The theory of chain fractions is associated with the theory of approximations of real numbers rational, with the theory of dynamical systems, as well as with many other branches of mathematics. The brochure describes the relationship between chain fractions and the geometry of convex polygons. From this connection it follows, for example, that the chain fraction is periodic if and only if the number expressed by it is the root of the quadratic equation with integer coefficients .. It is also told how often a unit is found among the elements of a chain fraction expressing an arbitrary real number (two, three, .).. ). The final section of the brochure provides an overview of the results related to the multidimensional generalizations of the classical theory of chain fractions obtained recently. The text of the brochure is an additional processing of the recording of the lecture given by the author for schoolchildren of grades 9 - 11 on December 2, 2000 at the Small Mekhmat of Moscow State University .. The brochure is intended for a wide range of readers interested in mathematics: high school students, junior students, teachers, and in part it will be of interest to professional mathematicians.