A Course in Modern GeometriesSpringer Science & Business Media, 9. mars 2013 - 441 sider 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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Side xi
... geometric explorations and fractal geometry in the courses I teach makes them more rewarding for both my students and myself . I sincerely hope that materials in this text will be rewarding for you and your students as well . Toward ...
... geometric explorations and fractal geometry in the courses I teach makes them more rewarding for both my students and myself . I sincerely hope that materials in this text will be rewarding for you and your students as well . Toward ...
Side xii
... Geometry's Future , Arlington MA : COMAP . - King , James and Schattschneider , Doris ( 1997 ) . Geometry Turned On ! Dynamic Software in Learning , Teaching , and Research , MAA Notes 41 . – NRC . ( 1989 ) . Everybody Counts , National ...
... Geometry's Future , Arlington MA : COMAP . - King , James and Schattschneider , Doris ( 1997 ) . Geometry Turned On ! Dynamic Software in Learning , Teaching , and Research , MAA Notes 41 . – NRC . ( 1989 ) . Everybody Counts , National ...
Side xiii
Judith N. Cederberg. Preface to the First Edition The origins of geometry are lost in the mists of ancient history , but geometry was already the preeminent area of Greek mathemat- ics over 20 centuries ago . As such , it became the ...
Judith N. Cederberg. Preface to the First Edition The origins of geometry are lost in the mists of ancient history , but geometry was already the preeminent area of Greek mathemat- ics over 20 centuries ago . As such , it became the ...
Side xiv
Judith N. Cederberg. graphics . The geometry of the artists , projective geometry , has be- come the tool of computer scientists and engineers as they work on the frontiers of CAD / CAM ( computer - aided design / computer - aided ...
Judith N. Cederberg. graphics . The geometry of the artists , projective geometry , has be- come the tool of computer scientists and engineers as they work on the frontiers of CAD / CAM ( computer - aided design / computer - aided ...
Side xv
... geometry and demonstrates that this geometry provides a general framework within which the geometries of Chapters 2 and 3 can be placed . Mathematically , the next logical step in this process is the study of topology , which is usually ...
... geometry and demonstrates that this geometry provides a general framework within which the geometries of Chapters 2 and 3 can be placed . Mathematically , the next logical step in this process is the study of topology , which is usually ...
Innhold
1 | |
5 | |
17 | |
Geometric Transformations of the Euclidean Plane | 99 |
4 | 116 |
6 | 128 |
7 | 135 |
13 | 175 |
Projective Geometry | 213 |
10 | 269 |
Appendices | 389 |
Geometry | 399 |
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AABC affine transformation algebra analytic angle sum APQR assume asymptotic triangles axiomatic system axis collineation congruent Construct contains Corollary corresponding Definition determined dimension direct isometry distance distinct points elements elliptic geometry equation equilateral triangle Euclid's Euclidean geometry Euclidean plane exactly Exercise fifth postulate FIGURE Find the matrix fractal frieze group frieze pattern glide reflection H(AB homogeneous coordinates homogeneous parameters hyperbolic geometry ideal points incident invariant points label maps Mathematics matrix representation midpoint non-Euclidean geometry Note P₁ pair parallel lines pencil of points pencils of lines perpendicular perspective plane of order Playfair's axiom point conic point set points and lines polar projective geometry Proof Let proof of Theorem properties prototile Prove Theorem real numbers result rotation Saccheri quadrilateral segment self-similarity sensed parallel set of points sides Sierpinski triangle similar straight lines symmetry groups tiling translation ultraparallel unique vector verify vertices