A Course in Modern GeometriesSpringer Science & Business Media, 9 Μαρ 2013 - 441 σελίδες 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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Σελίδα x
... affine . " • Chapter 4 : Several exercise sets now include pointers to web - based instructions for carrying out specific exercises using dynamic geometry software . Appreciation For the additions to this second edition , I owe much to ...
... affine . " • Chapter 4 : Several exercise sets now include pointers to web - based instructions for carrying out specific exercises using dynamic geometry software . Appreciation For the additions to this second edition , I owe much to ...
Σελίδα xviii
... Affine Transformations 173 183 190 3.14 Exploring 3 - D Isometries 198 3.15 Suggestions for Further Reading . 207 4 Projective Geometry 213 4.1 Gaining Perspective 213 4.2 The Axiomatic System and Duality 214 4.3 Perspective Triangles ...
... Affine Transformations 173 183 190 3.14 Exploring 3 - D Isometries 198 3.15 Suggestions for Further Reading . 207 4 Projective Geometry 213 4.1 Gaining Perspective 213 4.2 The Axiomatic System and Duality 214 4.3 Perspective Triangles ...
Σελίδα 2
... affine geometries to the build- ing of Latin squares is equally intriguing . Since Latin squares are clearly described in several readily accessible sources , the reader is encouraged to explore this topic by consulting the resources ...
... affine geometries to the build- ing of Latin squares is equally intriguing . Since Latin squares are clearly described in several readily accessible sources , the reader is encouraged to explore this topic by consulting the resources ...
Σελίδα 17
... affine plane of order n are given below . The unde- fined terms and definitions are identical to those for a finite projective plane . Axioms for Finite Affine Planes Axiom A.1 . There exist at least four distinct points , no three of ...
... affine plane of order n are given below . The unde- fined terms and definitions are identical to those for a finite projective plane . Axioms for Finite Affine Planes Axiom A.1 . There exist at least four distinct points , no three of ...
Σελίδα 18
... affine plane does not satisfy the principle of duality . 11. Find models of affine planes of orders 2 and 3 . The following exercises ask you to prove a series of theorems about finite affine planes . You should prove these in the order ...
... affine plane does not satisfy the principle of duality . 11. Find models of affine planes of orders 2 and 3 . The following exercises ask you to prove a series of theorems about finite affine planes . You should prove these in the order ...
Περιεχόμενα
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