Exploring Geometry

Exploring Geometry

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This text promotes student engagement with the beautiful ideas of geometry. Every major concept is introduced in its historical context and connects the idea with real-life. A system of experimentation followed by rigorous explanation and proof is central. Exploratory projects play an integral role in this text. Students develop a better sense of how to prove a result and visualize connections between statements, making these connections real. They develop the intuition needed to conjecture a theorem and devise a proof of what they have observed.
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Product details

  • Hardback | 538 pages
  • 156 x 235 x 31.75mm | 1,111g
  • Productivity Press
  • Portland, United States
  • English
  • New edition
  • 2nd New edition
  • 4 Tables, black and white; 557 Illustrations, black and white
  • 1498760805
  • 9781498760805
  • 2,061,183

Table of contents

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


Euclidean Constructions

Project 6 - Euclidean Eggs


Transformational Geometry

Euclidean Isometries




Project 7 - Quilts and Transformations

Glide Reflections

Structure and Representation of Isometries

Project 8 - Constructing Compositions


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


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
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