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 vii
... explorations The explorations offer opportunities for collaborative construction of knowledge through activities that emphasize visualization . They are designed to lead to student discovery before formal presentation of concepts , or ...
... explorations The explorations offer opportunities for collaborative construction of knowledge through activities that emphasize visualization . They are designed to lead to student discovery before formal presentation of concepts , or ...
Side viii
... explorations is given in the table below . Geometric Explorations Title Location Exploring Dynamic Geometry Software ( DGS ) Web Exploring the Hyperbolic Plane with DGS Web Exploring the Double Elliptic Plane 2.9 Exploring Line and ...
... explorations is given in the table below . Geometric Explorations Title Location Exploring Dynamic Geometry Software ( DGS ) Web Exploring the Hyperbolic Plane with DGS Web Exploring the Double Elliptic Plane 2.9 Exploring Line and ...
Side ix
... explorations in Chapters 2-5 which contain specific directions for using either of these programs . • Chapter 2 : ( 1 ) Parallel web - based explorations of the Poincaré model using Cabri Geometry II and Geometer's Sketchpad enable ...
... explorations in Chapters 2-5 which contain specific directions for using either of these programs . • Chapter 2 : ( 1 ) Parallel web - based explorations of the Poincaré model using Cabri Geometry II and Geometer's Sketchpad enable ...
Side xi
... Explorations : The explorations are intended to supplement and enhance student understanding ( but a course based on the text can be taught without these sections ) . They can be used ei- ther in scheduled classes / labs or as ...
... Explorations : The explorations are intended to supplement and enhance student understanding ( but a course based on the text can be taught without these sections ) . They can be used ei- ther in scheduled classes / labs or as ...
Side xii
... explorations and a list of other suggested resources : http://www.stolaf.edu/people/cederj/geotext/info.htm . Judith N. Cederberg cederj@stolaf.edu http://www.stolaf.edu/people/cederj/ Preface to the First Edition The origins of ...
... explorations and a list of other suggested resources : http://www.stolaf.edu/people/cederj/geotext/info.htm . Judith N. Cederberg cederj@stolaf.edu http://www.stolaf.edu/people/cederj/ Preface to the First Edition The origins of ...
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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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Vanlige uttrykk og setninger
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