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.- 3.1. First acquaintance.- 3.2. How to measure distances.- 3.3. The study of geometry on a cylinder.- 4. A world in which right and left are indistinguishable.- 5. A bounded world.- 5.1. Description of the geometry.- 5.2. Lines on the torus.- 5.3. Some applications.- 6. What does it mean to specify a geometry?.- 6.1. The definition of a geometry.- 6.2. Superposing geometries.- II. The theory of 2-dimensional locally Euclidean geometries.- 7. Locally Euclidean geometries and uniformly discontinuous groups of motions of the plane.- 7.1. Definition of equivalence by means of motions.- 7.2. The geometry corresponding to a uniformly discontinuous group.- 8. Classification of all uniformly discontinuous groups of motions of the plane.- 8.1. Motions of the plane.- 8.2. Classification: generalities and groups of Type I and II.- 8.3. Classification: groups of Type III.- 9. A new geometry.- 10. Classification of all 2-dimensional locally Euclidean geometries.- 10.1. Constructions in an arbitrary geometry.- 10.2. Coverings.- 10.3. Construction of the covering.- 10.4. Construction of the group.- 10.5. Conclusion of the proof of Theorem 1.- III. Generalisations and applications.- 11. 3-dimensional locally Euclidean geometries.- 11.1. Motions of 3-space.- 11.2. Uniformly discontinuous groups in 3-space: generalities.- 11.3. Uniformly discontinuous groups in 3-space: classification.- 11.4. Orientability of the geometries.- 12. Crystallographic groups and discrete groups.- 12.1. Symmetry groups.- 12.2. Crystals and crystallographic groups.- 12.3. Crystallographic groups and geometries: discrete groups.- 12.4. A typical example: the geometry of the rectangle.- 12.5. Classification of all locally Cn or Dn geometries.- 12.6. On the proof of Theorems 1 and 2.- 12.7. Crystals and their molecules.- IV. Geometries on the torus, complex numbers and Lobachevsky geometry.- 13. Similarity of geometries.- 13.1. When are two geometries defined by uniformly discontinuous groups the same?.- 13.2. Similarity of geometries.- 14. Geometries on the torus.- 14.1. Geometries on the torus and the modular figure.- 14.2. When do two pairs of vectors generate the same lattice?.- 14.3. Application to number theory.- 15. The algebra of similarities: complex numbers.- 15.1. The geometrical definition of complex numbers.- 15.2. Similarity of lattices and the modular group.- 16. Lobachevsky geometry.- 16.1. 'Motions'.- 16.2. 'Lines'.- 16.3. Distance.- 16.4. Construction of the geometry concluded.- 17. The Lobachevsky plane, the modular group, the modular figure and geometries on the torus.- 17.1. Discreteness of the modular group.- 17.2. The set of all geometries on the torus.- Historical remarks.- List of notation.- Additional Literature.