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 4
... determine consistency of the axiomatic system for four - point geometry . Axioms for Four - Point Geometry Undefined Terms . Point , line , on . Axiom 4P.1 . There exist exactly four points . Axiom 4P.2 . Two distinct points are on ...
... determine consistency of the axiomatic system for four - point geometry . Axioms for Four - Point Geometry Undefined Terms . Point , line , on . Axiom 4P.1 . There exist exactly four points . Axiom 4P.2 . Two distinct points are on ...
Side 12
... determine if finite projective planes exist with more points and lines , it is clearly im- practical to employ trial - and - error procedures . Instead we develop a series of theorems that lead to a general result regarding the number ...
... determine if finite projective planes exist with more points and lines , it is clearly im- practical to employ trial - and - error procedures . Instead we develop a series of theorems that lead to a general result regarding the number ...
Side 15
... determine the total number of points and lines in a projective plane of order n . Theorem P.7 A projective plane of order n contains exactly. FIGURE 1.5 Proof P.5 , P on l ... determined by binary triples . 1.3 . Finite Projective Planes 15.
... determine the total number of points and lines in a projective plane of order n . Theorem P.7 A projective plane of order n contains exactly. FIGURE 1.5 Proof P.5 , P on l ... determined by binary triples . 1.3 . Finite Projective Planes 15.
Side 22
... 000 would be those at distance 101 100 001 000 111 110 011 010 001 110 100 000 111 011 010 101 FIGURE 1.7. FIGURE 1.6 Cube determined by binary triples . FIGURE 1.8 A Desargues ' configuration . 22 1. Axiomatic Systems and Finite ...
... 000 would be those at distance 101 100 001 000 111 110 011 010 001 110 100 000 111 011 010 101 FIGURE 1.7. FIGURE 1.6 Cube determined by binary triples . FIGURE 1.8 A Desargues ' configuration . 22 1. Axiomatic Systems and Finite ...
Side 43
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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