A Simple Non-Euclidean Geometry and Its Physical Basis

I. M. Yaglom

ISBN 10: 0387903321 / ISBN 13: 9780387903323

Editore: Springer New York, Springer New York Feb 1979, 1979

Riguardo questo articolo

Neuware -There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the Bibliography. This geometry, also called hyperbolic geometry, is part of the required subject matter of many mathematics departments in universities and teachers' colleges-a reflec tion of the view that familiarity with the elements of hyperbolic geometry is a useful part of the background of future high school teachers. Much attention is paid to hyperbolic geometry by school mathematics clubs. Some mathematicians and educators concerned with reform of the high school curriculum believe that the required part of the curriculum should include elements of hyperbolic geometry, and that the optional part of the curriculum should include a topic related to hyperbolic geometry. I The broad interest in hyperbolic geometry is not surprising. This interest has little to do with mathematical and scientific applications of hyperbolic geometry, since the applications (for instance, in the theory of automorphic functions) are rather specialized, and are likely to be encountered by very few of the many students who conscientiously study (and then present to examiners) the definition of parallels in hyperbolic geometry and the special features of configurations of lines in the hyperbolic plane. The principal reason for the interest in hyperbolic geometry is the important fact of 'non-uniqueness' of geometry; of the existence of many geometric systems.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 332 pp. Englisch.

Codice articolo 9780387903323

Dati bibliografici

• Titolo: A Simple Non-Euclidean Geometry and Its Physical Basis
• Editore: Springer New York, Springer New York Feb 1979
• Data di pubblicazione: 1979
• Lingua: Inglese
• ISBN 10: 0387903321
• ISBN 13: 9780387903323
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 19 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 19 millimetri di profondità
• Peso: 505 grammi

Descrizione articolo

Riassunto

There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the Bibliography. This geometry, also called hyperbolic geometry, is part of the required subject matter of many mathematics departments in universities and teachers' colleges-a reflec­ tion of the view that familiarity with the elements of hyperbolic geometry is a useful part of the background of future high school teachers. Much attention is paid to hyperbolic geometry by school mathematics clubs. Some mathematicians and educators concerned with reform of the high school curriculum believe that the required part of the curriculum should include elements of hyperbolic geometry, and that the optional part of the curriculum should include a topic related to hyperbolic geometry. I The broad interest in hyperbolic geometry is not surprising. This interest has little to do with mathematical and scientific applications of hyperbolic geometry, since the applications (for instance, in the theory of automorphic functions) are rather specialized, and are likely to be encountered by very few of the many students who conscientiously study (and then present to examiners) the definition of parallels in hyperbolic geometry and the special features of configurations of lines in the hyperbolic plane. The principal reason for the interest in hyperbolic geometry is the important fact of "non-uniqueness" of geometry; of the existence of many geometric systems.

Contenuti

1. What is geometry?.- 2. What is mechanics?.- I. Distance and Angle; Triangles and Quadrilaterals.- 3. Distance between points and angle between lines.- 4. The triangle.- 5. Principle of duality; coparallelograms and cotrapezoids.- 6. Proof s of the principle of duality.- II. Circles and Cycles.- 7. Definition of a cycle; radius and curvature.- 8. Cyclic rotation; diameters of a cycle.- 9. The circumcycle and incycle of a triangle.- 10. Power of a point with respect to a circle or cycle; inversion.- Conclusion.- 11. Einstein’s principle of relativity and Lorentz transformations.- 12. Minkowskian geometry.- 13. Galilean geometry as a limiting case of Euclidean and Minkowskian geometry.- Supplement A. Nine plane geometries.- Supplement B. Axiomatic characterization of the nine plane geometries.- Supplement C. Analytic models of the nine plane geometries.- Answers and Hints to Problems and Exercises.- Index of Names.- Index of Subjects.