MATHEMATICS: THE FOUR PILLARS OF GEOMETRY

By EVELYN MARTIN

CONTENTS STRAIGHTEDGE AND COMPASS EUCLID’S CONSTRUCTION AXIOMS EUCLID’S CONSTRUCTION OF THE EQUILATERAL TRIANGLE SOME BASIC CONSTRUCTIONS MULTIPLICATION AND DIVISION SIMILAR TRIANGLES DISCUSSION EUCLID’S APPROACH TO GEOMETRY THE PARALLEL AXIOM CONGRUENCE AXIOMS AREA AND EQUALITY AREA OF PARALLELOGRAMS AND TRIANGLES THE PYTHAGOREAN THEOREM PROOF OF THE THALES THEOREM ANGLES IN A CIRCLE THE PYTHAGOREAN THEOREM REVISITED COORDINATES THE NUMBER LINE AND THE NUMBER PLANE LINES AND THEIR EQUATIONS DISTANCE INTERSECTIONS OF LINES AND CIRCLES ANGLE AND SLOPE ISOMETRIES THE THREE RE?ECTIONS THEOREM VECTORS AND EUCLIDEAN SPACES VECTORS DIRECTION AND LINEAR INDEPENDENCE MIDPOINTS AND CENTROIDS THE INNER PRODUCT INNER PRODUCT AND COSINE THE TRIANGLE INEQUALITY ROTATIONS, MATRICES, AND COMPLEX NUMBERS PERSPECTIVE PERSPECTIVE DRAWING DRAWING WITH STRAIGHTEDGE ALONE PROJECTIVE PLANE AXIOMS AND THEIR MODELS HOMOGENEOUS COORDINATES PROJECTION LINEAR FRACTIONAL FUNCTIONS THE CROSS-RATIO WHAT IS SPECIAL ABOUT THE CROSS-RATIO? PROJECTIVE PLANES PAPPUS AND DESARGUES REVISITED COINCIDENCES VARIATIONS ON THE DESARGUES THEOREM PROJECTIVE ARITHMETIC THE ?ELD AXIOMS THE ASSOCIATIVE LAWS THE DISTRIBUTIVE LAW TRANSFORMATIONS THE GROUP OF ISOMETRIES OF THE PLANE VECTOR TRANSFORMATIONS TRANSFORMATIONS OF THE PROJECTIVE LINE SPHERICAL GEOMETRY THE ROTATION GROUP OF THE SPHERE REPRESENTING SPACE ROTATIONS BY QUATERNIONS A ?NITE GROUP OF SPACE ROTATIONS THE GROUPS S3 AND RP3 NON-EUCLIDEAN GEOMETRY EXTENDING THE PROJECTIVE LINE TO A PLANE COMPLEX CONJUGATION RE?ECTIONS AND M¨OBIUS TRANSFORMATIONS PRESERVING NON-EUCLIDEAN LINES PRESERVING ANGLE NON-EUCLIDEAN DISTANCE NON-EUCLIDEAN TRANSLATIONS AND ROTATIONS THREE RE?ECTIONS OR TWO INVOLUTIONS REFERENCES