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Задачи на клетчатой бумаге
The book is a collection of important theorems and problems from the theory of point lattices and its applications, drawn by the authors from numerous monographs, textbooks, popular science books, and journal articles. Some of the problems are new.
The fundamental properties of lattices and the calculation of areas of polygons located on lattices (Pick's formula) form the main content of the initial chapters. Subsequent extensive materials are devoted to questions of the arrangement of circles, regular and semiregular polygons on lattices. Topics related to the geometric theory of numbers are also widely represented: the "nose stretching" algorithm, Klein polygons, theorems of Minkowski, Dirichlet, Hurwitz, etc. The topic "Discrete Harmonic Functions" is highlighted in a separate chapter and includes interesting analogues of the classical potential theory on the plane: the maximum principle, Liouville's theorem, the Dirichlet problem, Poisson's formula, etc.
The book as a whole or its individual chapters can be useful for conducting regular and elective classes in secondary schools, for teaching special courses and running special seminars in universities, and for preparing student term papers and theses.
The work was carried out within the framework of a scientific grant allocated in 2006 by the "Kolmogorov FMS Club," which brings together students, alumni, teachers, and veterans of the A.N. Kolmogorov School of the Lomonosov Moscow State University Advanced Educational Scientific Center.
