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Thinking Geometrically a Survey of Geometries AMS / MAA TEXTBOOKS VOL 26 Thinking Geometrically A Survey of Geometries Thomas Q. Sibley Thinking Geometrically A Survey of Geometries c 2015 by The Mathematical Association of America (Incorporated) Library of Congress Control Number: 2015936100 Print ISBN: 978-1-93951-208-6 Electronic ISBN: 978-1-61444-619-4 Printed in the United States of America Current Printing (last digit): 10987654321 10.1090/text/026 Thinking Geometrically A Survey of Geometries Thomas Q. Sibley St. John’s University Published and distributed by The Mathematical Association of America Council on Publications and Communications Jennifer J. Quinn, Chair Committee on Books Fernando Gouvea,ˆ Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor Matthias Beck Richard E. Bedient Otto Bretscher Heather Ann Dye Charles R. Hampton Suzanne Lynne Larson John Lorch Susan F. Pustejovsky MAA TEXTBOOKS Bridge to Abstract Mathematics, Ralph W. 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Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-240-396-5647 Contents Preface xv 1 Euclidean Geometry 1 1.1 Overview and History........................................................................... 1 1.1.1 The Pythagoreans and Zeno .......................................................... 2 1.1.2 Plato and Aristotle...................................................................... 4 1.1.3 Exercises for Section 1.1.............................................................. 5 1.2 Euclid’s Approach to Geometry I: Congruence and Constructions................... 10 1.2.1 Congruence............................................................................... 11 1.2.2 Constructions............................................................................ 12 1.2.3 Equality of Measure.................................................................... 15 1.2.4 The Greek Legacy....................................................................... 16 1.2.5 Exercises for Section 1.2.............................................................. 17 1.2.6 Archimedes............................................................................... 25 1.3 Euclid’s Approach II: Parallel Lines......................................................... 26 1.3.1 Exercises for Section 1.3.............................................................. 30 1.4 Similar Figures.................................................................................... 33 1.4.1 Exercises for Section 1.4.............................................................. 37 1.5 Three-Dimensional Geometry................................................................. 42 1.5.1 Polyhedra.................................................................................. 42 1.5.2 Geodesic Domes ........................................................................ 46 1.5.3 The Geometry of the Sphere.......................................................... 48 1.5.4 Exercises for Section 1.5.............................................................. 51 1.5.5 Buckminster Fuller ..................................................................... 59 1.5.6 Projects for Chapter 1.................................................................. 59 1.5.7 Suggested Readings .................................................................... 65 2 Axiomatic Systems 67 2.1 From Euclid to Modern Axiomatics......................................................... 67 2.1.1 Overview and History.................................................................. 67 2.1.2 Axiomatic Systems..................................................................... 68 2.1.3 A Simplified Axiomatic System..................................................... 71 2.1.4 Exercises for Section 2.1.............................................................. 73 vii viii Contents 2.2 Axiomatic Systems for Euclidean Geometry.............................................. 75 2.2.1 SMSG Postulates........................................................................ 76 2.2.2 Hilbert’sAxioms........................................................................ 77 2.2.3 Exercises for Section 2.2.............................................................. 78 2.2.4 David Hilbert............................................................................. 81 2.3 Models and Metamathematics................................................................. 82 2.3.1 Exercises for Section 2.3.............................................................. 88 2.3.2 Kurt Godel.¨ ............................................................................... 94 2.3.3 Projects for Chapter 2.................................................................. 94 2.3.4 Suggested Readings .................................................................... 96 3 Analytic Geometry 97 3.1 Overview and History........................................................................... 98 3.1.1 The Analytic Model .................................................................... 98 3.1.2 Exercises for Section 3.1.............................................................. 100 3.1.3 Rene´ Descartes.......................................................................... 104 3.2 Conics and Locus Problems ................................................................... 104 3.2.1 Exercises for Section 3.2.............................................................. 110 3.2.2 Pierre de Fermat......................................................................... 114 3.3 Further Topics in Analytic Geometry........................................................ 114 3.3.1 Parametric Equations................................................................... 114 3.3.2 Polar Coordinates....................................................................... 116 3.3.3 Barycentric Coordinates............................................................... 118 3.3.4 Other Analytic Geometries............................................................ 120 3.3.5 Exercises for Section 3.3.............................................................. 121 3.4 Curves in Computer-Aided Design.......................................................... 126 3.4.1 Exercises for Section 3.4.............................................................. 131 3.5 Higher Dimensional Analytic Geometry.................................................... 133 3.5.1 Analytic Geometry in Rn .............................................................. 133 3.5.2 Regular Polytopes....................................................................... 137 3.5.3
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