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 t
he 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.
