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 4
... obtained by using interpre- tations that are objects and relations adapted from the real world , we say we have ... obtain . We illustrate the use of models to determine consistency of the axiomatic system for four - point geometry ...
... obtained by using interpre- tations that are objects and relations adapted from the real world , we say we have ... obtain . We illustrate the use of models to determine consistency of the axiomatic system for four - point geometry ...
Σελίδα 6
... obtained from S by replacing Axiom A with a negation of A. Thus , to demonstrate that a system consisting of n axioms is independent , n models must be exhibited - one for each axiom . The independence of the axiomatic system for four ...
... obtained from S by replacing Axiom A with a negation of A. Thus , to demonstrate that a system consisting of n axioms is independent , n models must be exhibited - one for each axiom . The independence of the axiomatic system for four ...
Σελίδα 9
... obtained by replacing each occurrence of the term " point " in p by the term " line " and each occurrence of the term " line " in p by the term " point . " 7. Obtain an axiomatic system for four - line geometry by dualizing the axioms ...
... obtained by replacing each occurrence of the term " point " in p by the term " line " and each occurrence of the term " line " in p by the term " point . " 7. Obtain an axiomatic system for four - line geometry by dualizing the axioms ...
Σελίδα 10
... obtained by von Staudt in 1856 , but it wasn't until early in this century that finite geometries assumed a prominent role in mathematics . Since then , the study of these geometries has grown considerably and there are still a number ...
... obtained by von Staudt in 1856 , but it wasn't until early in this century that finite geometries assumed a prominent role in mathematics . Since then , the study of these geometries has grown considerably and there are still a number ...
Σελίδα 12
... obtained by replacing each occurrence of the word " point " by the word " line " and vice versa ( consequently , the words " concurrent ” and “ collinear " must also be interchanged ) . Definition 1.5 An axiomatic system in which the ...
... obtained by replacing each occurrence of the word " point " by the word " line " and vice versa ( consequently , the words " concurrent ” and “ collinear " must also be interchanged ) . Definition 1.5 An axiomatic system in which the ...
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