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This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincar� that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue---not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics.

The models subsequently discovered by Klein and Poincar� brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology.

By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincar� in their full brilliance.

- Sales Rank: #1456216 in Books
- Published on: 1996-10-29
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x 7.25" w x .25" l, .75 pounds
- Binding: Paperback
- 153 pages

Review

"Translations are well done and very readable ... papers ... are well chosen ... an extremely attractive and valuable book to have and to read ... fills an important niche in the mathematical literature by making these papers available to a contemporary audience ... allows the modern reader to see how the great mathematicians of another time viewed both their subject and mathematics in general, a view which can still be inspirational." ---- Bulletin of the London Mathematical Society

Most helpful customer reviews

3 of 3 people found the following review helpful.

This could be the beginning of a beautiful seminar course

By Viktor Blasjo

Hyperbolic geometry is mathematics at its best: deep classical roots; stunning intrinsic beauty and conceptual simplicity; diverse and profound applications. In this source book we see how three great masters worked to understand this new and exciting geometry.

First, Beltrami's two 1868 papers. The geodesic geometry of surfaces of constant negative curvature such as the pseudosphere capture much of the essence of hyperbolic geometry. However, one does not find the actual hyperbolic plane lying around in three-space. But Beltrami has a way of mapping a surface of constant curvature into the Euclidean plane such that geodesics go to lines. From this point of view the previously intractable step--how to go from a hyperbolic surface to the hyperbolic plane--suggests itself immediately, and we obtain the projective disc model. Now, one way of looking at this construction is to say that it consists of putting a constant-curvature metric on a disc. This point of view is sufficiently abstract to work in n dimensions, as Beltrami shows in his second paper. As a bonus he exploits two other constant-curvature metrics to obtain the other two fundamental models of hyperbolic geometry: the conformal disc model and the half plane model. (Especially for the second paper one is very grateful for Stillwell's introductions.)

Next, Felix Klein. Instead of differential geometry, Klein approches the subject from the point of view of projective geometry. Indeed, Beltrami's projective disc metric begs to be interpreted in terms of projective geometry: the distance between two points in the circle is easily expressed in terms of the cross-ratio of these two points and the two colinear points on the circle. Similarly, projective geometry subsumes spherical and Euclidean geometry as well.

Lastly, there are three little texts by Poincar�, from a third viewpoint: complex function theory. The isometries of Beltrami's half plane model are readily described in terms of linear fractional transformations (in fact, the harmony is even more marked in three dimensions, as Poincar� soon realises). But we can also go "backwards", i.e. we can deduce Beltrami's metric from the isometry group. This proves to be a very rewarding shortcut indeed, since we can employ the built-in geometry of complex function theory.

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