Exploring Geometry

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