A Participatory Approach to Modern Geometry

The following sections are included:

• CONTENTS
• PREFACE

The following sections are included:

• Introduction
• Triangle-Circle and Triangle Grids
• Square-Circle and Square Grids
• Star Designs Based on the Triangle Grid
• Star Exploration
• The Two Great Systems of Ancient Geometry

The following sections are included:

• Introduction
• Five Proofs of the Pythagorean Theorem
• Pythagorean Triples and the Brunes Star
• Application of the Pythagorean Theorem

The following sections are included:

• Introduction
• Lines
• Lines and Calculus
• Scan Converting of a Line
• An Example of Scan Converting

The following sections are included:

• Geometric Constructions
• The Fundamental Constructions
• The World Within a Triangle
• Special Points in an Equilateral Triangle
• A Theorem about Inscribed Circles Within a Triangle
• Inscribing Regular Polygons in Circle
• The Centroid of a Polygon
• Additional Constructions

The following sections are included:

• Introduction
• The Sum of the Angles of a Triangle
• Conditions for Congruent Triangles
• Triangle Inequality
• Constructing a Triangle from Partial Information
• Two Applications of Congruent Triangles

The following sections are included:

• Introduction
• The Trigonometry of a Right Triangle
• Given the Value of One Trigonometric Function to Derive the Others
• Trigonometry of a Triangle with Angles Greater than 90 Degrees
• Periodic Functions and their Inverses
• Application of Trigonometry to Compute the Angles and Sides of a Triangle Given Partial Information About the Triangle
• Calculation of SSA
• Additional Problems to Solve Triangles with Partial Information
• An Application of Trigonometry to Finding the Radius of Earth
• Appendix 6A. Proof of the Law of Sines

The following sections are included:

• Introduction
• Three Important Theorems
• Converses
• Equivalence Tautologies and Definitions
• Proof

The following sections are included:

• Introduction
• Interior Angles of a Triangle
• The Exterior Angles of a Triangle
• A Useful Relationship Between Interior and Exterior Angles
• Interior Angles of a Regular Polygon
• Vertical Angles
• Radians and Degrees
• Angles Within a Circle
• Appendix 8A. A Theorem of Importance to Non-Euclidean Geometry

The following sections are included:

• Introduction
• Construction of Voronoi Domains
• Voronoi Domains and Pattern Recognition
• Additional Problems

The following sections are included:

• Introduction
• Euclid’s Parallel Axiom
• Parallel Lines in Euclidean Geometry
• Conditions for Two Lines Being Parallel
• The Sum of the Angles of a Triangle
• An Application of Parallel Lines to Finding the Circumference of the Earth
• Appendix 10A. The Geometry of Lines in the Poincare Plane

The following sections are included:

• Introduction
• Frameworks
• An Algorithm to Determine Rigidity of a Framework
• An Introduction to Graph Theory
• Determining Rigidity by Graph Theory
• Appendix 11A. Proof of the Graphical Method for Determining the Rigidity of Frameworks

The following sections are included:

• Similarity
• Similar Triangles
• Surgery on a Right Triangle
• The Pythagorean Theorem
• Chords, Secant Lines and Tangents to Circles
• Problems

The following sections are included:

• Introduction
• Construction of nl, a + b, and a − b
• Construction of and ab
• Line Chopper
• Doing Algebra with Geometry
• Square Roots
• Construction of a Sequence of Rectangles with Proportions

The following sections are included:

• Introduction
• Axioms of Area
• Area of a Parallelogram
• Area of a Triangle
• Area of a Polygon
• Area of a Trapezoid
• Area of an Equilateral Triangle
• Areas of a Family of Similarity Figures
• Pythagorean Theorem
• Geoboards
• Problems

The following sections are included:

• Introduction
• What is a Vector?
• The Magnitude of a Vector
• The Sum and Difference of Vectors
• Triangles, Parallelograms and Vectors
• Problems
• The Law of Cosines
• Perpendicular Vectors
• Invariance of the Length of a Vector and the Angle between Vectors Under Rotation
• Another Way to Name Vectors
• Appendix 15A. Proof of the Law of Cosines

The following sections are included:

• Introduction
• From Right Triangle to Logarithmic Spiral
• A Simple Way to Generate a Dense Set of Points on a Logarithmic Spiral
• The Law of Repetition of Ratios
• Whirling Squares
• Additional Constructions
• Appendix 16A. The Four Turtle Problem
• Appendix 16B. The Baravelle Spiral

The following sections are included:

• Introduction
• The Golden Mean and the Golden Section
• The Silver Mean
• The Golden Spiral
• A Golden Rectangle
• A Regular Pentagon
• Golden Triangles
• Application of the Golden Section to Art and Architecture
• A Regular Decagon and Regular Pentagon Within a Circle
• Sacred Cut

The following sections are included:

• Introduction
• Isometries
• Translations
• Rotations
• Reflections
• Glide Reflections
• Identity Transformation
• Compounding Isometries
• Proper and Improper Isometries
• Additional Problems
• Appendix 18A. Proof that an Isometry is Determined by the Transformation of Three Points

The following sections are included:

• Introduction
• Reflections in One Mirror
• Reflections in Two Mirrors
• Reflections in Three Mirrors
• Multiple Rotations
• Proof that Any Isometry Can be Carried Out with No More than Three Mirrors
• Three Mirrors in a Configuration of that Results in a Glide Reflection

The following sections are included:

• Introduction
• Exercises

The following sections are included:

• Groups and Symmetry
• Symmetry of a Square
• Multiplications of Symmetry Transformations
• Symmetries and Mirrors
• Symmetry Groups as an Algebraic System
• Group Theory
• Appendix 21A. Multiplication Table for the Symmetries of a Square

The following sections are included:

• Introduction
• Explanation of the Seven Frieze Patterns
• Identifying Frieze Patterns
• Additional Problems

The following sections are included:

• Introduction
• The Koch Snowflake
• A Fractal Tree
• A Moonscape
• A Cauliflower
• A Fractal Wallhanging
• Finding the “DNA” of an Iterative Function System
• Symmetry Finder
• Construction Fractals from the Iterative Function System

The following sections are included:

• Introduction
• Matrices for Rotations and Reflections in the Plane
• Transformations of Mr. Flatlands
• Homogeneous Coordinates
• Translations
• Scaling
• Fractals
• Projections
• An Introduction to Matrices

The following sections are included:

• APPENDIX A PROJECTIVE GEOMETRY CONSTRUCTIONS
  • A1. Introduction: Projective Transformations
  • A2. Pappus' Theorem
  • A3. The Dual of Pappus' Theorem
  • A4. Pascal's Theorem
  • A5. Branchion's Theorem
  • A6. Construction of a Conic Section
  • A7. Desargues Theorem
  • A8. A Hexagonal Net
  • A9. Quadrilateral Net
  • A10. Points of the Quadrilateral Net Going to Infinity I
  • A11. Points Going to Infinity II
  • A12. Pole and Polar of an Ellipse or Circle
  • A13. Inversion in a Circle
• APPENDIX B GLOSSARY
• APPENDIX C ESSAYS

The following sections are included:

• Bibliography
• Index