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Modern Geometries

Modern Geometries

Non-Euclidean, Projective, and Discrete Geometry
2nd Edition

Michael Henle

Apr 2001, Hardback, 389 pages
ISBN13: 9780130323132
ISBN10: 0130323136
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£62.99
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For sophomore/senior-level courses in Geometry.

Engaging and accessible, this text describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically a non-Euclidean geometry book, it provides a brief, but solid, introduction to modern geometry using analytic methods. It relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane and building on skills already known and extensively practiced there. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout.



Dependency Chart.


Introduction.

I. BACKGROUND.

1. Some History.
2. Complex Numbers.
3. Geometric Transformations.
4. The Erlanger Program.

II. PLANE GEOMETRY.

5. Möbius Geometry.
6. Steiner Circles.
7. Hyperbolic Geometry.
8. Cycles.
9. Hyperbolic Length.
10. Area.
11. Elliptic Geometry.
12. Absolute Geometry.

III. PROJECTIVE GEOMETRY.

13. The Real Projective Plane.
14. Projective Transformations.
15. Multidimensional Projective Geometry.
16. Universal Projective Geometry.

IV. SOLID GEOMETRY.

17. Quaternions.
18. Euclidean and Pseudo-Euclidean Solid Geometry.
19. Hyperbolic and Elliptic Solid Geometry.

V. DISCRETE GEOMETRY.

20. Matroids.
21. Reflections.
22. Discrete Symmetry.
23. Non-Euclidean Symmetry.

VI. AXIOM SYSTEMS.

24. Hilbert's Axioms.
25. Bachmann's Axioms.
26. Metric Absolute Geometry.

VII. CONCLUSION.

27. The Cultural Impact of Non-Euclidean Geometry.
28. The Geometric Idea of Space.
Bibliography.
Index.

  • A modern approach based on the systematic use of transformations—Uses the complex plane and geometric transformations to present a wide variety of geometries.

    • Reflects a major theme in modern geometry. Ex.___

  • Coverage of a great variety of geometries—Both non-Euclidean and nonmetric—e.g., Möbius geometry, hyperbolic plane geometry, elliptic plane geometry, absolute geometry, and projective geometry.

    • Gives students a comprehensive understanding of geometry. Ex.___

  • Coverage of solid geometries—While most texts confine themselves to plane geometry, this text covers several different solid geometries, including multivariable projective geometry and the geometry of relativity (Pseudo-Euclidean Geometry).
  • Significant modern applications of geometry—Including the geometry of relativity, symmetry, art and crystallography, finite geometry and computation.
  • Synthetic methods—e.g., axiom systems for Euclidean and absolute geometry.

    • Demonstrates how axiomatics can clarify logical relationships among geometric concepts and among different geometries. Ex.___

  • Flexible format and organization—Parts I, II, and VII are the core of the text; while the remaining Parts are almost independent of each other. A Dependency Chart indicates the relationship of various topics.

    • Enables instructors to construct a wide variety of different courses. Material from the independent Parts can be added to a syllabus, depending on the time available, and the interests of the instructor and students. Ex.___

  • Accessible—Except for a few specific topics, only a level of mathematical sophistication of a standard course in multivariable calculus is required.

    • Makes content accessible at the undergraduate level. Ex.___

  • Questions embedded in the text.

    • Test students' immediate understanding of topics under discussion to encourages active reading of the text. Ex.___

  • Extensive problem sets—Over 500 problems, ranging from routine to difficult, many with accompanying hints and/or worked examples. Arranges problems by topic and subtopic within each chapter.

    • Provides a chance to test students' understanding. Ex.___

  • NEW - A chapter on Matroids (Ch. 20).
  • NEW - New coverage of Discrete Geometry—Includes finite projective geometry and combinational geometry.

    • Gives examples of the kinds of geometry particularly important in the contemporary computer age. Ex.___

  • Abundant illustrations—Contains over 200 line drawings plus 7 plates.

    • Illustrates connections between geometry and art. Ex.___

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