Transformational Plane Geometry - Millersville University of Pennsylvania

[Pages:6]Transformational Plane Geometry

Ronald N. Umble Department of Mathematics Millersville University of Pennsylvania Millersville, PA, 17551 USA

Spring 2006

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Contents

1 Isometries

1

1.1 Transformations of the Plane . . . . . . . . . . . . . . . . . . . . 1

1.2 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Glide Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Halfturns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Properties of Translations and Halfturns . . . . . . . . . . . . . . 26

1.7 General Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Congruence

39

2.1 The Three Points Theorem . . . . . . . . . . . . . . . . . . . . . 39

2.2 Translations as Products of Reflections . . . . . . . . . . . . . . . 41

2.3 Rotations as Products of reflections . . . . . . . . . . . . . . . . . 46

2.4 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . 51

3 Classification of Isometries

59

3.1 The Angle Addition Theorem, part I . . . . . . . . . . . . . . . . 59

3.2 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 The Geometry of Conjugation . . . . . . . . . . . . . . . . . . . . 68

3.4 The Angle Addition Theorem . . . . . . . . . . . . . . . . . . . . 77

3.5 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . 79

4 Symmetry

85

4.1 Groups of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Groups of Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 The Rosette Groups . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.4 The Frieze Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.5 The Wallpaper Groups . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Similarity

113

5.1 Similarities and Dilatations . . . . . . . . . . . . . . . . . . . . . 113

5.2 Similarities as an Isometry and a Stretch . . . . . . . . . . . . . . 119

5.3 The Classification of Similarities . . . . . . . . . . . . . . . . . . 121

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Introduction

Euclidean plane geometry is the study of size and shape of objects in the plane. It is one of the oldest branches of mathematics. Indeed, by 300 BC Euclid had deductively derived the theorems of plane geometry from his five postulates. More than 2000 years later in 1628, Rene' Descartes introduced coordinates and revolutionized the discipline by using analytical tools to attack geometrical problems. To quote Descartes, "Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain lines is sufficient for its construction."

About 250 years later, in 1872, Felix Klein capitalized on Descartes' analytical approach and inaugurated his so called Erlangen Program, which views plane geometry as the study of those properties of plane figures that remain unchanged under some set of transformations. Klein's startling observation that plane geometry can be completely understood from this point of view is the guiding principle of this course and provides an alternative to Eucild's axiomatic/synthetic approach. In this course, we shall consider two such families of transformations: (1) the isometries (distance-preserving transformations), which include the translations, rotations, reflections and glide reflections and (2) the plane similarities, which include the isometries, stretches, stretch rotations and stretch reflections. Our goal is to understand congruence or similarity of plane figures in terms of some transformation in one of these two families.

Finally, a word about methodology before we begin. Geometry is a visual science, i.e., each concept should be contemplated in terms of a picture (often mental). Consequently, there will be ample opportunity throughout this course for the student to create and (quite literally) manipulate pictures that express the geometrical content of the concepts. This happens in two settings: (1) Daily homework assignments include several problems from the ancillary text Geometry: Constructions and Transformations, by Dayoub and Lott (Dale Seymour Publications, 1977 ISBN 0-86651-499-6); each construction requires a reflecting instrument such as a MIRA. (2) Biweekly laboratory assignments use the software package Geometer's Sketchpad lead the student through exploratory activities that reinforce the geometric principles presented in this text. Several of these assignments have been selected from the ancillary text Rethinking Proof with Geometer's Sketchpad by Michael de Villiers (Key Curriculum Press, 1999 ISBN 1-55953-323-4). This text complements the visualization skills gained using the MIRA and Geometer's Sketchpad by presenting

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each concept both synthetically (coordinate free) and analytically. Exercises throughout the text accommodate both points of view. The power of abstract algebra is introduced gently and slowly; prior knowledge of abstract algebra is not assumed.

April 28, 2004

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