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. |
Inni boken
Resultat 1-5 av 87
Side v
... result , the geometric content of several NSF school curriculum projects includes coverage of transformations and symmetry ( for ex- ample , Contemporary Mathematics in Context , Interactive Mathematics Program and Mathematics ...
... result , the geometric content of several NSF school curriculum projects includes coverage of transformations and symmetry ( for ex- ample , Contemporary Mathematics in Context , Interactive Mathematics Program and Mathematics ...
Side vi
... ( Results suggested by inductive approaches should be proved . ) • Recent developments in geometry should be included . • A wide variety of computer environments should be explored ... both as exploratory tools and for concept development ...
... ( Results suggested by inductive approaches should be proved . ) • Recent developments in geometry should be included . • A wide variety of computer environments should be explored ... both as exploratory tools and for concept development ...
Side viii
... results recently discovered by living mathematicians and scientists . Also , since fractal geometry is now used in many video and movie scenes , its study has contem- porary appeal , and can enhance understanding of the visual images ...
... results recently discovered by living mathematicians and scientists . Also , since fractal geometry is now used in many video and movie scenes , its study has contem- porary appeal , and can enhance understanding of the visual images ...
Side xi
... result of suggestions of others . In addition to those already thanked in the third printing ( David Flesner , Ockle Johnson , and Myrtle Lewin ) , I must add my thanks to Steven Hetzler , Janelle Sharoni , and John Wolfskill . Note to ...
... result of suggestions of others . In addition to those already thanked in the third printing ( David Flesner , Ockle Johnson , and Myrtle Lewin ) , I must add my thanks to Steven Hetzler , Janelle Sharoni , and John Wolfskill . Note to ...
Side xiv
... results that previously appeared to be completely abstract in nature . Since Euclid's geometry is historically the first major example of an axiomatic system and since one of the major goals of teaching geometry in high school is to ...
... results that previously appeared to be completely abstract in nature . Since Euclid's geometry is historically the first major example of an axiomatic system and since one of the major goals of teaching geometry in high school is to ...
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