## Geometry: Our Cultural HeritageThis book is based on lectures on geometry at the University of Bergen, Norway. Over the years these lectures have covered many different aspects and facets ofthis wonderful field.Consequently it has ofcourse never been possible to give a full and final account ofgeometry as such, at an undergraduate level: A carefully considered selection has always been necessary.The present book constitutes the main central themes of these selections. One of the groups I am aiming at, is future teachers of mathematics. All too often the geometry which goes into the syllabus for teacher-students present the material as pedantic and formalistic, suppressing the very pow erful and dynamic character of this old - and yet so young! - field. A field of mathematical insight, research, history and source of artistic inspiration. And not least important, a foundation for our common cultural heritage. Another motivation is to provide an invitation to mathematics in gen eral. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics is more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. An im portant task is seriously attempting to heal these ills. Ills perhaps inflicted on students at an early age, through deficient or even harmful teaching prac tices. Thus the book also aims at an informed public, interested in making a new beginning in math. And in doing so, learning more about this part of our cultural heritage. |

### Contents

II | 3 |

III | 4 |

IV | 7 |

VI | 11 |

VII | 12 |

VIII | 15 |

IX | 27 |

X | 29 |

LVII | 201 |

LVIII | 207 |

LIX | 211 |

LX | 212 |

LXI | 214 |

LXII | 221 |

LXIII | 222 |

LXIV | 224 |

XI | 38 |

XII | 40 |

XIII | 45 |

XIV | 48 |

XV | 50 |

XVI | 51 |

XVII | 54 |

XVIII | 56 |

XIX | 59 |

XX | 67 |

XXI | 69 |

XXII | 76 |

XXIII | 78 |

XXIV | 93 |

XXV | 95 |

XXVI | 102 |

XXVII | 106 |

XXVIII | 111 |

XXIX | 113 |

XXX | 115 |

XXXI | 119 |

XXXII | 120 |

XXXIII | 126 |

XXXIV | 127 |

XXXV | 131 |

XXXVI | 134 |

XXXVII | 135 |

XXXVIII | 138 |

XXXIX | 139 |

XL | 140 |

XLI | 144 |

XLII | 151 |

XLIII | 153 |

XLIV | 156 |

XLV | 159 |

XLVI | 165 |

XLVII | 167 |

XLVIII | 168 |

XLIX | 170 |

L | 171 |

LI | 176 |

LII | 177 |

LIII | 182 |

LIV | 185 |

LV | 195 |

LVI | 200 |

LXV | 225 |

LXVI | 233 |

LXVIII | 235 |

LXIX | 236 |

LXXI | 237 |

LXXII | 245 |

LXXIII | 248 |

LXXIV | 254 |

LXXV | 257 |

LXXVI | 261 |

LXXVII | 269 |

LXXVIII | 270 |

LXXIX | 274 |

LXXX | 277 |

LXXXI | 281 |

LXXXII | 282 |

LXXXIII | 285 |

LXXXIV | 287 |

LXXXV | 296 |

LXXXVI | 300 |

LXXXVII | 301 |

LXXXVIII | 304 |

LXXXIX | 307 |

XC | 308 |

XCI | 311 |

XCII | 314 |

XCIII | 316 |

XCIV | 318 |

XCV | 322 |

XCVI | 329 |

XCVII | 330 |

XCVIII | 331 |

XCIX | 338 |

C | 339 |

CI | 340 |

CII | 348 |

CIII | 353 |

CIV | 354 |

CV | 355 |

CVI | 356 |

CVII | 357 |

CVIII | 359 |

CIX | 360 |

363 | |

365 | |

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### Common terms and phrases

affine curve affine restriction Alexandria algebraic Archimedes Archytas assume axiomatic axioms Babylonians Caesar Cambyses circle coefficients compass and straightedge compute conchoid cone conic section construction coordinate system cube cubic cubical parabola cylinder defined definition degree denote distance Doubling the Cube elements ellipse equal equation equivalence relation Euclid's Euclid's Elements Euclidian plane Euclidian tools F(Xo formula geometry given Greek geometers hyperbola integers line segments mathematician mathematics metric minimal polynomial multiplicity namely natural numbers non-degenerate conic section number field origin Pē(R parabola point of intersection points at infinity polynomial problem projective closure projective curve projective geometry projective plane proof Proposition prove Pythagoras Pythagorean triple Pythagoreans radius real numbers Roman shown in Figure singular point sphere square straight line straightedge and compass surface tangent Thales Theorem theory triangle Trisecting x-axis yields zero