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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Resultat 1-5 av 64
Side v
... example , Standards 7 and 8 in the 9-12 section of the NCTM Curriculum and Evaluation Standards for School Mathematics ( 1989 ) and the comparable focus areas for grades 9-12 in the updated 1998 version recommend both synthetic and ...
... example , Standards 7 and 8 in the 9-12 section of the NCTM Curriculum and Evaluation Standards for School Mathematics ( 1989 ) and the comparable focus areas for grades 9-12 in the updated 1998 version recommend both synthetic and ...
Side xiii
... example of a formal axiomatic system and became a model for mathematical reasoning . However , the eventual discoveries of non - Euclidean ge- ometries profoundly affected both mathematical and philosophical understanding of the nature ...
... example of a formal axiomatic system and became a model for mathematical reasoning . However , the eventual discoveries of non - Euclidean ge- ometries profoundly affected both mathematical and philosophical understanding of the nature ...
Side xiv
... example of an axiomatic system and since one of the major goals of teaching geometry in high school is to expose ... examples of specific systems . These finite geometries not only demonstrate some of the concepts that oc- cur in the ...
... example of an axiomatic system and since one of the major goals of teaching geometry in high school is to expose ... examples of specific systems . These finite geometries not only demonstrate some of the concepts that oc- cur in the ...
Side 3
... example , in Euclidean geometry we substitute the term “ parallel lines " for the phrase “ lines that do not intersect . " Furthermore , it is impossible to prove all statements constructed from the defined and undefined terms of the ...
... example , in Euclidean geometry we substitute the term “ parallel lines " for the phrase “ lines that do not intersect . " Furthermore , it is impossible to prove all statements constructed from the defined and undefined terms of the ...
Side 8
... example of the four - point geometry , it is clear that models 4P.1 and 4P.2 are isomorphic . The verification that all models of this system are isomorphic follows readily once the following theorem is verified ( see Exercises 5 and 6 ) ...
... example of the four - point geometry , it is clear that models 4P.1 and 4P.2 are isomorphic . The verification that all models of this system are isomorphic follows readily once the following theorem is verified ( see Exercises 5 and 6 ) ...
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