A Course in Modern Geometries
Springer Science & Business Media, 9 mars 2013 - 441 sidor
A Course in Modern Geometries is designed for a junior-senior level course for mathematics majors, including those who plan to teach in secondary school. Chapter 1 presents several finite geometries in an axiomatic framework. Chapter 2 continues the synthetic approach as it introduces Euclid's geometry and ideas of non-Euclidean geometry. In Chapter 3, a new introduction to symmetry and hands-on explorations of isometries precedes the extensive analytic treatment of isometries, similarities and affinities. A new concluding section explores isometries of space. Chapter 4 presents plane projective geometry both synthetically and analytically. The extensive use of matrix representations of groups of transformations in Chapters 3-4 reinforces ideas from linear algebra and serves as excellent preparation for a course in abstract algebra. 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. Each chapter includes a list of suggested resources for applications or related topics in areas such as art and history. The second edition also includes pointers to the web location of author-developed guides for dynamic software explorations of the Poincaré model, isometries, projectivities, conics and fractals. Parallel versions of these explorations are available for "Cabri Geometry" and "Geometer's Sketchpad".
Judith N. Cederberg is an associate professor of mathematics at St. Olaf College in Minnesota.
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AABC affine transformations algebra analytic angle sum APQR assume axiomatic system axis Cantor set chaos collinear points collineation congruent Construct contains Corollary corresponding Definition determined direct isometry distance distinct points dynamic geometry software elements elliptic geometry equation equilateral triangle Euclid's Euclidean geometry Euclidean plane exactly Exercise Explain explorations FIGURE Find the matrix fractal geometry frieze group frieze pattern glide reflection H(AB homogeneous coordinates homogeneous parameters hyperbolic geometry ideal points invariant point iteration Julia set label maps Mathematics matrix representation midpoint non-Euclidean geometry Note orbits pair pencil of points pencils of lines perpendicular perspective Playfair's axiom point conic points and lines polar projective geometry Proof Let proof of Theorem properties prototile Prove Theorem real numbers result rotation Saccheri quadrilateral segment self-conjugate self-similarity sensed parallel set of points sides Sierpinski triangle similar straight lines tangent tiling translation ultraparallel unique vector verify vertices