM ...............................394 16 SHARPENING THE SWORD OF ALGEBRA ..........................................397 16.1 ON RATIONAL POLYNOMIALS .............................................397 16.2 THE MINIMAL POLYNOMIAL .............................................400 16.3 THE EUCLIDIAN ALGORITHM .............................................402 16.4 NUMBER FIELDS AND FIELD EXTENSIONS.................................404 16.5 MORE ON FIELD EXTENSIONS ............................................408 17 CONSTRUCTIONS WITH STRAIGHTEDGE AND COMPASS ...........................413 17.1 REVIEW OF LEGAL CONSTRUCTIONS .......................................413 17.2 CONSTRUCTIBLE POINTS ..................................................414 17.3 WHAT IS POSSIBLE? .....................................................415 17.4 TRISECTING ANY ANGLE .................................................420 17.5 DOUBLING THE CUBE AND CONSTRUCTING THEREGULARHEPTAGON.................................................422 17.6 SQUARING THE CIRCLE ...................................................423 CONTENTS XVII 17.7 REGULAR POLYGONS .....................................................424 17.8 CONSTRUCTIONS BY FOLDING .............................................431 18 FRACTAL GEOMETRY .............................................................435 18.1 FRACTALS AND THEIR DIMENSIONS ........................................435 18.2 THE VON KOCH SNOWFLAKE CURVE......................................436 18.3 FRACTAL SHAPES IN NATURE ..............................................437 18.4 THE SIERPINSKI TRIANGLES ..............................................437 18.5 A CANTOR SET...........................................................439 19 CATASTROPHE THEORY ..........................................................441 19.1 THE CUSP CATASTROPHE: GEOMETRY OF A CUBIC SURFACE ..............441 19.2 RUDIMENTS OF CONTROL THEORY ........................................443 20 GENERAL POLYHEDRA AND TESSELLATIONS, AND THEIR GROUPS OF SYMMETRY 445 20.1 ISOMETRIES OF R N ......................................................445 20.2 TOPOLOGICAL SPACES AND TOPOLOGICAL GROUPS ........................447 20.3 DISCRETE TRANSFORMATION GROUPS OF METRIC SPACES..................449 20.4 ISOMETRIES OF R 2 ......................................................450 20.5 SYMMETRY OF PLANE ORNAMENTS .......................................451 20.5.1 ROSETTE GROUPS ..............................................451 20.5.2 FRIEZE GROUPS................................................454 20.5.3 WALLPAPER GROUPS ...........................................458 20.5.4 THE 17 TYPES OF WALLPAPER GROUPS ........................460 20.6 SYMMETRIES IN SPACE..................................................468 20.6.1 SYSTEMS OF ROTATIONAL SYMMETRIES IN SPACE ...............469 20.6.2 REFLECTION SYMMETRY ........................................472 20.6.3 PRISMATIC SYMMETRY TYPES .................................473 20.6.4 COMPOUND SYMMETRY AND THE S 2N SYMMETRY TYPE .......475 20.6.5 CUBIC SYMMETRY TYPES .....................................475 20.6.6 THE POSSIBLE SYMMETRY TYPES..............................478 21 HINTS AND SOLUTIONS TO SOME OF THE EXERCISES .............................483 REFERENCES ...........................................................................503 INDEX .................................................................................507
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title	Geometry
title_auth	Geometry
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title_full	Geometry Serge Lang ; Gene Murrow
title_fullStr	Geometry Serge Lang ; Gene Murrow
title_full_unstemmed	Geometry Serge Lang ; Gene Murrow
title_short	Geometry
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topic	Geometry Geometrie (DE-588)4020236-7 gnd
topic_facet	Geometry Geometrie Einführung Lehrbuch
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