Item description for Geometries and Groups (Universitext) by M. Reid Viacheslav V. Nikulin...
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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Est. Packaging Dimensions: Length: 9.44" Width: 6.3" Height: 0.63" Weight: 0.91 lbs.
Release Date Jun 21, 2002
ISBN 3540152814 ISBN13 9783540152811
Availability 114 units. Availability accurate as of Mar 26, 2017 01:02.
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Reviews - What do customers think about Geometries and Groups (Universitext)?
Interesting material, perhaps a bit loosely held together Dec 4, 2005
The book begins gently with some background labelled "forming intuition" (although defining a geometry as a point set with a metric feels more like forming formalism to me). Then we get to the main theme of the book: isometry groups. They help us classify locally Euclidean geometries in two dimensions. The presentation is very elementary and explicit with details, and therefore quite tedious (we are already by page 120). Next we do the same thing in 3 dimensions, which is especially interesting, the authors argue, since the universe in which we live is three-dimensional. But it is hard to imagine these potential universes, except to say that there are 18 types and decide which of them are bounded or orientable. Perhaps sensing our dissapointment, the authors seem to say: well, yes, but at least these ideas pay off in physics as you can see here in our next chapter on the marvelously interesting theory of crystallography. We are not too impressed: Crystals?! Who cares about ****ing crystals? Anyway, after that the authors decide that it is interesting to study different locally Euclidean geometries on the torus. This leads to the modular group, and now we should be convinced that it is interesting to look for a geometry to accommodate the modular group as a group of motions. Lo and behold, hyperbolic geometry falls out, and the book ends triumphantly since we only wanted hyperbolic geometry in order to understand the modular group and torus geometries. Apparently, our previous interest in the true geometry of the universe, repeatedly appealed to in the discussion of locally Euclidean geometries, is gone without a trace.