Geometries and Groups
Springer-Verlag, 1987 - Mathematics - 251 pages
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".
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Generalisations and applications
12 Crystallographic groups and discrete groups
Geometries on the torus
5 other sections not shown
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3-space already angle assertion axis belongs bounded called centre circle closed complex composite condition consider consists constructed contained coordinates corresponding course covering crystal curve cylinder defined definition denote determined disc discrete groups distance distinct endpoint equal equivalent points exactly example Exercise exists fact Figure Finally fixed follows give given glide reflection group of motions groups of Type hence identity inequality integer joining kind lattice Lemma length line segment Lobachevsky locally Euclidean geometries means modular motion F Note obtained obviously opposite pair parallel perpendicular plane position possible proof properties prove question radius reader rectangle regions relation represented result rotation satisfies sides similar space sphere spherical square strip sufficiently superposition Suppose surface symmetry Theorem torus translation triangle uniformly discontinuous group upper half-plane vector whole