Geometry With Geometry Explorer - With CD

1 Geometry and the Axiomatic Method

1.1 Early Origins of Geometry

1.2 Thales and Pythagoras

1.2.1 Thales

1.2.2 Pythagoras

1.3 Project 1 - The Ratio Made of Gold

1.3.1 Golden Section

1.3.2 Golden Rectangles

1.4 The Rise of the Axiomatic Method

1.5 Properties of Axiomatic Systems

1.5.1 Consistency

1.5.2 Independence

1.5.3 Completeness

1.5.4 Gödel's Incompleteness Theorem

1.6 Euclid's Axiomatic Geometry

1.6.1 Euclid's Postulates

1.7 Project 2 - A Concrete Axiomatic System

2 Euclidean Geometry

2.1 Angles, Lines, and Parallels

2.2 Congruent Triangles and Pasch's Axiom

2.3 Project 3 - Special Points of a Triangle

2.3.1 Circumcenter

2.3.2 Orthocenter

2.3.3 Incenter

2.4 Measurement and Area

2.4.1 Mini-Project - Area in Euclidean Geometry

2.4.2 Cevians and Areas

2.5 Similar Triangles

2.5.1 Mini-Project - Finding Heights

2.6 Circle Geometry

2.7 Project 4 - Circle Inversion

2.7.1 Orthogonal Circles Redux

3 Analytic Geometry

3.1 The Cartesian Coordinate System

3.2 Vector Geometry

3.3 Project 5 - Bézier Curves

3.4 Angles in Coordinate Geometry

3.5 The Complex Plane

3.5.1 Polar Form

3.5.2 Complex Functions

3.5.3 Analytic Functions and Conformal Maps (Optional)

3.6 Birkhoff's Axiomatic System

4 Constructions

4.1 Euclidean Constructions

4.2 Project 6 - Euclidean Eggs

4.3 Constructibility

4.4 Mini-Project - Origami Construction

5 Transformational Geometry

5.1 Euclidean Isometries

5.2 Reflections

5.2.1 Mini-Project - Isometries through Reflection

5.2.2 Reflection and Symmetry

5.3 Translations

5.3.1 Translational Symmetry

5.4 Rotations

5.4.1 Rotational Symmetry

5.5 Project 7 - Quilts andTransformations

5.6 Glide Reflections

5.6.1 Glide Reflection Symmetry

5.7 Structure and Representation of Isometries

5.7.1 Matrix Form of Isometries

5.7.2 Compositions of Rotations and Translations

5.7.3 Compositions of Reflections and Glide Reflections

5.7.4 Isometries in Computer Graphics

5.7.5 Summary of Isometry Compositions

5.8 Project 8 - Constructing Compositions

6 Symmetry

6.1 Finite Plane Symmetry Groups

6.2 Frieze Groups

6.3 Wallpaper Groups

6.4 Tiling the Plane

6.4.1 Escher

6.4.2 Regular Tessellations of the Plane

6.5 Project 9 - Constructing Tessellations

7 Non-Euclidean Geometry

7.1 Background and History

7.2 Models of Hyperbolic Geometry

7.2.1 Poincaré Model

7.2.2 Mini-Project - The Klein Model

7.3 Basic Results in Hyperbolic Geometry

7.3.1 Parallels in Hyperbolic Geometry

7.3.2 Omega Points and Triangles

7.4 Project 10 - The Saccheri Quadrilateral

7.5 Lambert Quadrilaterals and Triangles

7.5.1 Lambert Quadrilaterals

7.5.2 Triangles in Hyperbolic Geometry

7.6 Area in Hyperbolic Geometry

7.7 Project 11 - Tiling the Hyperbolic Plane

7.8 Models and Isomorphism

8 Non-Euclidean Transformations

8.1 Möbius Transformations

8.1.1 Fixed Points and the Cross Ratio

8.1.2 Geometric Properties of Möbius Transformations

8.2 Isometries in the Poincaré Model

8.3 Isometries in the Klein Model

8.4 Mini-Project - The Upper Half-Plane Model

8.5 Weierstrass Model

8.6 Hyperbolic Calculation

8.6.1 Arclength of Parameterized Curves

8.6.2 Geodesics

8.6.3 The Angle of Parallelism

8.6.4 Right Triangles

8.6.5 Area

8.7 Project 12 - Infinite Real Estate?

9 Fractal Geometry

9.1 The Search for a "Natural" Geometry

9.2 Self-Similarity

9.2.1 Sierpinski's Triangle

9.2.2 Cantor Set

9.3 Similarity Dimension

9.4 Project 13 - An Endlessly Beautiful Snowflake

9.5 Contraction Mappings

9.6 Fractal Dimension

9.7 Project 14 - IFS Ferns

9.8 Algorithmic Geometry

9.8.1 Turtle Geometry

9.9 Grammars and Productions

9.9.1 Space-filling Curves

9.10 Project 15 - Words into Plants

A Book I of Euclid's Elements

A.1 Definitions

A.2 The Postulates (Axioms)

A.3 Common Notions

A.4 Propositions (Theorems)

B Brief Guide to Geometry Explorer

B.1 The Main Geometry Explorer Window

B.2 Selecting Objects

B.3 Active vs. Inactive Tools

B.4 Labels

B.5 Object Coloring

B.6 Online Help

B.7 Undo/Redo of Actions

B.8 Clearing and Resizing the Canvas

B.9 Saving Files as Images

B.10 Main Window Button Panels

B.10.1 Create Panel

B.10.2 Construct Panel

B.10.3 Transform Panel

B.11 Measurement in Geometry Explorer

B.11.1 Neutral Measurements

B.11.2 Euclidean-only Measurements

B.11.3 Hyperbolic-only Measurements

B.11.4 User Input Measurements

B.12 Using Tables

B.13 Using the Calculator

B.14 Hyperbolic Geometry

B.15 Analytic Geometry

B.16 Turtle Geometry

C Birkhoff's Axioms

D Hilbert's Axioms

E The 17 Wallpaper Groups