Exploring Geometry

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.

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