Geometry and Symmetry

Dr. L. Christine Kinsey is in the Mathematics and Statistics department at Canisius University.Teresa E. Moore is the author of Geometry and Symmetry, published by Wiley.Efstratios Prassidis is the author of Geometry and Symmetry, published by Wiley.

• Preface xiI Euclidean geometry 11 A brief history of early geometry 31.1 Prehellenistic mathematics 31.2 Greek mathematics before Euclid 51.3 Euclid 91.4 The Elements 111.5 Projects 142 Book I of Euclid’s The Elements 152.1 Preliminaries 152.2 Propositions I.5–I.26: Triangles 392.3 Propositions I.27–I.32: Parallel lines 542.4 Propositions I.33–I.46: Area 592.5 The Pythagorean Theorem 642.6 Hilbert’s axioms for euclidean geometry 712.7 Distance and geometry 752.8 Projects 783 More euclidean geometry 803.1 Circletheorems 803.2 Similarity 883.3 More triangle theorems 923.4 Inversioninacircle 1013.5 Projects 1074 Constructions 1094.1 Straightedge and compass constructions 1094.2 Trisections 1214.3 Constructions with compass alone 1244.4 Theoretical origami 1294.5 Knots and star polygons 1394.6 Linkages 1444.7 Projects 153II Noneuclidean Geometries 1555 Neutral geometry 1575.1 Viewsongeometry 1575.2 Neutralgeometry 1595.3 Alternate parallel postulates 1695.4 Projects 1766 Hyperbolic geometry 1786.1 The history of hyperbolic geometry 1786.2 Strangenewuniverse 1816.3 Models of the hyperbolic plane 1866.4 Consistency of geometries 1976.5 Asymptoticparallels 1996.6 Biangles 2036.7 Divergentparallels 2086.8 Triangles in hyperbolic space 2106.9 Projects 2187 Other geometries 2207.1 Exploring the geometry of a sphere 2207.2 Ellipticgeometry 2267.3 Comparative geometry 2397.4 Areaanddefect 2427.5 Taxicab geometry 2547.6 Finite geometries 2587.7 Projects 264III Symmetry 2658 Isometries 2678.1 Transformationgeometry 2678.2 Rosette groups 2898.3 Frieze patterns 2948.4 Wallpaper patterns 3018.5 Isometries in hyperbolic geometry 3148.6 Projects 3199 Tilings 3209.1 Tilings on the plane 3209.2 Tilings by irregular tiles 3279.3 Tilings of noneuclidean spaces 3419.4 Penrose tilings 3459.5 Projects 35510 Geometry in three dimensions 35710.1 Euclidean geometry in three dimensions 35710.2 Polyhedra 36910.3 Volume 38710.4 Infinite polyhedra 39310.5 Isometries in three dimensions 39710.6 Symmetries of polyhedra 40610.7 Four-dimensional figures 41210.8 Projects 417Appendix A Logic and proofs 419A 1 Mathematical systems 419A 2 Logic 420A 3 Structuringproofs 424A 4 Inventingproofs 427A 5 Writingproofs 428A 6 Geometric diagrams 429A 7 Using geometric software 431A 8 Van Hiele levels of geometric thought 431Appendix B Postulates and theorems 434B 1 Postulates 434B 2 Book I of Euclid’s The Elements 436B 3 Moreeuclideangeometry 439B 4 Constructions 441B 5 Neutralgeometry 442B 6 Hyperbolic geometry 443B 7 Othergeometries 445B 8 Isometries 446B 9 Tilings 448B.10 Geometry in three dimensions 448Bibliography 450Index 455