# Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry

Springer Science & Business Media, Jul 23, 2010 - Mathematics - 831 pages

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.

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### Contents

 I Points and lines in the plane 1 II Circles and spheres 61 can we distribute points on it evenly? 140 IV Conics and quadrics 181 V Plane curves 249 VI Smooth surfaces 341 VII Convexity and convex sets 408 VIII Polygons polyhedra polytopes 505
 X Lattices and packings in higher dimensions 623 billiards 675 geodesic flow on a surface 738 Selected Abbreviations for Journal Titles 785 Name Index 789 Subject Index 795 Symbol Index 827 Copyright

 IX Lattices packings and tilings in the plane 563