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A Course in Modern Geometries: Undergraduate Texts in Mathematics(2nd,2001)
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Judith N. Cederberg
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Springer-Verlag
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second edition
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442
ISBN
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0387989722
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Designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. The first chapter presents several finite geometries in an axiomatic framework, while Chapter 2 continues the synthetic approach in introducing both Euclids and ideas of non-Euclidean geometry. There follows a new introduction to symmetry and hands-on explorations of isometries that precedes an extensive analytic treatment of similarities and affinities. Chapter 4 presents plane projective geometry both synthetically and analytically, and the new Chapter 5 uses a descriptive and exploratory approach to introduce chaos theory and fractal geometry, stressing the self-similarity of fractals and their generation by transformations from Chapter 3. Throughout, each chapter includes a list of suggested resources for applications or related topics in areas such as art and history, plus this second edition points to Web locations of author-developed guides for dynamic software explorations of the Poincar� model, isometries, projectivities, conics and fractals. Parallel versions are available for "Cabri Geometry" and "Geometers Sketchpad".
1. Axiomatic Systems and Finite Geometries
2. Non-Euclidean Geometry
3. Geometric Transformations of the Euclidean Plane
4. Projective Geometry
5. Chaos to Symmetry: An Introduction to Fractal Geometry
Appendices
"I studied Dr. Cederberg's text a while ago while I was at St. Olaf College. Being an ambitious youth, I was always trying to seek out the "best" book in a field to study. However, it's certainly difficult to learn from the masters if one doesn't have a solid background in the basic materials. I learned Calculus from G. Hardy's "Pure Math" but found it extremely difficult to comprehend (though it was a rewarding try). Then I turned to Spivak for a more modern treatment. In geometry, I went the opposite way: studying Cederberg's book first before moving to the more advanced one. I like her clear presentation and especially the part on matrix representations of groups of transformations. This book would be a valuable source for teachers of geometry."
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