Exploring Geometry

Description

This text promotes student engagement with the beautiful ideas of geometry. Every major concept is introduced in its historical context and connects the idea with real-life. A system of experimentation followed by rigorous explanation and proof is central. Exploratory projects play an integral role in this text. Students develop a better sense of how to prove a result and visualize connections between statements, making these connections real. They develop the intuition needed to conjecture a theorem and devise a proof of what they have observed. Content: Geometry and the Axiomatic MethodEarly Origins of GeometryThales and PythagorasProject 1 - The Ratio Made of GoldThe Rise of the Axiomatic MethodProperties of the Axiomatic SystemsEuclid's Axiomatic GeometryProject 2 - A Concrete Axiomatic SystemEuclidean GeometryAngles, Lines, and Parallels ANGLES, LINES, AND PARALLELS 51Congruent Triangles and Pasch's AxiomProject 3 - Special Points of a TriangleMeasurement and AreaSimilar TrianglesCircle GeometryProject 4 - Circle Inversion and OrthogonalityAnalytic GeometryThe Cartesian Coordinate SystemVector GeometryProject 5 - Bezier CurvesAngles in Coordinate GeometryThe Complex PlaneBirkhoff's Axiomatic SystemConstructionsEuclidean ConstructionsProject 6 - Euclidean EggsConstructibilityTransformational GeometryEuclidean IsometriesReflectionsTranslationsRotationsProject 7 - Quilts and TransformationsGlide ReflectionsStructure and Representation of IsometriesProject 8 - Constructing CompositionsSymmetryFinite Plane Symmetry GroupsFrieze GroupsWallpaper GroupsTilting the PlaneProject 9 - Constructing TesselationsHyperbollic GeometryBackground and HistoryModels of Hyperbolic GeometryBasic Results in Hyperbolic GeometryProject 10 - The Saccheri QuadrilateralLambert Quadrilaterals and TrianglesArea in Hyperbolic GeometryProject 11 - Tilting the Hyperbolic PlaneElliptic GeometryBackground and HistoryPerpendiculars and Poles in Elliptic GeometryProject 12 - Models of Elliptic GeometryBasic Results in Elliptic GeometryTriangles and Area in Elliptic GeometryProject 13 - Elliptic TilingProjective GeometryUniversal ThemesProject 14 - Perspective and ProjectionFoundations of Projective GeometryTransformations and Pappus's TheoremModels of Projective GeometryProject 15 - Ratios and HarmonicsHarmonic SetsConics and CoordinatesFractal GeometryThe Search for a "Natural" GeometrySelf-SimilaritySimilarity DimensionProject 16 - An Endlessly Beautiful SnowflakeContraction MappingsFractal DimensionProject 17 - IFS FernsAlgorithmic GeometryGrammars and ProductionsProject 18 - Words Into PlantsAppendix A: A Primer on ProofsAppendix A A Primer on Proofs 497Appendix B Book I of Euclid's Elements Appendix C Birkhoff's Axioms Appendix D Hilbert's Axioms Appendix E Wallpaper Groups