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. |
Αναζήτηση στο βιβλίο
Αποτελέσματα 1 - 5 από τα 88.
Σελίδα 8
... theorem is verified ( see Exercises 5 and 6 ) . Theorem 4P.1 There are exactly six lines in the four - point geometry . Finally , any discussion of the properties of axiomatic systems must include mention of the important result ...
... theorem is verified ( see Exercises 5 and 6 ) . Theorem 4P.1 There are exactly six lines in the four - point geometry . Finally , any discussion of the properties of axiomatic systems must include mention of the important result ...
Σελίδα 9
... Theorem 4P.1 will be a theorem of four - line geometry . How would its proof differ from the proof of Theorem 4P.1 in Exercise 5 ? 1.3 Finite Projective Planes As indicated by the examples in the previous section , there are ge ...
... Theorem 4P.1 will be a theorem of four - line geometry . How would its proof differ from the proof of Theorem 4P.1 in Exercise 5 ? 1.3 Finite Projective Planes As indicated by the examples in the previous section , there are ge ...
Σελίδα 12
... theorem is also a theorem is said to satisfy the principle of duality . Thus , in an axiomatic system that satisfies the principle of dual- ity , the proof of any theorem can be " turned into " a proof of a dual theorem merely by ...
... theorem is also a theorem is said to satisfy the principle of duality . Thus , in an axiomatic system that satisfies the principle of dual- ity , the proof of any theorem can be " turned into " a proof of a dual theorem merely by ...
Σελίδα 15
... theorem in hand , the following theorem follows immediately by duality . Theorem P.6 In a projective plane of order n , each line is incident with exactly n + 1 points . Using these results , we can now determine the total number of ...
... theorem in hand , the following theorem follows immediately by duality . Theorem P.6 In a projective plane of order n , each line is incident with exactly n + 1 points . Using these results , we can now determine the total number of ...
Σελίδα 16
... Theorem P.5 there are exactly n + 1 lines through P and by Theorem P.6 each of these lines 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 ...
... Theorem P.5 there are exactly n + 1 lines through P and by Theorem P.6 each of these lines 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 ...
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 |
Άλλες εκδόσεις - Προβολή όλων
Συχνά εμφανιζόμενοι όροι και φράσεις
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