1st Edition

Transformational Plane Geometry

By Ronald N. Umble, Zhigang Han Copyright 2015
    234 Pages 91 B/W Illustrations
    by Chapman & Hall

    236 Pages 91 B/W Illustrations
    by Chapman & Hall

    Designed for a one-semester course at the junior undergraduate level, Transformational Plane Geometry takes a hands-on, interactive approach to teaching plane geometry. The book is self-contained, defining basic concepts from linear and abstract algebra gradually as needed.

    The text adheres to the National Council of Teachers of Mathematics Principles and Standards for School Mathematics and the Common Core State Standards Initiative Standards for Mathematical Practice. Future teachers will acquire the skills needed to effectively apply these standards in their classrooms.

    Following Felix Klein’s Erlangen Program, the book provides students in pure mathematics and students in teacher training programs with a concrete visual alternative to Euclid’s purely axiomatic approach to plane geometry. It enables geometrical visualization in three ways:

    1. Key concepts are motivated with exploratory activities using software specifically designed for performing geometrical constructions, such as Geometer’s Sketchpad.
    2. Each concept is introduced synthetically (without coordinates) and analytically (with coordinates).
    3. Exercises include numerous geometric constructions that use a reflecting instrument, such as a MIRA.

    After reviewing the essential principles of classical Euclidean geometry, the book covers general transformations of the plane with particular attention to translations, rotations, reflections, stretches, and their compositions. The authors apply these transformations to study congruence, similarity, and symmetry of plane figures and to classify the isometries and similarities of the plane.

    Axioms of Euclidean Plane Geometry
    The Existence and Incidence Postulates
    The Distance and Ruler Postulates
    The Plane Separation Postulate
    The Protractor Postulate
    The Side-Angle-Side Postulate and the Euclidean Parallel Postulate

    Theorems of Euclidean Plane Geometry
    The Exterior Angle Theorem
    Triangle Congruence Theorems
    The Alternate Interior Angles Theorem and the Angle Sum Theorem
    Similar Triangles

    Introduction to Transformations, Isometries, and Similarities
    Isometries and Similarities
    Appendix: Proof of Surjectivity

    Translations, Rotations, and Reflections
    Appendix: Geometer’s Sketchpad Commands Required by Exploratory Activities

    Compositions of Translations, Rotations, and Reflections
    The Three Points Theorem
    Rotations as Compositions of Two Reflections
    Translations as Compositions of Two Halfturns or Two Reflections
    The Angle Addition Theorem
    Glide Reflections

    Classification of Isometries
    The Fundamental Theorem and Congruence
    Classification of Isometries
    Orientation and the Isometry Recognition Problem
    The Geometry of Conjugation

    Symmetry of Plane Figures
    Groups of Isometries
    Symmetry Type
    Frieze Patterns
    Wallpaper Patterns

    Plane Similarities
    Classification of Dilatations
    Classification of Similarities and the Similarity Recognition Problem
    Conjugation and Similarity Symmetry

    Appendix: Hints and Answers to Selected Exercises




    Ronald N. Umble is a professor of mathematics at Millersville University of Pennsylvania. He has directed numerous undergraduate research projects in mathematics. He received his Ph.D. in algebraic topology under the supervision of James D. Stasheff from the University of North Carolina at Chapel Hill.

    Zhigang Han is an assistant professor of mathematics at Millersville University of Pennsylvania. He earned his Ph.D. in symplectic geometry and topology under the supervision of Dusa McDuff from Stony Brook University.

    "This book is designed for a one-semester course at the junior undergraduate level and turns especially to future educators in the USA. … The arrangement and clarity of the text meet the most demanding pedagogical and mathematical requirements. Highlights of the book are the classification of isometries and similarities of the Euclidean plane. … a wonderful first step into transformational plane geometry …"
    Zentralblatt MATH 1311