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Modern Geometries: Non-Euclidean, Projective, and Discrete Geometry, 2nd edition

  • Michael Henle

Published by Pearson (November 13th 2019) - Copyright © 2001

2nd edition

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Overview

Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. KEY TOPICS: Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. MARKET: For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.

Table of contents



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.

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