This book (2nd edition) is an unusual phenomenon in domestic and foreign scientific literature. It focuses on a graphic, visual depiction of the basic concepts and objects of modern geometry and topology. All the illustrations in the book, and they occupy about 50% of its volume in the book, were made by the author - Doctor of Physics - Mathematical Sciences, Academician of the Russian Academy of Sciences, Professor of Moscow State University A. T. Fomenko. Graphic sheets A. T. Fomenko have long attracted attention for their unusualness, artistic expressiveness, mathematical accuracy of the images behind him. For mathematicians, physicists, it will be interesting to a wide range of readers.