A Course in Modern Geometries

Framsida
Springer Science & Business Media, 23 sep. 2004 - 441 sidor
2 Recensioner
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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Axiomatic Systems and Finite Geometries
1
12 Axiomatic Systems
2
13 Finite Projective Planes
9
14 An Application to ErrorCorrecting Codes
18
15 Desargues Configurations
25
16 Suggestions for Further Reading
30
NonEuclidean Geometry
33
22 Euclids Geometry
34
Projective Geometry
213
42 The Axiomatic System and Duality
214
43 Perspective Triangles
221
44 Harmonic Sets
223
45 Perspectivities and Projectivities
229
46 Conies in the Projective Plane
240
47 An Analytic Model for the Projective Plane
250
48 The Analytic Form of Projectivities
258

23 NonEuclidean Geometry
47
24 Hyperbolic GeometrySensed Parallels
51
25 Hyperbolic GeometryAsymptotic Triangles
61
26 Hyperbolic GeometrySaccheri Quadrilaterals
68
27 Hyperbolic GeometryArea of Triangles
74
28 Hyperbolic GeometryUltraparallels
80
29 Elliptic Geometry
84
210 Significance of the Discovery of NonEuclidean Geometries
93
Geometric Transformations of the Euclidean Plane
99
32 Exploring Line and Point Reflections
103
33 Exploring Rotations and Finite Symmetry Groups
108
34 Exploring Translations and Frieze Pattern Symmetries
116
35 An Analytic Model of the Euclidean Plane
121
36 Transformations of the Euclidean Plane
129
37 Isometrics
136
38 Direct Isometries
144
39 Indirect Isometries
154
310 Frieze and Wallpaper Patterns
165
311 Exploring Plane Tilings
173
312 Similarity Transformations
183
313 Affine Transformations
190
314 Exploring 3D Isometries
198
315 Suggestions for Further Reading
207
49 Cross Ratios
264
410 Collineations
270
411 Correlations and Polarities
283
412 Subgeometries of Projective Geometry
298
413 Suggestions for Further Reading
311
Chaos to Symmetry An Introduction to Fractal Geometry
315
51 A Chaotic Background
316
52 Need for a New Geometric Language
334
53 Fractal Dimension
347
54 Iterated Function Systems
360
55 Finally What Is a Fractal?
377
56 Applications of Fractal Geometry
380
57 Suggestions for Further Reading
382
Euclids Definitions Postulates and the First 30 Propositions of Elements Book I
389
Hilberts Axioms for Plane Geometry
395
Birkhoffs Postulates for Euclidean Plane Geometry
399
The SMSG Postulates for Euclidean Geometry
401
Some SMSG Definitions for Euclidean Geometry
405
The ASA Theorem
409
References
413
Index
427
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