## Modern Geometries: The Analytic Approach, 1/e

Michael G. Henle, Oberlin College

Published September, 1996 by Prentice Hall Engineering/Science/Mathematics

Copyright 1997, 372 pp.
Cloth
ISBN 0-13-193418-X

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Geometry-Junior Level-Mathematics

Basically a non-Euclidean geometry book, this text updates the teaching of college geometry based upon three fundamental ideas: (A) geometries only approximate reality, (B) the best presentation of a geometry is by transformations and transformation groups, and (C) points and other geometric objects should be coordinatized. Modern Geometries is engaging, accessible, and describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists.
Approaches subject matter from a modern point of view, yet includes coverage of many traditional topics.
Covers several non Euclidean geometries in great detail.
Integrates the ideas of Felix Klein throughout the text, making systematic use of Klein's Erlanger Program (which is accepted as the right way to view geometry by research mathematicians and scientists).
Uses coordinates throughout the text—not only real numbers but also complex numbers (and other number systems).
Presents projective geometry, and includes a treatment of Hilbert's axioms for Euclidean geometry.
Effectively integrates over 200 line drawings plus 7 plates illustrating connections between geometry and art.
Features over 500 problems ranging from routine to difficult, many with accompanying hints and/or worked examples.
Includes a series of questions embedded in the text that test immediate understanding of topics under discussion to encourage active reading of the text.
Offers a flexible format and organization suitable for a wide variety of geometry courses.

I. BACKGROUND.

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

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.
l5. 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. EXTRAS.
20. Reflections.
21. Discrete Symmetry.
22. Non-Euclidean Symmetry.
23. Non-Euclidean Coordinate Systems.
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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