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…mehr
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".
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.Forming geometrical intuition; statement of the main problem. The theory of 2-dimensional locally Euclidean geometries. Generalisations and applications. Geometries on the torus, complex numbers and Lobachevsky geometry. Historical remarks. List of notation. Index.
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.Forming geometrical intuition; statement of the main problem. The theory of 2-dimensional locally Euclidean geometries. Generalisations and applications. Geometries on the torus, complex numbers and Lobachevsky geometry. Historical remarks. List of notation. Index.
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