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GEOMETRY OF LATTICE ANGLES, POLYGONS, AND CONES SUMMARY

When some interested person asks my PhD adviser Prof. Vladimir Arnold, who is one of the famous mathematicians of our times, to name his preferable branch of mathematics, he always says that mathematics is a single entity which is beautiful in total. It is very often that papers are in the intersection of several branches of mathematics, in some cases they give birth to new branches or modify the existing ones. In addition he usually says, smiling thoughtfully, that the difference between physics and mathematics is also very small, namely in the cost of experiments. Mathematicians do the experiments on paper or computers and therefore the experiments are usually relatively cheap, while in physics the experiments are very expensive like in the case of the Large Hadron Collider. As a former student of Prof. Vladimir Arnold I totally agree with his point of view on mathematics. Certainly I do not have works in many topics of mathematics, but still I have written papers on subjects in several distinct fields. Most of my papers are on geometric continued fractions [4], [5], [6], and [9]. This subject is in the intersection of number theory and geometry. In addition [H5] is touching the theory of dynamic systems. In [10], we use topological methods to prove consistency of some continued fraction algorithm. Further, in papers [2], [3], and [7] we study variational principles of functional of energies of knots, we apply principles of variational calculus to topology. The subjects of papers [12] and [13]