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.
