The Geometry of Transformations

Workshop, June 14-17, 1999
Eastern Illinois University

The purpose of this workshop is to explore the geometric properties of transformations of the Euclidean plane, particularly isometries. We will examine in detail the elementary isometries -- translations, rotations, and reflections; we will develop a complete classification of all isometries; we will see how the language of isometries can be used to describe symmetry in planar patterns; we will learn how to classify, identify and create repetitive frieze (in a strip) and wallpaper (in the whole plane) patterns.

The explorations will involve extensive use of The Geometer's Sketchpad, a powerful computer package for experimenting with geometric constructions.

Outline:

1. Introduction : What are transformations and why are we interested in them? What is an isometry?
2. The Basic Isometries: A detailed look at everyone's favorite isometries -- translations, rotations, and reflections. What happens when you combine two or more of these isometries?
3. Classification and Identification: A list of all possible isometries. What does it take to determine an isometry? Using isometries to describe symmetry.
4. Frieze Patterns: How to classify, identify, and create the 7 different types of repeating strip or border patterns.
5. Wallpaper Patterns: How to classify, identify, and create the 17 different types of repeating wallpaper patterns.