2nd Edition

Exploring Geometry

By Michael Hvidsten Copyright 2017
    558 Pages 557 B/W Illustrations
    by CRC Press

    558 Pages 557 B/W Illustrations
    by Chapman & Hall

    558 Pages 557 B/W Illustrations
    by Chapman & Hall

    Exploring Geometry, Second Edition 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.


  • Second edition of a successful textbook for the first undergraduate course

  • Every major concept is introduced in its historical context and connects the idea with real life

  • Focuses on experimentation

  • Projects help enhance student learning

  • All major software programs can be used; free software from author
  • 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


    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.