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 7
Judith N. Cederberg. Since this model contains only two points , the negation of Axiom 4P.1 is clearly true and it is ... containing unde- fined and defined terms of the system can be proved valid or invalid , or in other words , if it is ...
Judith N. Cederberg. Since this model contains only two points , the negation of Axiom 4P.1 is clearly true and it is ... containing unde- fined and defined terms of the system can be proved valid or invalid , or in other words , if it is ...
Side 8
... axiomatic system comprehensive enough to contain the results of elementary number theory is not complete . Exercises For Exercises 1-4 , consider the following axiomatic system 8 1. Axiomatic Systems and Finite Geometries.
... axiomatic system comprehensive enough to contain the results of elementary number theory is not complete . Exercises For Exercises 1-4 , consider the following axiomatic system 8 1. Axiomatic Systems and Finite Geometries.
Side 14
... contains , ” and “ through . " The meanings of these substitute terms should be obvious by their context . Second ... containing n + 1 points , P1 , P2 , ... , Pr + 1 . Then there are P 1 P2 P n + 1 R 1 m 1 E 2 m n + 1. FIGURE 1.4 Proof P ...
... contains , ” and “ through . " The meanings of these substitute terms should be obvious by their context . Second ... containing n + 1 points , P1 , P2 , ... , Pr + 1 . Then there are P 1 P2 P n + 1 R 1 m 1 E 2 m n + 1. FIGURE 1.4 Proof P ...
Side 15
... contains neither P nor Q ( see Exercise 7 ) . By case 1 , Q is on exactly n + 1 lines m1 , m2 , ... , Mn + 1 . But each of these lines intersects m in a point R ; for i = 1 , n + 1. It can easily be shown that these points are distinct ...
... contains neither P nor Q ( see Exercise 7 ) . By case 1 , Q is on exactly n + 1 lines m1 , m2 , ... , Mn + 1 . But each of these lines intersects m in a point R ; for i = 1 , n + 1. It can easily be shown that these points are distinct ...
Side 16
... contains exactly n + 1 points , that is , n points in addition to P. Thus , the total number of points is ( n + 1 ) n + 1 = n2 + n + 1 . A dual argument verifies that the total number of lines is also n2 + n + 1 . Thus , a finite ...
... contains exactly n + 1 points , that is , n points in addition to P. Thus , the total number of points is ( n + 1 ) n + 1 = n2 + n + 1 . A dual argument verifies that the total number of lines is also n2 + n + 1 . Thus , a finite ...
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