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

234 Pages
by Chapman & Hall

Also available as eBook on:

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
Transformations
Isometries and Similarities
Appendix: Proof of Surjectivity

Translations, Rotations, and Reflections
Translations
Rotations
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
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
Rosettes
Frieze Patterns
Wallpaper Patterns

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

Appendix: Hints and Answers to Selected Exercises

Bibliography

Index

### Biography

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