Overview of Modern Geometry

Hello

My name is David Thomas. I want to thank you for taking an interest in Modern Geometry and its Technology Resources. Audio

This presentation briefly previews the content and format of the Modern Geometry text and its Technology Resources. Audio

For whom was Modern Geometry written?

Modern Geometry was written with a particular audience in mind: Undergraduate mathematics education majors and practicing secondary school mathematics teachers. Audio

What is Modern Geometry about?

Modern Geometry provides a historically-grounded, applications-oriented, technology-rich survey of Euclidean, transformation, hyperbolic, fractal, and projective geometry. Audio

The Modern Geometry textbook presents concepts, relationships, and procedures in a systematic and deductive manner. Audio

1 Geometry Through the Ages 1

1.1 Greek Geometry Before Euclid 2

1.2 Euclid and the Elements 10

1.3 Neutral Geometry 20

1.4 Famous Open Problems in Geometry 28

2 Topics in Euclidean Geometry 45

2.1 Elementary Constructions 47

2.2 Exploring Relationships Between Objects 59

2.3 Formal Geometric Proof 73

3 Other Geometries 83

3.1 The Concept of Parallelism 86

3.2 Points, Lines, and Curves in Poincare's Disc Model 93

3.3 Polygons in Hyperbolic Space 113

3.4 Congruence in Hyperbolic Space 123

4 Transformation Geometry 129

4.1 An Analytic Model of the Euclidean Plane 130

4.2 Representing Linear Transformations in 2-space with Matrices 143

4.3 The Direct Isometries: Translations and Rotations 157

4.4 Indirect Isometries: Reflections 171

4.5 Composition and Analysis of Transformations 179

4.6 Other Linear Transformations 189

5 Fractal Geometry 203

5.1 Introduction to Self-similarity 206

5.2 Fractal Dimension 219

5.3 Iterated Function Systems 235

5.4 From Order to Chaos 245

5.5 The Mandelbrot Set 257

6 Projective Geometry 269

6.1 Elements of Perspective Drawing 269

6.2 Introduction to Projective Geometry 283

6.3 The Cross Ratio 299

6.4 Applications of the Cross Ratio 313

6.5 Matrix Methods for 3-point Perspective Transformations 319

6.6 Applications of Geometry in Remote Sensing 329

6.7 Applications of Geometry in Terrain Rendering 337

How are the Technology Resources used?

Using mathematical modeling tools such as the Geometers Sketchpad, NonEuclid, and MSW Logo, image processing tools such as Scion Image, and WWW-based information resources such as Euclid's Elements, students explore and extend concepts, skills, and relationships presented in the text. Audio

The Technology Resources include over 100 interactive activities for extending and applying concepts and procedures developed in the textbook. For instance ... Audio

How do the technology-based activities relate to the content and format of the textbook?

1. Where the textbook uses a figure to illustrate the concept of a two-point perspective view of a cube ... Audio

The Technology Resources provide an interactive Geometers Sketchpad sketch or JavaSketchpad applet to explore the range of perspectives possible in such a view. Audio

2. Where the textbook uses a figure to illustrate the concept of a fractal curve ... Audio

The Technology Resources provide an interactive MSW Logo model for generating different approximations to the curve. Audio

3. Where the textbook uses a figure to represent the features of triangles in hyperbolic space ... Audio

The Technology Resources provide an interactive model of hyperbolic space called NonEuclid in which students may create and measure triangles themselves. Audio

4. Where the textbook uses a figure to illustrate data visualization … Audio

The Technology Resources provide an interactive terrain model using dem3D. Audio

5. Where the textbook provides a historical reference for the life and achievements of the mathematician Henri Poincare ... Audio

The Technology Resources provide a link to Poincare’s biography in the MacTutor History of Mathematics Archive. Audio

Why should technology play any role in the preparation and continuing education of geometry teachers?

1. The Modern Geometry Technology Resources have been provided because of my strong belief that having convenient access to modeling tools and information resources such as these empowers students to think, act, and communicate mathematically in new and powerful ways. Audio

2. Modern Geometry was written to empower undergraduate mathematics education majors in this manner and to encourage teachers of mathematics to make similar opportunities available to their own students. Audio

3. My hope is that Modern Geometry will inspire you and your students to undertake voyages of mathematical discovery facilitated by these technologies. Bon voyage! Audio

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