"This is a quite exceptional book, a lively and approachable treatment of an important field of mathematics given in a masterly style. Assuming only a school background, the authors develop locally Euclidean geometries, going as far as the modular space of structures on the torus, treated in terms of Lobachevsky's non-Euclidean geometry. Each section is carefully motivated by discussion of the physical and general scientific implications of the mathematical argument, and its place in the history of mathematics and philosophy. The book is expected to find a place alongside classics such as Hilbert and Cohn-Vossen's ""Geometry and the imagination"" and Weyl's ""Symmetry""."
By:
Viacheslav V. Nikulin,
Igor R. Shafarevich
Translated by:
M. Reid
Imprint: Springer Verlag
Country of Publication: Germany
Edition: 1st ed. 1994. Corr. 2nd printing 0
Dimensions:
Height: 233mm,
Width: 155mm,
Spine: 14mm
Weight: 840g
ISBN: 9783540152811
ISBN 10: 3540152814
Series: Universitext
Pages: 254
Publication Date: 01 September 1994
Audience:
College/higher education
,
Professional and scholarly
,
Professional & Vocational
,
A / AS level
,
Further / Higher Education
Format: Paperback
Publisher's Status: Active
I. Forming geometrical intuition; statement of the main problem.- §1. Formulating the problem.- §2. Spherical geometry.- §3. Geometry on a cylinder.- §4. A world in which right and left are indistinguishable.- §5. A bounded world.- §6. What does it mean to specify a geometry?.- II. The theory of 2-dimensional locally Euclidean geometries.- §7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- §8. Classification of all uniformly discontinuous groups of motions of the plane.- §9. A new geometry.- §10. Classification of all 2-dimensional locally Euclidean geometries.- III. Generalisations and applications.- §11. 3-dimensional locally Euclidean geometries.- §12. Crystallographic groups and discrete groups.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- §13. Similarity of geometries.- §14. Geometries on the torus.- §15. The algebra of similarities: complex numbers.- §16. Lobachevsky geometry.- §17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.
