Geometry Revealed

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Product Description

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built 'above' the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended...In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called 'elementary' geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations.Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Geometry Revealed Review

This is a wonderful book, in a very literal sense. Its 800 pages are full of wonders: beautiful theorems, elegant proofs, and profound theories, placed in an intricate framework of thematic and historical connections.

The central objective of the book is to discuss geometric problems that are easily stated and visualized, but whose solution requires mathematical tools that were created for a different purpose and that lie at higher levels of abstraction. These increasing levels of abstraction are the ``Jacob's ladder'' of the subtitle. Each chapter takes as a point of departure a collection of basic problems in two- and three-dimensional Euclidean geometry. How can K points be distributed on the surface of a sphere so as to maximize the distance between the closest pair? The solution is known for K <= 12 and K=24. What is the trajectory of a ball caroming around a billiard table? It may be periodic or ergodic; it may fill the interior of the table, or only part of the interior; it may fill the space quickly or very slowly; for the three-dimensional problem, almost nothing is known. Given the combinatorial structure of a polytope, can it be instantiated with vertices with integer coordinates, and if so, how large do the integers have to be? In two dimensions, yes, using integers of size O(n^___noise___ 1000); in three dimensions, again yes, but the upper bound is very badly known; in four dimensions, not always, and if it can, the integers may have to be doubly exponential. If you set out from a starting point on a smooth surface following a geodesic, how far can you go before the geodesic ceases to be the unique shortest path from the starting point (the ``cut point'' of the geodesic)? The complete answer is known only for a few special surfaces, such as the sphere and the torus; it is not known for the ellipsoid. Berger explores the deep mathematical issues that such problems raise; he also tracks this history of these problems --- conjectures, partial solutions, errors --- in remarkable detail, and provides extensive chapter bibliographies ranging from Greek mathematics to current research.

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