Geometry and the Axiomatic Method
Early Origins of Geometry
Thales and Pythagoras
Project 1 - The Ratio Made of Gold
The Rise of the Axiomatic Method
Properties of the Axiomatic Systems
Euclid's Axiomatic Geometry
Project 2 - A Concrete Axiomatic System
Euclidean Geometry
Angles, Lines, and Parallels ANGLES, LINES, AND PARALLELS 51
Congruent Triangles and Pasch's Axiom
Project 3 - Special Points of a Triangle
Measurement and Area
Similar Triangles
Circle Geometry
Project 4 - Circle Inversion and Orthogonality
Analytic Geometry
The Cartesian Coordinate System
Vector Geometry
Project 5 - Bezier Curves
Angles in Coordinate Geometry
The Complex Plane
Birkhoff's Axiomatic System
Constructions
Euclidean Constructions
Project 6 - Euclidean Eggs
Constructibility
Transformational Geometry
Euclidean Isometries
Reflections
Translations
Rotations
Project 7 - Quilts and Transformations
Glide Reflections
Structure and Representation of Isometries
Project 8 - Constructing Compositions
Symmetry
Finite Plane Symmetry Groups
Frieze Groups
Wallpaper Groups
Tilting the Plane
Project 9 - Constructing Tesselations
Hyperbollic Geometry
Background and History
Models of Hyperbolic Geometry
Basic Results in Hyperbolic Geometry
Project 10 - The Saccheri Quadrilateral
Lambert Quadrilaterals and Triangles
Area in Hyperbolic Geometry
Project 11 - Tilting the Hyperbolic Plane
Elliptic Geometry
Background and History
Perpendiculars and Poles in Elliptic Geometry
Project 12 - Models of Elliptic Geometry
Basic Results in Elliptic Geometry
Triangles and Area in Elliptic Geometry
Project 13 - Elliptic Tiling
Projective Geometry
Universal Themes
Project 14 - Perspective and Projection
Foundations of Projective Geometry
Transformations and Pappus's Theorem
Models of Projective Geometry
Project 15 - Ratios and Harmonics
Harmonic Sets
Conics and Coordinates
Fractal Geometry
The Search for a "Natural" Geometry
Self-Similarity
Similarity Dimension
Project 16 - An Endlessly Beautiful Snowflake
Contraction Mappings
Fractal Dimension
Project 17 - IFS Ferns
Algorithmic Geometry
Grammars and Productions
Project 18 - Words Into Plants
Appendix A: A Primer on Proofs
Appendix A A Primer on Proofs 497
Appendix B Book I of Euclid’s Elements
Appendix C Birkhoff’s Axioms
Appendix D Hilbert’s Axioms
Appendix E Wallpaper Groups
Biography
Michael Hvidsten is Professor of Mathematics at Gustavus Adlophus College in St. Peter, Minnesota. He holds a PhD from the University of Illinois. His research interests include minimal surfaces, computer graphics and scientific visualizations, and software development. Geometry Explorer software is available free from his website.
