Visual Geometry and Topology (PDF)

Geometry and topology are strongly motivated by the
visualization of ideal objects that have certain special
characteristics. A clear formulation of a specific property
or a logically consistent proof of a theorem often comes
only after the mathematician has correctly "seen" what is
going on. These pictures which are meant to serve as
signposts leading to mathematical understanding, frequently
also contain a beauty of their own.
The principal aim of this book is to narrate, in an
accessible and fairly visual language, about some classical
and modern achievements of geometry and topology in both
intrinsic mathematical problems and applications to
mathematical physics. The book starts from classical notions
of topology and ends with remarkable new results in
Hamiltonian geometry. Fomenko lays special emphasis upon
visual explanations of the problems and results and
downplays the abstract logical aspects of calculations. As
an example, readers can very quickly penetrate into the new
theory of topological descriptions of integrable Hamiltonian
differential equations. The book includes numerous graphical
sheets drawn by the author, which are presented in special
sections of "Visual material". These pictures illustrate the
mathematical ideas and results contained in the book. Using
these pictures, the reader can understand many modern
mathematical ideas and methods.
Although "Visual Geometry and Topology" is about
mathematics, Fomenko has written and illustrated this book
so that students and researchers from all the natural
sciences and also artists and art students will find
something of interest within its pages.