Visual Geometry and Topology

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.